Jeff Bivin -- LZHS

Using FundamentalTrig Identities

Verifying IdentitiesAnd

Solving Trig Equations

By: Jeffrey BivinLake Zurich High School

jeff.bivin@lz95.org

Last Updated: December 29, 2009

Jeff Bivin -- LZHS

Reciprocal Identities

1sin

cscu

u

1cos

secu

u

1tan

cotu

u

1csc

sinu

u

1sec

cosu

u

1cot

tanu

u

Jeff Bivin -- LZHS

Quotient Identities

sintan

cos

uu

u

coscot

sin

uu

u

Jeff Bivin -- LZHS

Pythagorean Identities

2 2 1sin cos

2 21 tan sec

2 21 cot csc

Jeff Bivin -- LZHS

Cofunction Identities

2sin cosu u 2cos sinu u

2tan cotu u 2cot tanu u

2sec cscu u 2csc secu u

Complimentary Angles

Jeff Bivin -- LZHS

Even/Odd Identities

sin sinu u cos cosu u

tan tanu u

u u cot cot

sec secu u

csc cscu u

f x f x f x f x

Jeff Bivin -- LZHS

Use the given to evaluate all six trig functions

26265cot sinand

What quadrant is cotangent negative and sine positive??? II

First determine that quadrant the given information holds true…….

tangent

155cot tanif then

2

22626 1cos

226676 1cos 126

2 2526cos

5 2652626

cos

2626 26sin cscif then

526

265

cos

sec

if

then

2 2 1sin cos

Jeff Bivin -- LZHS

Simplify a Trig Expression2sin cos sinx x x

2 1sin cosx x

2x xsin sin

3sin x

2 2 1remember x x : sin cos2 21x x cos sin

Jeff Bivin -- LZHS

Verify a Trig Identity

1

sin coscsc

cos sin

1

1

sin sin cos coscsc

cos sin

2 2

1

sin cos coscsc

cos sin

1

1

coscsc

cos sin

1

cscsin

csc csc

Jeff Bivin -- LZHS

Verify a Trig Identity

1

sin coscsc

cos sin

11

sin cos

cos siny

Use the table feature and graphing utility to check your result.

2 cscy

Select the path style for y2 so you

can see the tracing

Jeff Bivin -- LZHS

Verify a Trig Identitysin cos cos sin

sec cscsin cos

cos sin cos sin cos sinsec csc

sin cos

2 2cos sin cos sin cos sinsec csc

sin cos

1 1sec csc

sin cos

csc sec sec csc

1sec csc

sin cos

Jeff Bivin -- LZHS

Factoring Trig Expressions2 1sin x

1 1sin sinx x

24 3tan tan

4 3 1tan tan

2 1a

1 1a a

24 3a a

4 3 1a a

Jeff Bivin -- LZHS

Factoring Trig Expressions22 6cos cos

2 3 2cos cos

3 1sec

21 1sec sec sec

22 6a a

2 3 2a a

3 1a

21 1a a a

Jeff Bivin -- LZHS

Factoring Trig Expressions2 3sec tan

2 2a a

2 1a a

21 3tan tan

2 2tan tan

2 1tan tan

2 21remember x x : tan sec

Jeff Bivin -- LZHS

Factoring Trig Expressions4 4csc cotx x

2 2 2 2csc cot csc cotx x x x 2 21: cot cscremember x x

2 21 csc cotx x 2 21 csc cotx x

2 2csc cotx x2 21 x x cot cot

21 2 x cot

Jeff Bivin -- LZHS

Factoring Trig Expressions3 2 1cos cos cosx x x

2 1 1 1cos cos cosx x x

21 1cos cosx x

1 1 1cos cos cosx x x

21 1cos cosx x

Jeff Bivin -- LZHS

Add & Subtract Trig Expressions1 1

1 1sin sinx x

1 1 1 1

1 1

sin sin

sin sin

x x

x x

2

1 1

1

sin sin

sin

x x

x

2

2sin

cos

x

x

Jeff Bivin -- LZHS

Add & Subtract Trig Expressions1

1

sin cos

cos sin

x x

x x

2 1 1

1

sin cos cos

cos sin

x x x

x x

2 21 1

1

cos cos

cos sin

x x

x x

22 2

1

cos

cos sin

x

x x

22 1

1

cos

cos sin

x

x x

2 1 1

1

cos cos

cos sin

x x

x x

2 1 cos

sin

x

x

Jeff Bivin -- LZHS

Add & Subtract Trig Expressions2csc

cotcot

xx

x

2 2cot csc

cot

x x

x

2 21csc csc

cot

x x

x

1

cot x

tan x

Jeff Bivin -- LZHS

Verify Trig Identities

Jeff Bivin -- LZHS

Verify Trig Identities - Guidelines

1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.

