Unit 6 Trigonometric Identities - CBRH - Corner Brook ... 6 Trig Id and eq 2016.pdf · Prove trigonometric identities, using: ... – Replace a “squared” term with a Pythagorean

Unit 6Trigonometric

Identities• Prove trigonometric identities

• Solve trigonometric equations

Prove trigonometric identities, using:

• Reciprocal identities

• Quotient identities

• Pythagorean identities

• Sum or difference identities– (restricted to sine, cosine and tangent)

• Double-angle identities – (restricted to sine, cosine and tangent)

TRIG IDENTITIES

• You should be able to explain the difference between a trigonometric equation and a trigonometric identity.

• An identity is true for all permissible values, whereas an equation is only true for a smaller subset of the permissible values.

• This difference can be demonstrated with the aid of graphing technology.

For example:

• This can be solved by using the graphs of

• The solutions to

are x = 30° and x = 150°, which are the

x-values of the

intersection points.

• Thus this is not an identity because it is only true for certain values of x.

1sin ,0 360

2x x

1sin and

2y x y

1sin ,0 360

2x x

Solve: sin x = tan x cos x.• This can be solved by using the graphs of:

y = sin x and y = tan x cos x

• These graphs are almost identical.– The only differences in the graphs occur at the

points (90°, 1) and (270°, −1).

– Why? They are non-permissible values of x.

• Therefore, sin x = tan x cos x is an identity since the expressions are equivalent for all permissible values.

• Note: There may be some points for which identities are not equivalent.

• These non-permissible values for identities occur where one of the expressions is undefined.

• In the previous example, y = tan x cos x is not defined when x = 90° + 180°n, n ∈ Ι since y = tan x is undefined at these values.

• Non-permissible values often occur when a trigonometric expression contains:– A rational expression, resulting in values that give

a denominator of zero

– Tangent, cotangent, secant and cosecant, since these expressions all have non-permissible values in their domains.

Practice:Determine graphically if the following

are identities. Use Technology

Identify the non-permissible values.

( ) sin cos tan 2sini 2 2( ) tan 1 secii

cos( ) sec

siniii

Non-permissible values? Is this an identity?

( ) sin cos tan 2sini

x

y

sin cos tany

x

y

2siny

x

y

x

y


2y=tan 1 2secy

2 2( ) tan 1 secii

x

y

x

y


cos

siny

secy

cos( ) sec

siniii

We can also verify numerically that an identity is valid by substituting numerical values into both sides of the equation.

• Example: Verify whether the following are identities.

A) B)(use degrees) (use radians)

sin cos 2 2sin cos 1

Example: Verify whether the following are identities.

C) D)(use degrees) (use radians)

NOTE:

• This approach is insufficient to conclude that the equation is an identity because only a limited number of values were substituted for θ, and there may be a certain group of numbers for which this identity does not hold.

• To prove the identity is true using this method would require verifying ALL of the values in the domain (an infinite number).

• This type of reasoning is called inductive reasoning.

2 2tan 1 sec 2 2cot 1 csc

Proofs!• A proof is a deductive argument that

is used to show the validity of a mathematical statement.

• Deductive reasoning occurs when general principles or rules are applied to specific situations.

• Deductive reasoning is the process of coming up with a conclusion based on facts that have already been shown to be true.

• The facts that can be used to prove your conclusion deductively may come from accepted definitions, properties, laws or rules.

• The truth of the premises guaranteesthe truth of the conclusion.

Find the fifth term in the sequence

Inductive Reasoning

1. 3, 5, 7, 9, . . .

2. 3, 12, 27, 48, ...

3. 7, 14, 21, 28, ...

Deductive Reasoning

1. tn =2n + 1

2. tn = 3n2

3. Dates of Sundays in Feb 2016 year

• 15, 16, 18, 19, 25, 26, 28, 29, ______

What is the next number in this sequence?

Trig Proofs• Trig proofs (and simplifications of

trig expressions) are based on the definition of the 6 trigonometric functions and the Fundamental Trigonometric Identities.

• Sine fn:

• Cosine fn

• Tangent fn

• Cotangent

• Secant fn

• Cosecant fn

sin1

y

P(x, y)

1

x

y cos1

x

tanyx

cotxy

1sec

x

1csc

y

Definition of the 6 trigonometric functions

Fundamental Trigonometric Identities.

• Reciprocal Quotient Pythagorean

1tan

cot

1cot

tan

1cos

sec

1sec

cos

1sin

csc

1csc

sin

coscot

sin

sintan

cos

2 2sin cos 1

2 2tan 1 sec

2 2cot 1 csc

Note: These identities can be proven using the definitions of the trig functions.

