Grade 12 Pre-Calculus Mathematics

[MPC40S]

Chapter 6

Trigonometric Identities

Outcomes

T5, T6

12P.T.5. Solve, algebraically first and second degree trigonometric equations with the domain expressed in degrees and radians. 12P.T.6. Prove trigonometric identities, using

- Reciprocal identities - Quotient identities - Pythagorean identities - Sum or difference (restricted to sine, cosine, and tangent) - Double-angle identities (restricted to sine, oscine, and tangent.

MPC40S Date: ____________________

Pg. #2

MPC40S Date: ____________________

Pg. #3

Chapter 6 – Homework

Section Page Questions

MPC40S Date: ____________________

Pg. #4

MPC40S Date: ____________________

Pg. #5

Chapter 6: TRIGONOMETRIC IDENTITIES

6.1 – Reciprocal, Quotient, and Pythagorean Identities

An equation like sin 𝜃 =1

2 is true for very few values of 𝜃, like 𝜃 =

𝜋

6 or 𝜃 =

5𝜋

6, or all

coterminal angles with those angles.

An identity like tan 𝑥 =sin 𝑥

cos 𝑥 is always true for all 𝜃 ∈ 𝑅 (or nearly all).

Recall the following identities:

tan 𝑥 =sin 𝑥

cos 𝑥 csc 𝑥 =

1

sin 𝑥 sec 𝑥 =

1

cos 𝑥

cot 𝑥 =cos 𝑥

sin 𝑥 cot 𝑥 =

1

tan 𝑥

Non-permissible values are any values that result in a denominator equal to 0. Example #1

Determine the non-permissible values of the following identities, over the interval 0 ≤ 𝑥 ≤ 2𝜋

a) cot 𝑥 =1

tan 𝑥

b) cot 𝑥

(tan 𝑥−1)(sin 𝑥)

T6

MPC40S Date: ____________________

Pg. #6

Example #2

Verify that the equation sec 𝑥

tan 𝑥+cot 𝑥= sin 𝑥 is true for the following values of x.

a) 𝑥 = 60°

b) 𝑥 =3𝜋

4

MPC40S Date: ____________________

Pg. #7

c) State the non-permissible values of the above identity over the interval 0 ≤ 𝑥 ≤ 2𝜋

MPC40S Date: ____________________

Pg. #8

Pythagorean Identities From the unit circle, we know that cos 𝜃 = 𝑥 and sin 𝜃 = 𝑦. We also know that the equation of the unit circle is 𝑥2 + 𝑦2 = 1 Therefore, upon substitution, 1). _____________________________ Dividing Equation 1) by cos2θ results in:

sin2𝜃

cos2𝜃+

cos2𝜃

cos2𝜃=

1

cos2𝜃 → 2). _____________________________

Dividing Equation 1) by sin2θ results in:

sin2𝜃

sin2𝜃+

cos2𝜃

sin2𝜃=

1

sin2𝜃 → 3). _____________________________

We can manipulate these equations to create other equations. sin2𝜃 + cos2𝜃 = 1 tan2𝜃 + 1 = sec2𝜃 1 + cot2𝜃 = csc2𝜃

______________________ ______________________ ______________________ ______________________ ______________________ ______________________

Note: These identities are only true when the trigonometric functions are squared. (i.e. sin 𝜃 + cos 𝜃 ≠ 1)

MPC40S Date: ____________________

Pg. #9

Example #3

Simplify the following expressions completely.

a) tan 𝜃 cos 𝜃

sec 𝜃 cot 𝜃 b)

cot 𝑥

csc 𝑥 cos 𝑥 c)

sec2𝜃 cos 𝜃

csc 𝜃

MPC40S Date: ____________________

Pg. #10

MPC40S Date: ____________________

Pg. #11

Chapter 6: TRIGONOMETRIC IDENTITIES

6.3 – Proving Identities Part 1 To PROVE AN IDENTITY means to manipulate both sides of an equation independently until they are identical. Helpful Hints

• Rewrite all functions in terms of sin 𝜃 and cos 𝜃

• Factor

• Multiply by a form of 1 (for example: sin θ

sin θ)

• Simplify

• Expand

• Multiply by the conjugate (1 + cos 𝑥 → 1 − cos 𝑥)

Things to Remember

• Identities are different from equations. Keep side separate!

• Use the correct variable (𝜃, 𝑥, 𝛼, 𝛽, etc. )

• Left Hand Side = Right Hand Side (LHS = RHS)

• Show all of your work (marks are allocated for work shown)

• Do not invent new math. It is better to not arrive at the same answer than to make something up!

