5-1 Study Guide and Intervention
Trigonometric Identities
Basic Trigonometric Identities An equation is an identity if the left side is equal to the right side for all values of the
variable for which both sides are defined. Trigonometric identities are identities that involve trigonometric functions.
Reciprocal Identities
1
sin θ =
cos θ =
csc θ
csc θ =
1
sec θ =
sin 𝜃
1
Pythagorean Identities
tan θ =
sec 𝜃
1
cot θ =
cos 𝜃
sin2 θ + cos 2 θ = 1
1
cot 𝜃
tan2 θ + 1 = sec 2 θ
1
tan 𝜃
cot 2 θ + 1 = csc 2 θ
𝟑
Example: If sin θ = 𝟓 and 0° < θ < 90°, find tan θ.
Use two identities to relate sin θ and tan θ.
sin2 θ + cos 2 θ = 1
3 2
( )
5
Now find tan θ.
Pythagorean Identity
sin θ
+ cos θ = 1
2
cos θ =
2
16
25
cos θ = ± √
sin θ =
16
25
or ±
4
5
tan θ = cos θ
3
5
Quotient identity
Simplify.
tan θ =
3
5
4
5
sin θ = , cos θ =
Take the square root of each side.
tan θ =
3
4
Simplify.
3
4
5
5
Since 0° < θ < 90°, cos θ is positive.
4
Thus, cos θ = 5.
Exercises
Find the value of each expression using the given information.
1. If cot θ =
12
,
5
find tan θ.
1
2. If sin θ = – 4, find csc θ.
4
5. If cot α = – 3 and sin α < 0, find cos α and csc α.
Chapter 5
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Glencoe Precalculus
5-1 Study Guide and Intervention (continued)
Trigonometric Identities
Simplify and Rewrite Trigonometric Expressions You can apply trigonometric identities and algebraic techniques
such as substitution, factoring, and simplifying fractions to simplify and rewrite trigonometric expressions.
Example: Simplify each expression.
a. sec x – cos x
1
sec x − cos x = cos 𝑥 − cos x
=
1 − cos2 𝑥
cos 𝑥
=
sin2 𝑥
cos 𝑥
Reciprocal Identity
Add.
Pythagorean Identity
sin 𝑥
)
cos 𝑥
= sin x (
Factor.
= sin x tan x
Quotient Identity
𝟏
b. csc x 𝐜𝐨𝐭 𝟐 x + 𝐬𝐢𝐧 𝒙
1
csc x cot 2 x + sin 𝑥 = csc x cot 2 x + csc x
Reciprocal Identity
= csc x (csc 2 x – 1) + csc x
Pythagorean Identity
= csc 3 x – csc x + csc x
Distributive Property
3
= csc x
Simplify.
Exercises
Simplify each expression.
1. cos x (tan x + cot x)
Chapter 5
2. sin x + cos x cot x
6
Glencoe Precalculus
5-2 Study Guide and Intervention
Verifying Trigonometric Identities
Verify Trigonometric Identities To verify an identity means to prove that both sides of the equation are equal for all
values of the variable for which both sides are defined.
Example: Verify that
𝐬𝐞𝐜 𝟐 𝒙 – 𝟏
𝐬𝐞𝐜 𝟐 𝒙
= 𝐬𝐢𝐧𝟐 x.
The left-hand side of this identity is more complicated, so start with that expression first.
sec2 𝑥 – 1
sec2 𝑥
=
(tan2 𝑥 + 1) − 1
sec2 𝑥
tan2 𝑥
= sec2 𝑥
sin2 𝑥
)
cos2 𝑥
1
cos2 𝑥
Pythagorean Identity
Simplify.
(
=
sin2 𝑥
Quotient Identity and Reciprocal Identity
= cos2 𝑥 · cos 2 x
Simplify.
= sin2 x
Multiply.
Notice that the verification ends with the expression on the other side of the identity.
Exercises
Verify each identity.
1. sec θ – cos θ = sin θ tan θ
5-3 Study Guide and Intervention
Solving Trigonometric Equations
Use Algebraic Techniques to Solve To solve a trigonometric equation, you may need to apply algebraic methods.
These methods include isolating the trigonometric expression, taking the square root of each side, factoring and applying
the Zero-Product Property, applying the quadratic formula, or rewriting using a single trigonometric function. In this
lesson, we will consider conditional trigonometric equations, or equations that may be true for certain values of the
variable but false for others.
Example 1: Find all solutions of tan x cos x – cos x = 0 on the interval [0, 2π).
tan x cos x – cos x = 0
cos x (tan x – 1) = 0
Chapter 5
Original equation
Factor.
6
Glencoe Precalculus
tan x – 1 = 0
cos x = 0 or
𝜋
2
x = or
3𝜋
2
tan x = 1
𝜋
x = 4 or
𝜋
Set each factor equal to 0.
5𝜋
4
3𝜋
𝜋
5𝜋
When x = 2 or 2 , tan x is undefined, so the solutions of the original equation are 4 or 4 . When you solve for all values of
x, the solution should be represented as x + 2nπ for sin x and cos x and x + nπ for tan x, where n is any integer. The
𝜋
5𝜋
solutions are 4 + nπ or 4 + nπ.
Example 2: Find all solutions of sin x + √𝟑 = –sin x.
sin x + √3 = –sin x
Original equation
2 sin x + √3 = 0
Add sin x to each side.