Caution• The Pythagorean identities can be

expressed in different ways:

2 2sin cos 1

2 21 cos sin

2 21 sin cos

2 2tan 1 sec 2 2cot 1 csc

Simplify expressions using the Pythagorean identities, the reciprocal identities, and the

quotient identities

• Strategies that you might use to begin the simplifications:

– Replace a “squared” term with a Pythagorean identity

– Write the expression in terms of sine or cosine

– For expressions involving addition or subtraction, it may be necessary to use a common denominator to simplify a fraction

– Factor

– Multiply by a conjugate to obtain a Pythagorean identity

• You may also be asked to determine any non-permissible values of the variable in an expression.

• For example, identify the non-permissible values of θ in , and then simplify the expression.

• Solution:– The non-permissible values are when sin 0.

Why?

sin cos cot

coscot

sin

,k k

• In this case we write the expression in terms of sine or cosine

Simplify : sin cos cot

NOTE:

• Students often find simplifying trigonometric expressions more challenging than proving trigonometric identities because they may be uncertain of when an expression is simplified as much as possible.

• However, developing a good foundation with simplifying expressions makes the transition to proving trigonometric identities easier.

Simplify the following

• In this case we write the expression in terms of sine or cosine

A) sin secx x

Simplify the following

• In this case we use a Pythagorean Identity and simplify

21 cosB)

sin

• In this case we factor and then use a Pythagorean Identity

2C) sec sec sin

• In this case we have choices• We could use a Pythagorean

Identity and simplify• Or we could change each

term to sin and cos

2secD)

tan

xx

• What do we do here?• Multiply by a conjugate to obtain a Pythagorean

identity

sin cosE)

1 cos sin

• What do we do here?• First we change everything to sin x and cos x

• Now we have choice.

1. We can add the numerator by finding a lowest common denominator and then simplify.

2. We can multiply both the numerator and denominator by the LCD of all of the fractions WITHIN the overall fraction.

tan sinF)

1 cos

x xx

sinsin

cos1 cos

xx

xx

1. We can multiply both the numerator and denominator by the LCD of all of the fractions WITHIN the overall fraction.sin

sincos

1 cos

xx

xx

1. We can add the numerator by finding a lowest common denominator and then simplify

sinsin

cos1 cos

xx

xx

• Page 296

• # 1 a) d) 3b) c)

4, 7, 8c), 9, 10

Warm UP• Factor and simplify

2

2

sin sin cos1.

sin

x x x

x

• Factor and simplify2tan 3tan 4

2.sin tan sin

x xx x x

Proving Identities• The fundamental trigonometric

identities are used to establish other relationships among trigonometric functions.

• To verify an identity, we show that one side is equal to the other side.

Left Side = Right Side

LS = RS

• Each side is manipulated independently of the other side. – It is incorrect to perform operations

across the equal sign, such as:• multiplying or dividing, adding or subtracting

each side by an expression

• or cross multiplying

• or raising both sides to an exponent.

– These operations are only possible if the equation is true.

– Until we verify, or prove the identity to be true, we do not know if both sides are equal

Prove that the following are Identities using the definitions of the trig function on the unit circle

1

) cossec

A

sinB) tan

cos

2 2C) sin cos 1 2 2D) tan 1 sec

Guidelines for Proving Trigonometric Identities

• We usually start with side that contains the more complicatedexpression.

• If you substitute one or more fundamental identities on the more complicated side you will often be able to rewrite it in a form identical to that of the other side.

• Rewriting the complicated side in terms of sines and cosines is often helpful.

• If sums or differences appear on one side, use the least common denominatorand combine fractions

• In other cases factoring is useful.

• It may be necessary to multiply a fraction by a conjugate to obtain a Pythagorean Identity

• There is no one method that can be used to prove every identity.

• In fact there are often many different methods that may be used.

• However, one method may be shorterand more efficient than another.

• The more identities you prove, the

more confident and efficient you will become.

• Practice! Practice! Practice!

• DON’T BE AFRAID to stop and start over again if you are not getting anywhere.

• Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas.

Prove the following

• Which side is the more complicated side?– Left

• Lets work on the left side and change to sinesand cosines.

A) sec cot cscx x x

• Which side is the more complicated side?– Left

• Lets work on the left side and change to sinesand cosines.

B) sin tan cos secx x x x

• Which side is the more complicated side?– Right

• Lets work on the right side and factor out the Greatest Common Factor

3 2C) cos cos cos sin

• Which side is the more complicated side?– Right

• Lets work on the right side add the fractions by using the LCD

cos 1 sinD) 2sec

1 sin cos

• Which side is the more complicated side?– same

• Lets work on the left side and multiply by the conjugate

sin 1 cosE)

1 cos sin

2 2 2F) cos sin 2cos 1

• In this case we change the sine into cosine using a Pythagorean Identity

• In this case we change everything to sin and cos

sin cosG) sin cos

tan cot

t tt t

t t

sinH) cot csc

1 cos

tt t

t

21 1I) 2 cot

1 cos 1 cost

t t

2

2

1 sec sec 1J)

5sec 22 3sec 5sec

• Text Page 314-5

• #1 b) c), 2-4, 7b), 9, 10b), 11 b)c)

OTHER TRIG STUFF• Even-Odd Identities (Negative Angle):

sin sinx x csc cscx x

cos cosx x sec secx x

tan tanx x cot cotx x

OTHER TRIG STUFF• Addition and Subtraction Rules:

OR

sin sin cos sin cosa b a b b a



• Addition and Subtraction Rules:

OR

cos cos cos sin sina b a b a b



PROOF:

• This one of those “interesting proofs”.

