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Double- and Half-Angle Formulas
Learning Objective: To derive and apply double- and half- angle formulas
Ex 5:Find tan if sin x = and Ex 6: Find the exact value of tan 67.5
Double Formulas To derive and apply double angle formulas
Ex 1: Find sin2x and cos2x if and Ex2:Findsin2xandcos2xif and
Ex 3: Find if sin x = and Ex4:Find if and
cos x = 1213
sin x = − 14
0 < x < π2
π2< x < 3π
2
tan2x −45
π < x < 3π2
tan2x csc x = 4
7
π2< x < 3π
2
3.4
Goal
Double-Angle Formulas for Sine and Cosine
DoubleAngleformulaforTangent:
Double Formulas To derive and apply double angle formulas
Ex 1: Find sin2x and cos2x if and Ex2:Findsin2xandcos2xif and
Ex 3: Find if sin x = and Ex4:Find if and
cos x = 1213
sin x = − 14
0 < x < π2
π2< x < 3π
2
tan2x −45
π < x < 3π2
tan2x csc x = 4
7
π2< x < 3π
2
3.4
Goal
Double-Angle Formulas for Sine and Cosine
DoubleAngleformulaforTangent:
Double Formulas To derive and apply double angle formulas
Ex 1: Find sin2x and cos2x if and Ex2:Findsin2xandcos2xif and
Ex 3: Find if sin x = and Ex4:Find if and
cos x = 1213
sin x = − 14
0 < x < π2
π2< x < 3π
2
tan2x −45
π < x < 3π2
tan2x csc x = 4
7
π2< x < 3π
2
3.4
Goal
Double-Angle Formulas for Sine and Cosine
DoubleAngleformulaforTangent:
tan3x
x2
−1213
3π2< x < 2π °
Double-Angle Formulas
V. Sum and Difference Identities
9) a. b. c. 10) a. b. c. 11) a. b. c. 18. 19. 20.
21. 22. 23.`
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
sin −θ( ) = − sinθ cos −θ( ) = cosθ tan −θ( ) = − tanθ
csc −θ( ) = − cscθ sec −θ( ) = secθ cot −θ( ) = − cotθ
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
IV. Even/Odd Identities
VI. Double Angle Identities
30.
31.
32.
VII. Half Angle Identities
sin2θ = 2sinθcosθ
cos2θ = cos2 θ − sin2 θcos2θ = 1− 2sin2 θcos2θ = 2cos2 θ − 1
tan2θ = 2 tanθ1− tan2 θ
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
cscθ = 1sinθ cotθ = 1
tanθ
Trigonometric Identities
Law of Sine
Law of Cosine
IV. Even/Odd Identities
V. Sum and Difference Identities
VI. Double Angle Identities
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
VII. Other Trig Formulas VIII. Half Angle Identities
Law of Sines
Law of Cosines
Area of Triangle
IV. Even/Odd Identities
V. Sum and Difference Identities
VI. Double Angle Identities
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
VII. Other Trig Formulas VIII. Half Angle Identities
Law of Sines
Law of Cosines
Area of Triangle
Area of Triangle
1.
2.
3.
4.
5.
k = 12bh
k = 12bcsin A
k = 12absinC
k = 12acsinB
k = s s − a( ) s − b( ) s − c( )s = a + b + c
2
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
6.
7.
8.
k = a2 sinBsinC2sin A
k = b2 sin AsinC2sinB
k = c2 sin AsinB2sinC
V. Sum and Difference Identities
9) a. b. c. 10) a. b. c. 11) a. b. c. 18. 19. 20.
21. 22. 23.`
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
sin −θ( ) = − sinθ cos −θ( ) = cosθ tan −θ( ) = − tanθ
csc −θ( ) = − cscθ sec −θ( ) = secθ cot −θ( ) = − cotθ
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
IV. Even/Odd Identities
VI. Double Angle Identities
30.
31.
32.