• Work vertically ↓

T6

MPC40S Date: ____________________

Pg. #12

Example #1

Prove the following identities for all permissible values of the variable. a) (sin 𝑥)(sec 𝑥)(cot 𝑥) = 1

LHS RHS

b) cos2𝑥

cot 𝑥= sin 𝑥 cos 𝑥

LHS RHS

MPC40S Date: ____________________

Pg. #13

c) sec 𝑥

tan 𝑥+cot 𝑥= sin 𝑥

LHS RHS

MPC40S Date: ____________________

Pg. #14

d) 1 − cos2𝜃 = cos2𝜃 tan2𝜃

LHS RHS

e) sin 𝛼

1−cos 𝛼=

1+cos 𝛼

sin 𝛼

LHS RHS

MPC40S Date: ____________________

Pg. #15

f) cos4𝜃 − sin4𝜃 = 1 − 2sin2𝜃

LHS RHS

MPC40S Date: ____________________

Pg. #16

MPC40S Date: ____________________

Pg. #17

Chapter 6: TRIGONOMETRIC IDENTITIES

6.2 – Sum, Difference, and Double-Angle Identities Along with the Pythagorean Identities, we have additional identities we can work with. They are the Sum and Difference Identities and the Double Angle Identities. Sum and Difference Identities sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽 sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽 cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽

tan(𝛼 − 𝛽) =tan 𝛼−tan 𝛽

1+tan 𝛼 tan 𝛽 tan(𝛼 + 𝛽) =

tan 𝛼+tan 𝛽

1−tan 𝛼 tan 𝛽

Double Angle Identities sin 2𝛼 = 2 sin 𝛼 cos 𝛼

cos 2𝛼 = cos2𝛼−sin2𝛼

cos 2𝛼 = 1 − 2sin2𝛼

cos 2𝛼 = 2cos2𝛼 − 1

tan 2𝛼 =2 tan 𝛼

1 − tan2𝛼

With the help of these identities, we can now we can find the coordinates of any point on the unit circle, provided it is a sum or difference of our special angels.

(𝜋

3,

𝜋

4,

𝜋

6 family members)

T6

MPC40S Date: ____________________

Pg. #18

Example #1

Simplify the following expressions and give the exact value.

a) cos𝜋

3cos

𝜋

12+ sin

𝜋

3sin

𝜋

12

b) cos2(47°) + sin2(47°)

c) 1 − 2sin2 𝜋

6

MPC40S Date: ____________________

Pg. #19

We can use these formulas to determine the exact value of a trigonometric ratio that is not on the unit circle. Example #2

Determine the exact value of each expression below.

a) sin7𝜋

12 b) cos 195°

MPC40S Date: ____________________

Pg. #20

We can simplify expressions using identities. Example #3

Consider the expression 1−cos 2𝜃

sin 2𝜃

a) Determine the non-permissible values for this expression between [0, 2𝜋]

b) Simplify the expression to one of the three primary trigonometry functions.

MPC40S Date: ____________________

Pg. #21

Example #4

If 0 < 𝜃 <𝜋

2 and sin 𝜃 =

3

5, determine the exact value of cos 2𝜃.

MPC40S Date: ____________________

Pg. #22

Example #5

If cos 𝛼 = −4

5 where

𝜋

2≤ 𝛼 ≤ 𝜋 and cos 𝛽 = −

12

13 where 𝜋 ≤ 𝛽 ≤

3𝜋

2, determine the exact

value of cos(𝛼 − 𝛽).

MPC40S Date: ____________________

Pg. #23

Chapter 6: TRIGONOMETRIC IDENTITIES

6.3 – Proving Identities Part 2 Example #1

Prove the following identities for all permissible values of the variable.

a) cot 𝜃 − csc 𝜃 =cos 2𝜃−cos 𝜃

sin 2𝜃+sin 𝜃

LHS RHS

T6

MPC40S Date: ____________________

Pg. #24

b) 1−tan2𝜃

1+tan2𝜃= cos 2𝜃

LHS RHS

MPC40S Date: ____________________

Pg. #25

Chapter 6: TRIGONOMETRIC IDENTITIES

6.4 – Solving Trigonometric Equations Using Identities We will be solving trigonometric equations just like in the past, but now we need to make substitutions using trigonometric identities. Example #1

Solve the following trigonometric equation sin 2𝜃 = sin 𝜃 over the interval [0, 2𝜋] Example #2

Solve the following trigonometric equation over the interval [0, 2𝜋). tan 𝑥 cos 𝑥 sin 𝑥 cot 𝑥 csc 𝑥 − 1 = 0

T5

MPC40S Date: ____________________

Pg. #26

Example #3

a) Determine the non-permissible values of the following equation over the interval [0, 2𝜋).

tan 𝜃 = 2 sin 𝜃 b) Solve the above trigonometric equation over the given interval

MPC40S Date: ____________________

Pg. #27

Example #4

Solve the following trigonometric equations over the interval [0, 2𝜋]. a) cos 2𝑥 + 1 = cos 𝑥 b) 1 − cos2𝑥 = 3 sin 𝑥 − 2

MPC40S Date: ____________________

Pg. #28

c) 2 sin 𝑥 = 7 − 3 csc 𝑥