• We need to use the:– Law of Cosines

• And the distance formula between 2 points


PROOF: cos cos cos sin sina b a b a b

PROOF: cos cos cos sin sina b a b a b

PROOF:

• Replace b by –b in



PROOF:

• Replace a by in


2a


PROOF:

• Replace b by –b in



Addition Formula for Tan

PROOF:

tan tan

tan1 tan tan

a ba b

a b

Subtraction Formula for Tan

tan tan

tan1 tan tan

a ba b

a b

Applications of theAngle Addition Formulae

• Finding exact values

• Deriving double and half angle formula

• Proving Identities

• In Calculus:– Trig derivatives (3208)

– Trig substitution in integration. (1001)

EXAMPLES:1. Find the exact values of:

A) cos 15o

B) sin 75o

C) sin12

D) 7

tan12

E) sin 60o cos 30o + sin 30o cos 60o

How can we verify that this is true?

F) tan15 tan30

1 tan15 tan30

o o

o o

G) and are both in Quadrant II, and . Determine the

exact value of .

A B513

cosA 35

sinB

cos A B

2. SimplifyA) sin sin

2 2

B) tan

Identities3. A)Prove: sin cos cos

6 3x x x

B) Prove: cos cos 2cos cos

PROVE:

sin) tan tan

cos cos

a bA a b

a b

) cos cos 2sin cosB a b a b a b

3 4 4 3) sin cos sin cos sin

7 7 7 7

x x x xC x

Find the exact value of:

o9 23) sin ) cos )sin225

12 12D E F

Double Angle Formulaesin2

cos2

tan2

Examples

1. Find the exact values of:

A) 2sin15ocos15o 2 2) cos sin8 8

B

2. Simplify:

2

4tan) sin cos )

2 2 1 tan

x xA B

2cos2 sin)

sin2

x xC

x

3. PROVE:

21 cos2) tan

1 cos2

AA A

A

tan2 tan) sin2

tan2 tan

B BB B

B B

) sin sinC

) 2sin sin 23 3 3

D cos

4 2) cos4 8cos 8cos 1E x x x

sin2)Show that can be

1 cos2 simplfied tocot

xF

xx

4. Suppose:

Find the exact value of:

A) sin 2 B) cos 2 C) tan2

1 3sin

4 2 2and

Half Angle FormulaeNot on Public but good to know

Consider:

cos2 = 2cos2 - 1

Let 22

xx

Half Angle FormulaeConsider:

cos2 = 1 – 2sin2

Let 22

xx

Examples:Find the exact value of:

A) sin 15o

B) cos 75o

Page 314-5#7A), 8, 9, 10A)C), 11A) 12, 13, 15,16,17

SOLVE, ALGEBRAICALLY ANDGRAPHICALLY, FIRST AND SECONDDEGREE TRIGONOMETRIC EQUATIONS

Last Section for Chapter 6 (6.4)

• The identities encountered earlier in this unit can now be applied to solve trigonometric equations.

Examples:

1. Find the solutions of

for 0° ≤ x < 360°.Solution: Graphically

A) Identify each curve B) What are the

points of intersection?

sin2 3cosx x

1. Find the solutions of

for 0° ≤ x < 360°.Solution: Algebraically

What are the solutions with an unrestricted domain, in radians?

sin2 3cosx x

2. Solve for 0° ≤ x ≤ 360°, giving exact solutions

where possible.

• Write the general solution in degrees and radian measure.

cos2 1 cosx x

3. Solve the trigonometric equation shown below for : 0 2x

32

sin3 cos cos3 sinx x x x

:

4. Solve: cos 2x + sin2x = 0.7311, for the domain 0° ≤ x < 360°.

Identifying and Repairing Errors

1. Identify and repair the mistake

A solution has been lost as a result of dividing both sides of the equation by sin x.

2. A student’s solution for tan2 x = sec x tan2 x for 0 ≤ x < π is shown below:

• Identify and explain the error(s).

• How many mark should the student get if this question was worth 4 marks?

2. tan2 x = sec x tan2 x for 0 ≤ x < π

• Provide the correct solution

Page 320 #1, 2A) B) C), 3, 5Page 321 #6,9,11,14,

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Unit 6 Trigonometric Identities - CBRH - Corner Brook ... 6 Trig Id and eq 2016.pdf · Prove trigonometric identities, using: ... – Replace a “squared” term with a Pythagorean