2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.

3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants with tangents, and cosecants with cotangents.

4. If the preceding guidelines do not help, try converting all terms to sines and cosines.

5. Always try something. Even making an attempt that leads to a dead end provides insight.

Jeff Bivin -- LZHS

Verify Trig Identities2

22

1xx

x

secsin

sec

22

11 x

x sinsec

2 21 x x cos sin

2 2sin sinx x

Jeff Bivin -- LZHS

Verify Trig Identities21 1

21 1

csccos cos

21 1

21 1

cos cos

csccos cos

22

22

1

csc

cos

22

22

x csc

sin

22

12 2

x csc

sin

2 22 2 csc csc

Jeff Bivin -- LZHS

Verify Trig Identitiestan cot sec csc

sin cos

sec csccos sin2 2

sin cossec csc

cos sin1

sec csc

cos sin

1 1

sec csccos sin

sec csc sec csc

Jeff Bivin -- LZHS

Verify Trig Identities

1

cossec tan

sin

1

1 1

cos sin

sec tansin sin

2

1

1

cos sinsec tan

sin

2

1

cos sin

sec tancos

1

sin

sec tancos

1

sin

sec tancos cos

sec tan sec tan

Jeff Bivin -- LZHS

Verify Trig Identities21

1

cos tan

cos sec

x x

x x

21 1

1

x x

x x

cos sec

cos sec

1 11

1

x xx

x x

sec seccos

cos sec

11

xx

x

cossec

cos

1 11

x

x x

cos

cos cos

1 1cos cos

cos cos

x x

x x

Jeff Bivin -- LZHS

Verify Trig Identities 2 22 1sec cotx x

2 2 1x x sec tan

2 21 1x x tan tan

1 1

Jeff Bivin -- LZHS

Verify Trig Identities1 1

1

sin sin

sin cos

x x

x x

1 1

1 1

sin sin

sin sin

x x

x x

2

2

1

1

x

x

sin

sin

22

1 sin

cos

x

x

1 1sin sin

cos cos

x x

x x

Jeff Bivin -- LZHS

Verify Trig Identities3 3

1sin cos

sin cossin cos

x xx x

x x

2 2sin cos sin sin cos cos

sin cos

x x x x x x

x x

2 2sin sin cos cosx x x x

1 1sin cos sin cosx x x x

Jeff Bivin -- LZHS

Verify Trig Identities

2 4 51 2cos sin sin cosx x x x

22 51x x x cos sin cos

22 5x x xcos cos cos

4 5x x xcos cos cos

5 5cos cosx x

2 4 52x x x x x x cos cos sin cos sin cos

Jeff Bivin -- LZHS

Verify Trig Identities4 4 2 41 2 2sin cos cos cosx x x x

2 2 4sin cosx x

2 2 41 cos cosx x

2 4 41 2cos cos cosx x x

2 4 2 41 2 2 1 2 2cos cos cos cosx x x x

Jeff Bivin -- LZHS

Solving Trig Equations

Jeff Bivin -- LZHS

Solving Trig Equations

2 1 0cos x

2 1cos x 12cos x

0 2,on 5

3 3x or x

,on 5

3 32 2x n or x n

Jeff Bivin -- LZHS

Solving Trig Equations2 24 1tan tanx x

23 1tan x 2 1

3tan x

0 2,on 7

6 6x or x

,on 5

6 6x n or x n

33tan x

5 116 6or x or x

Jeff Bivin -- LZHS

Solving Trig Equations2 0sec tanx x

21 0sincos cos

xx x 1 2 0sincos

xx

,on 5

6 62 2x n or x n

1 2 0sincoscos cosx

xx x 1 2 0sin x

12sin x

0cos x 3

2 2,x

Jeff Bivin -- LZHS

Solving Trig Equations0cos cotx x 0cos

sincos xxx

11 0sincos xx

,on

2x n

0cos x 3

2 2,x

0sin x 0,x

11 0sin x 11 sin x1sin x

2x

