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Chapter 13: Trigonometric Identities and Equations
13.1: Trigonometric Identities
Trigonometric identity: an equation involving trigonometric functions that is true for all values for which every
expression in the equation is defined
Examples:
1. Find the exact value of each expression if 0o < 𝜃 < 90o Using the Trigonometric Identities.
a. if cot 𝜃 = 2, find tan 𝜃
2
c. if cos 𝜃 = 3 find sin 𝜃
4
b. if sin 𝜃 = 5 , find cos 𝜃
2
d. if cos 𝜃 = 3 find csc 𝜃
2. Find the exact value of each expression if 270o < 𝜃 < 360o Using the Trigonometric Identities.
1
a. If cos 𝜃 = 2, find sin 𝜃
3
b. If cot 𝜃 = − 2, find cos 𝜃.
2. Simplify each expression.
a. tan 𝜃 cos2 𝜃
c.
cos 𝜃 csc 𝜃
tan 𝜃
b. csc 2 𝜃 − cot 2 𝜃
cot2 𝜃
d. 1 + 1−𝑐𝑠𝑐 2 𝜃
3. When unpolarized light passes through polarized sunglass lenses, the intensity of the light is cut in half. If
the light passes through another polarized lens with its axis at an angle of 𝜃 to the first, the intensity of the
𝐼
light is again diminished. The intensity of the emerging light can be found by using the formula 𝐼 = 𝐼𝑜 − csc𝑜2 𝜃,
where 𝐼𝑜 is the intensity of the light incoming to the second polarized lens, I is the intensity of the emerging
light, and 𝜃 is the angle between the axes of polarization.
a. Simplify the formula in terms of cos 𝜃.
b. Use the simplified formula to determine the intensity of light that passes through a second
polarizing lens with axis at 30o to the original.
13.2: Verifying Trigonometric Identities
Verifying Identities by Transforming One Side:
1. Simplify one side of an equation until the two sides of the equation are the same. It is often easier
to work with the more complicated side of the equation.
2. Transform that expression into the form of the simpler side.
Suggestions for Verifying Identities:
 Substitute one or more basic trigonometric identities to simplify the expression.
 Factor or multiply as necessary. You may have to multiply both the numerator and
denominator by the same trigonometric expression.
 Write each side of the identity in terms of sine and cosine only. Then simplify each side as
much as possible.
 The properties of equality do not apply to identities as with equations. Do not perform
operations to the quantities on each side of an unverified identity.
Examples:
1. Verify that each equation is an identity.
a.
c.
sec2 𝜃
tan 𝜃
= cot 𝜃 + tan 𝜃
sec 𝜃
= sin 𝜃
tan 𝜃+cot 𝜃
e. tan2 𝜃 csc 2 𝜃 = 1 + tan2 𝜃
b. (1 + sin 𝜃)(1 − sin 𝜃) = cos2 𝜃
d.
f.
sin2 (𝑥)
tan2 𝑥
1−cos 𝜃
1+cos 𝜃
= cos 2 𝑥
= (csc 𝜃 − cot 𝜃)2
13.3: Sum and Differences of Angles Identities
Sum Identities:


sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B – sin A sin B

tan (A + B) =
tan 𝐴+tan 𝐵
1−tan 𝐴 tan 𝐵
Difference Identities:


sin(A – B) = sin A cos B – cos A sin B
cos(A – B) = cos A cos B + sin A sin B

tan (A – B) =
tan 𝐴 − tan 𝐵
1+tan 𝐴 tan 𝐵
Examples:
1. Find the exact value of each expression. Using Sum and Difference of Angles Identities.
5𝜋
a. cos 165o
b. cos 105o
c. cos 12
d. sin (–15o)
e. sin 135o
f. sin (–210o)
5𝜋
g. tan 75 °
h. csc 12
2. Verify that each equation is an identity.
a. sin (90 + 𝜃) = cos 𝜃
3𝜋
b. cos (
2
− 𝜃) = − sin 𝜃
𝜋
c. tan (𝜃 + ) = − cot 𝜃
2
13.4: Double-Angle and Half-Angle Identities
Double-Angle Identities: the following hold true for all values of 𝜃
Tan2𝜃=
cos 2𝜃 = cos2 𝜃 – sin2 𝜃
cos 2𝜃 = 2cos2 𝜃 – 1
cos 2𝜃 = 1 – 2sin2 𝜃
sin 2𝜃 = 2sin 𝜃 cos 𝜃
2 tan 𝜃
1−𝑡𝑎𝑛2 𝜃
Half-Angle Identities: the following identities hold true for all values of 𝜃
𝜃
1−cos 𝜃
2
2
sin = ±√
𝜃
1+cos 𝜃
2
2
cos = ±√
𝜃
1−cos 𝜃
2
1+cos 𝜃
tan = ±√
Examples:
𝜃
𝜃
1. Find the exact values of sin 2𝜃, cos 2𝜃, sin 2, and cos 2 .
5
𝜋
a. cos 𝜃 = − 13 ; 2 < 𝜃 < 𝜋
c. sin 𝜃 =
1
4
; 0° < 𝜃 < 90°
b. sin 𝜃 =
d. cos 𝜃 =
4
5
3
5
; 90° < 𝜃 < 180°
; 270° < 𝜃 < 360°
, cos 𝜃 ≠ – 1
8
e. tan 𝜃 = − 15 ; 90° < 𝜃 < 180°
2. Find the exact value of each expression.
𝜋
a. sin 8
b. tan 15°
3. Verify that each equation is an identity.
a. tan 𝜃 =
1−cos 2𝜃
sin 2𝜃
b. (sin 𝜃 + cos 𝜃)2 = 1 + 2 sin 𝜃 cos 𝜃
13.5: Solving Trigonometric Equations
trigonometric equations: trig identities are equations that are true for certain values of the variable
Examples:
1. Solve each equation if 0o < 𝜃 < 360o.
a. 2sin 𝜃 + 1 = 0
b. cos 𝜃 =
√3
2
c. sin 2𝜃 = −
√3
2
2. Solve each equation for all values of 𝜃 if 𝜃 is measured in radians.
a. 4sin2 𝜃 – 1 = 0
𝜃
𝜃
b. sin 2 + cos 2 = √2
c. cos 2𝜃 + 4cos 𝜃 = –3
3. Solve each equation for all values of 𝜃 if 𝜃 is measured in degrees.
a. cos 2𝜃 – sin2𝜃 + 2 = 0
b. cos 𝜃 – 2cos 𝜃sin 𝜃 = 0
4. Solve each equation for all values of 𝜃 if 𝜃 is measured in radians.
a. sin2 2𝜃 + cos2 𝜃 = 0
b. cos 8𝜃 = 1
c. sin 𝜃 tan 𝜃 – tan 𝜃 = 0
c. 2cos2 𝜃 = cos 𝜃