VII. Half Angle Identities
sin2θ = 2sinθcosθ
cos2θ = cos2 θ − sin2 θcos2θ = 1− 2sin2 θcos2θ = 2cos2 θ − 1
tan2θ = 2 tanθ1− tan2 θ
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
cscθ = 1sinθ cotθ = 1
tanθ
Trigonometric Identities
Law of Sine
Law of Cosine
IV. Even/Odd Identities
V. Sum and Difference Identities
VI. Double Angle Identities
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
VII. Other Trig Formulas VIII. Half Angle Identities
Law of Sines
Law of Cosines
Area of Triangle
IV. Even/Odd Identities
V. Sum and Difference Identities
VI. Double Angle Identities
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
VII. Other Trig Formulas VIII. Half Angle Identities
Law of Sines
Law of Cosines
Area of Triangle
Area of Triangle
1.
2.
3.
4.
5.
k = 12bh
k = 12bcsin A
k = 12absinC
k = 12acsinB
k = s s − a( ) s − b( ) s − c( )s = a + b + c
2
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
6.
7.
8.
k = a2 sinBsinC2sin A
k = b2 sin AsinC2sinB
k = c2 sin AsinB2sinC
HalfAngleFormulas
Trigonometry3.5 Formulas for TangentTeacher’s Note
Introduce sum and di↵erence formulas for tangent.
(a) tan(x+ y) =tanx+ tan y
1� tanx tan y
(b) tan(x� y) =tanx� tan y
1 + tanx tan y
Example 1 Find tan(x� y) if tanx = 13 and tanx = 6
5 .
tan(x� y) =tanx� tan y
1 + tanx tan y=
13 � 6
5
1 + ( 13 )(65 )
=5� 18
15 + 6= �13
21
Checkpoint Find the exact value of tan 165�.
tan 165� = tan(120� + 45�) =tan 120� + tan 45�
1� tan 120� tan 45�=
�p3 + 1
1� (�p3)(1)
=1�
p3
1 +p3=
(1�p3)2
1� 3=
4� 2p3
�2=
p3� 2
Introduce double-angle formula for tangent.
(c) tan 2x =2 tanx
1� tan2 x
Example 2 Find tan 2x if sinx = � 45 and ⇡ < x < 3⇡
2 ..
(1) cosx = � 35 , tanx = 4
3
(2) tan 2x =2 tanx
1� tan2 x=
2( 43 )
1� 169
= (8
3)(�9
7) = �24
7
Introduce half-angle formula for tangent.
(d) tan x2 =
sinx
1 + cosx
(e) tan x2 =
1� cosx
sinx
(f) tan x2 =
r1� cosx
1 + cosx
1
V. Sum and Difference Identities
9) a. b. c. 10) a. b. c. 11) a. b. c. 18. 19. 20.
21. 22. 23.`
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
sin −θ( ) = − sinθ cos −θ( ) = cosθ tan −θ( ) = − tanθ
csc −θ( ) = − cscθ sec −θ( ) = secθ cot −θ( ) = − cotθ
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
IV. Even/Odd Identities
VI. Double Angle Identities
30.
31.
32.
VII. Half Angle Identities
sin2θ = 2sinθcosθ
cos2θ = cos2 θ − sin2 θcos2θ = 1− 2sin2 θcos2θ = 2cos2 θ − 1
tan2θ = 2 tanθ1− tan2 θ
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
Reference
Reciprocal Identities Quotient Identities 1sin
cscT
T
1cscsin
TT
1cossec
TT
1seccos
TT
1tancot
TT
1cottan
TT
sintancos
TTT
coscotsin
TTT
Pythagorean Identities 2 2sin cos 1T T� 2 2tan 1 secT T� 2 2cot 1 cscT T�
Cofunction Identities Even-Odd Identities
sin cos2ST T§ · �¨ ¸© ¹
cos sin2ST T§ · �¨ ¸© ¹
sin sin( )T T� � csc csc( )T T� �
csc sec2ST T§ · �¨ ¸© ¹
sec csc2ST T§ · �¨ ¸© ¹
cos cos( )T T� sec sec( )T T�
tan cot2ST T§ · �¨ ¸© ¹
cot tan2ST T§ · �¨ ¸© ¹
tan tan( )T T� � cot cot( )T T� �
Sum of Angles Identities Difference of Angles Identities
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
sin sin cos cos sin( )A B A B A B� � � �
cos cos cos sin sin( )A B A B A B� � � �
tan tantan1 tan tan
( ) A BA BA B�
� � �
Double-Angle Identities
sin2 2sin cosT T T � 2 2
2
2
cos2 cos sincos2 1 2sincos2 2cos 1
T T TT TT T
� � �
2
2tantan21 tan
TTT
�
Half-Angle Identities
1 cossin2 2T T� r 1 coscos
2 2T T� r 1 costan
2 1 cosT T
T�
r�
1 costan2 sinT T
T�
sintan2 1 cosT T
T
�
Power-Reducing Identities
2 1 cos2sin2
TT � 2 1 cos2cos
2TT �
2 1 cos2tan1 cos2
TTT
�
�
Product-to-Sum Identities Sum-to-Product Identities
> @1sin sin cos cos2
( ) ( )A B A B A B� � � � sin sin 2 sin cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos cos cos cos2
( ) ( )A B A B A B� � � � cos cos 2 cos cos2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1sin cos sin sin2
( ) ( )A B A B A B� � � � sin sin 2 cos sin2 2
A B A BA B � �§ · § ·� � �¨ ¸ ¨ ¸© ¹ © ¹
> @1cos sin sin sin2
( ) ( )A B A B A B� � � � cos cos 2 sin sin2 2
A B A BA B � �§ · § ·� � � �¨ ¸ ¨ ¸© ¹ © ¹
© Gina Wilson (All Things Algebra®, LLC), 2018
sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
cot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
cscθ = 1sinθ cotθ = 1
tanθ
Trigonometric Identities
Law of Sine
Law of Cosine
IV. Even/Odd Identities
V. Sum and Difference Identities
VI. Double Angle Identities
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
VII. Other Trig Formulas VIII. Half Angle Identities
Law of Sines
Law of Cosines
Area of Triangle
IV. Even/Odd Identities
V. Sum and Difference Identities
VI. Double Angle Identities
I. Reciprocal Identities
II. Quotient Identities
III. Pythagorean Identities
VII. Other Trig Formulas VIII. Half Angle Identities
Law of Sines
Law of Cosines
Area of Triangle
Area of Triangle
1.
2.
3.
4.
5.
k = 12bh
k = 12bcsin A
k = 12absinC
k = 12acsinB
k = s s − a( ) s − b( ) s − c( )s = a + b + c
2
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
9) a. b. c. 10) a. b. c. 11) a. b. c.
sin2 θ + cos2 θ = 1sin2 θ = 1− cos2 θcos2 θ = 1− sin2 θ
tan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1sec2 θ − tan2 θ = 1
1+ cot2 θ = csc2 θcot2 θ = csc2 θ − 1csc2 θ − cot2 θ = 1
6.
7.
8.
k = a2 sinBsinC2sin A
k = b2 sin AsinC2sinB
k = c2 sin AsinB2sinC
Ex 7: Given that cos x = , Ex 8: Prove
Find
Ex 9: Prove Ex 10: Prove
Ex 11: Prove Ex 12: Prove
23
π < x < 2π
tan 2x − x2
⎛
⎝⎜⎞
⎠⎟
Prove the following: Example 5 Example 6
Example7 Example8
Example 9
cot x = sin2x1− cos2x
1− cos2xcos2x + 1
= tan2 x
sin x + cos x( )2 = 1+ sin2x 11− tan x
− 11+ tan x
= tan2x
cot x − tan x = 2cot 2x
Prove the following: Example 5 Example 6
Example7 Example8
Example 9
cot x = sin2x1− cos2x
1− cos2xcos2x + 1
= tan2 x
sin x + cos x( )2 = 1+ sin2x 11− tan x
− 11+ tan x
= tan2x
cot x − tan x = 2cot 2x
Prove the following: Example 5 Example 6
Example7 Example8
Example 9
cot x = sin2x1− cos2x
1− cos2xcos2x + 1
= tan2 x
sin x + cos x( )2 = 1+ sin2x 11− tan x
− 11+ tan x
= tan2x
cot x − tan x = 2cot 2x
11− tan x
− 11+ tan x
= tan2x
Prove the following: Example 5 Example 6
Example7 Example8
Example 9
cot x = sin2x1− cos2x
1− cos2xcos2x + 1
= tan2 x
sin x + cos x( )2 = 1+ sin2x 11− tan x
− 11+ tan x
= tan2x
cot x − tan x = 2cot 2x