Jeff Bivin -- LZHS

Solving Trig Equations22 5 3sin sinx x

22 5 3 0sin sinx x

,on 7 116 62 2x n or x n

2 1 3 0sin sinx x

7 116 6x or x

2 1 0sin x 3 0sin x 2 1sin x

12sin x

3sin x

Jeff Bivin -- LZHS

Solving Trig Equations2 2 2 0x cos

2 2 2cos x 222cos x 7

4 42 2 2 2x n or x n 7

8 8x n or x n 0 2,on

9 7 158 8 8 8 , , ,

Jeff Bivin -- LZHS

Sum & Difference Formulas

Jeff Bivin -- LZHS

Sum & Difference Formulas

1tan tantan tantan x y

x yx y

1tan tantan tantan x y

x yx y

sin sin cos cos sinx y x y x y sin sin cos cos sinx y x y x y

cos cos cos sin sinx y x y x y cos cos cos sin sinx y x y x y

Jeff Bivin -- LZHS

S (x3, y3)

P (1, 0)

Q (x1, y1)

R (x2, y2 ) A

B

-B

PR SQ 2 22 22 2 1 3 1 31 0x y x x y y

2 22 22 2 1 3 1 31 0x y x x y y

2 1 3 1 32 2 2x x x y y 2 1 3 1 3x x x y y

cos( ) cos cos( ) sin sin( )A B A B A B cos( ) cos cos sin sinA B A B A B

S (x3, y3) (cos(-B), sin(-B) )Q (x1, y1) (cosA, sinA)

R (x2, y2 ) (cos(A+B), sin(A+B) )

2 2 2 2 2 2

2 2 2 1 1 3 3 1 3 1 32 1 2 2x y x x y x y x x y y

2 2 2 2 2 2

2 2 2 1 1 3 3 1 1 3 32 1 2 2x x y x x x x y y y y

2 1 3 1 31 2 1 1 1 2 2x x x y y

Proof of cos(A+B) = cosA•cosB - sinA•sinB

Jeff Bivin -- LZHS

Proof ofsin(A+B) = sinA•cosB + cosA•sinB

2 2cos cos sin sinA B A B

2sin cosA B A B

2cos A B 2cos A B

A B A B sin cos cos sin

Note: This proof uses the cofunction identities.

Jeff Bivin -- LZHS

Use Sum & Difference Formulas

Find the exact value of sin(750)

75 30 45 30 45sin sin cos cos sino o o o o

75 30 45sin sino o o

32 212 2 2 275sin o

624 475sin o

2 6475sin o

Jeff Bivin -- LZHS

Use Sum & Difference FormulasFind the exact value of:

712 4 3 4 3cos cos cos sin sin

7 3 412 12 12cos cos

32 27 112 2 2 2 2cos

62712 4 4cos

2 6712 4cos

712cos

4 3cos

Jeff Bivin -- LZHS

Use Sum & Difference FormulasFind the exact value of:

3 1

1 3 1

( )

( )

13 13 4 912 12 12 12tan tan tan

1312tan

33 4tan

3 1

1 3

1 3 1 3 1 2 3 3 4 2 31 3 21 3 1 3

2 3

33 4

33 41

tan tan

tan tan

Rationalize the

denominator

Jeff Bivin -- LZHS

1 1 1 1cos sin cos cos sin sin sin cosx x x x

Use Sum & Difference FormulasSimplfy:

2 21 1x x x x

22 1x x

1 1cos sin cosx x

21 x

x1 21 x

x2 21 1x x x x

Jeff Bivin -- LZHS

sin sin cos cos sinu v u v u v

Use Sum & Difference FormulasFind the exact value of sin(u+v) given the following:

5 313 5u and v u v areinQuad IV sin cos , &

3

54

5 313 5sin cos sinu v u v

513

12u v

5 3 12 413 5 13 5sin u v

15 4865 65sin u v

6365sin u v

Jeff Bivin -- LZHS

Verify Trig Identities 2 2sin sin sin sinx y x y x y

sin cos cos sin sin cos cos sinx y x y x y x y

2 2 2 2sin cos cos sinx y x y

2 2 2 21 1sin sin sin sinx y x y

2 2 2 2 2 2sin sin sin sin sin sinx x y y x y

2 2 2 2sin sin sin sinx y x y

Jeff Bivin -- LZHS

Solving Trig Equations on [0, 2π)