2 sin x = – √3
Subtract √3 from each side.
sin x = –
√3
2
Divide each side by 2.
4𝜋
3
5𝜋
3
Solve for x.
x=
or
The solutions are
4𝜋
3
+ 2nπ or
5𝜋
3
+ 2nπ.
Exercises
Solve each equation for all values of x.
1. cos x = –1
2. sin3 x – 4 sin x = 0
--------------------------------------------------------------------------------------------------------------------------------------------**Use Trigonometric Identities to Solve You can use trigonometric identities along with algebraic methods to solve
trigonometric equations. Be careful to check all solutions in the original equation to make sure they are valid solutions.
Example 1: Find all solutions of 2 𝐭𝐚𝐧𝟐 x – 𝐬𝐞𝐜 𝟐 x + 3 = 1 – 2 tan x on the interval [0, 2π).
2 tan2 x – sec 2 x + 3 = 1 – 2 tan x
2 tan2 x – (tan2 x + 1) + 3 = 1 – 2 tan x
tan2 x + 2 = 1 – 2 tan x
2
tan x + 2 tan x + 1 = 0
Chapter 5
Simplify.
Factor.
tan x = –1
3𝜋
4
sec 2 x = tan2 x + 1
Simplify.
(tan 𝑥 + 1)2 = 0
x=
Original equation
or
Take the square root of each side.
7𝜋
4
Solve for x on [0, 2π).
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Glencoe Precalculus
Example 2: Find all solutions of 1 + cos x = sin x on the interval [0, 2π).
1 + cos x = sin x
Original equation
(1 + cos 𝑥)2 = (sin x)2
1 + 2 cos x + cos 2 x = sin2 x
2
Square each side.
Multiply.
2
1 + 2 cos x + cos x = 1 – cos x
Pythagorean Identity
2 cos 2 x + 2 cos x = 0
Simplify.
2 cos x (cos x + 1) = 0
Factor.
cos x = 0 or cos x = –1
x=
𝜋
,
2
Zero Product Property
π,
3𝜋
2
Solve for x on [0, 2π).
5-4 Study Guide and Intervention
Sum and Difference Identities
Evaluate Trigonometric Functions You can use the sum and difference identities and the values of trigonometric
functions of common angles to find the exact values of less common angles.
Sum Identities
Difference Identities
cos (α + β) = cos α cos β – sin α sin β
cos (α – β) = cos α cos β + sin α sin β
sin (α + β) = sin α cos β + cos α sin β
sin (α – β) = sin α cos β – cos α sin β
tan 𝛼 + tan 𝛽
tan 𝛼 − tan 𝛽
tan (α + β) = 1 − tan 𝛼 tan 𝛽
tan (α – β) = 1+ tan 𝛼 tan 𝛽
Example: Find the exact value of cos 375°.
cos 375° = cos (330° + 45°)
330° and 45° are common angles with a sum of 375°.
= cos 330° cos 45° – sin 330° sin 45°
Chapter 5
=
√3
2
·
√2
2
=
√6
4
+
=
√6 + √2
4
√2
4
1
– (− 2) ·
√2
2
Cosine Sum Identity
cos 330° =
√3
,
2
cos 45° =
√2
,
2
1
sin 330° = – , sin 45° =
2
√2
2
Multiply.
Combine the fractions.
6
Glencoe Precalculus
Exercises
Find the exact value of each trigonometric expression.
2. tan 15°
1. cos (–15°)
3. cos (−
7𝜋
)
12
4. cos
5. sin 20° cos 10° + cos 20° sin 10°
6.
11𝜋
12
π
5𝜋
+ tan
9
36
π
5𝜋
1 − tan tan
9
36
tan
Simplify each expression.
𝜋
𝜋
7. cos 70° cos 20° – sin 70° sin 20°
8. sin 12 cos y – sin y cos 12
---------------------------------------------------------------------------------------------------------------------------------------------------
Solve Trigonometric Equations You can solve trigonometric equations using the sum and difference identities along
with algebraic methods and the same techniques you used before.
𝝅
𝝅
Example: Find the solutions of sin (𝟐 + 𝒙) + cos (𝟐 + 𝒙) = 0 on the interval [0, 2π).
𝜋
𝜋
sin ( 2 + 𝑥) + cos ( 2 + 𝑥) = 0
𝜋
2
𝜋
2
𝜋
2
Original equation
𝜋
2
sin cos x + cos sin x + cos cos x – sin sin x = 0
Cosine Sum Identity
1 (cos x) + 0 (sin x) + 0 (cos x) – 1 (sin x) = 0
Substitute.
cos x – sin x = 0
Simplify.
cos x = sin x
𝜋
On the interval [0, 2π), cos x = sin x when x = 4 and x =
Add.
5𝜋
.
4
Exercises
Find the solution of each equation on the interval [0, 2π).
𝜋
𝜋
1. cos ( 4 − 𝑥) – sin ( 4 − 𝑥) = –1
Chapter 5
2. sin (π + x) + sin (π + x) = 1
6
Glencoe Precalculus
Answer Key:
Chapter 5
6
Glencoe Precalculus
Chapter 5
6
Glencoe Precalculus
Chapter 5
6
Glencoe Precalculus
Chapter 5
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Glencoe Precalculus