4 4 1x x cos cos

74 4x or x

4 4 4 4 1x x x x cos cos sin sin cos cos sin sin

4 4 1x x cos cos cos cos

42 1x cos cos

222 1x cos

2 1x cos2122

x cos

Jeff Bivin -- LZHS

Double-Angle Formulas

Jeff Bivin -- LZHS

Double-Angle Formulas 2x x x sin sin

2x x x cos cos

2x x x x x sin sin cos cos sin

2 2x x xsin sin cos

2x x x x x cos cos cos sin sin

2 22x x x cos cos sin

2 22 1x x x cos sin sin

22 1 2x x cos sin

2 22 1x x x cos cos cos

2 22 1x x x cos cos cos

22 2 1x x cos cos

2 21x x cos sin

2 21x x sin cos

Jeff Bivin -- LZHS

Double-Angle Formulas 2x x x tan tan

12 x xx xx

tan tantan tantan

221

2 xx

x

tantan

tan

Jeff Bivin -- LZHS

Double-Angle Formulas

2 2x x xsin sin cos

2 22x x x cos cos sin

22 1 2x x cos sin

22 2 1x x cos cos

221

2 xx

x

tantan

tan

Jeff Bivin -- LZHS

Half-Angle Formulas

1

2 2

u u

cossin

1

2 2

u u

coscos

1

2 1

u u u

u u

cos sin

tansin cos

2 2

2

u u

u

Note the signs of and

depend on the quadrant in which lies

: sin cos

.

Jeff Bivin -- LZHS

Solve for x

2 0x x sin cos

2 0x x x sin cos cos

2 1 0x x cos sin

0x cos 2 1 0x sin3

2 2x , 2 1x sin12x sin

76 2x n 116 2x n

76 2x n 116 2x n

2x n

2x n

Jeff Bivin -- LZHS

3 4 245 5 25sin 2 2u

sin 2 2sin cosu u u

Use Double-Angle FormulasFind the exact value of sin(2u), cos(2u) & tan(2u)

3

4

5u

35 2sin ,given u u

2cos 2 1 2sinu u 23

5cos 2 1 2u 9

25cos 2 1 2u 18 7

25 25cos 2 1u

221

2 xx

x

tantan

tan

3423

4

2

12x

tan

329161

2x

tan

32716

2xtan

3 16 242 7 72x tan

Jeff Bivin -- LZHS

565

12sin sin2

Use Half-Angle FormulasFind the exact value of: 5

12sin

1

2 2

u u

cossin

561

2

cos

321

2

321

2

2 3

4

2 3

2

Jeff Bivin -- LZHS

747

8tan tan2

Use Half-Angle FormulasFind the exact value of: 7

8tan

74

74

1

cos

sin

1

2 1

u u u

u u

cos sin

tansin cos

2222

1

22

22

1 2

2

2 2 2

2 2

2 2 2

2

2 1 1 2

Jeff Bivin -- LZHS

Solve for x 2 1 0x x sin cos

12 1 0x x cos cos

12 1x x cos cos

2 21

2 1x x cos cos

212 1 2x x x cos cos cos

21 2 4 2x x x cos cos cos20 1 3 2x x cos cos

0 1 2 1x x cos cos

0 1 2 x cos

0 1 x cos

1 2 x cos1 2 x cos12 x cos

3 2n x 53 2n x

1x cos2x n

Jeff Bivin -- LZHS

Product-to-Sum Formulas

12u v u v u v sin sin cos( ) cos( )

12u v u v u v cos cos cos( ) cos( )

12u v u v u v sin cos sin( ) sin( )

12u v u v u v cos sin sin( ) sin( )

Jeff Bivin -- LZHS

Sum-to-Product Formulas

2 22 u v u vu v sin sin sin cos

2 22 u v u vu v sin sin cos sin

2 22 u v u vu v cos cos cos cos

2 22 u v u vu v cos cos sin sin

Jeff Bivin -- LZHS

Power-Reducing Formulas

2 1 2

2

uu

cossin 2 1 2

2

uu

coscos

2 1 2

1 2

uu

u

cos

tancos

Jeff Bivin -- LZHS

24 2sin sinx x

Use Power-Reducing FormulasRewrite sin4x in terms of the first power of cosine.

21 cos22

x

214 1 cos 2x

214 1 2cos 2 cos 2x x

1 cos414 21 2cos 2 xx

18 2 4cos 2 1 cos4x x

18 3 4cos 2 cos 4x x