7-1

7.3 Sum and Difference Identities

Cosine Sum and Difference Identities: cos A B does NOT equal cos cos .A B

Cosine of a Sum or Difference

cos ____________________________________A B

cos ____________________________________A B

EXAMPLE 1 Finding Exact Cosine Function Values Find the exact value of each expression.

(a) cos15 (b) cos 75°

Sine of a Sum or Difference

sin ____________________________________A B

sin ____________________________________A B

Tangent of a Sum or Difference

tan A B

tan A B

EXAMPLE 3 Finding Exact Sine and Tangent Function Values

Find the exact value of each expression.

(a) sin 75° (b) tan 105°

(c) sin 40° cos 160° − cos 40° sin 160° (d) cos87 cos93 sin87 sin93

EXAMPLE 4 Writing Functions as Expressions Involving Functions of

Write each function as an expression involving functions of .

(a) cos 30 (b) tan 45

(c) sin 180

EXAMPLE 5 Finding Function Values and the Quadrant of A + B

Suppose that A and B are angles in standard position, with 4

sin , ,5 2

A A

and 5 3

cos , .13 2

B B

Find each of the following.

(a) sin (A + B) (b) tan (A + B)

7-3

Verifying an Identity

EXAMPLE 7 Verifying an Identity

Verify that the following equation is an identity.

sin cos cos6 3

7.4 Double-Angle and Half-Angle Identities

■ Double-Angle Identities ■ Verifying an Identity

Double-Angle Identities

cos 2A = sin 2A =

cos 2A =

cos 2A = tan 2A =

EXAMPLE 1 Finding Function Values of 2 Given Information about

Given 3

cos5

and sin 0, find sin2 , cos2 , and tan2 .

EXAMPLE 2 Finding Function Values of Given Information about 2

Find the values of the six trigonometric functions of if 4

cos25

and 90 180 .

7-5

EXAMPLE 3 Simplifying Expressions Using Double-Angle Identities

Simplify each expression.

(a) 2 2cos 7 sin 7x x (b) sin15 cos15

EXAMPLE 4 Deriving a Multiple-Angle Identity

Write sin 3x in terms of sin x.

Half-Angle Identities

In the following identities, the symbol _____________ indicates that the sign is chosen based on the function

under consideration and the _____________ of .2

A

cos (𝐴

2) = ___________ sin (

𝐴

2) = ___________ tan (

𝐴

2) = ___________

EXAMPLE 9 Using a Half-Angle Identity to Find an Exact Value

Find the exact value of tan 22.5° using the identity sin

tan .2 1 cos

A A

A

7.5 Inverse Circular Functions

■ Review of Inverse Functions ■ Inverse Sine, Cosine, Tangent Functions ■ Remaining Inverse Circular

Functions ■ Inverse Function Values

Review of Inverse Functions

If a function is defined so that each ____________ element is used __________________

__________________, then it is called a one-to-one function.

Do not confuse the −1 in 1f with a negative exponent. The symbol 1f x does NOT represent

1.

f x It

represents the ____________ ____________ of f.

Review of Inverse Functions

1. In a one-to-one function, each x-value corresponds to ____________ ____________

y-value and each y-value corresponds to ____________ ____________ x-value.

2. If a function f is one-to-one, then f has an ____________ ____________ 1.f

3. The domain of f is the ____________ of 1,f and the range of f is the ____________ of 1.f That is, if the

point (a, b) is on the graph of f, then ____________ is on the graph of 1.f

4. The graphs of f and 1f are ____________ of each other across the line y = x.

5. To find 1f x from ,f x follow these steps.

Step 1 Replace f x with y and interchange x and y.

Step 2 Solve for y.

Step 3 Replace y with 1 .f x

Inverse Sine Function

1siny x

or arcsiny x means that sin , for .2 2

x y y

We can think of 1

siny x or arcsiny x as

“y is the angle in the interval ,2 2

whose sine is x.”

EXAMPLE 1 Finding Inverse Sine Values

Find y in each equation.

(a) 1

arcsin2

y (b) 1sin 1y (c) 1sin 2y

Be certain that the number given for an inverse function value is in the range of the particular inverse

function being considered.

7-7

Domain: _______________ Range: _________

Inverse Cosine Function

1cosy x

or arccosy x means that cos , for 0 .x y y

Domain: _______________ Range: _________

EXAMPLE 2 Finding Inverse Cosine Values

Find y in each equation.

(a) y = arccos (-1) (b) 1 2

cos2

y

(c) y = arccos (1/2)

x y

x y

Inverse Tangent Function

1tany x

or arctany x means that tan , for .2 2

x y y

Domain: _______________ Range: _________

Remaining Inverse Circular Functions

Inverse Cotangent, Secant, and Cosecant Functions

1coty x

or arccoty x means that cot , for 0 .x y y

1secy x

or arcsecy x means that sec , for 0 , .2

x y y y

1cscy x

or arccscy x means that csc , for , 0.2 2

x y y y

x y

7-9

Inverse Function Domain

Range

Interval

Quadrants of the

Unit Circle

1siny x

1cosy x

1tany x

1coty x

1secy x

1cscy x

EXAMPLE 3 Finding Inverse Function Values (Degree-Measured Angles)

Find the degree measure of in the following.

(a) arctan1 (b) 1sec 2

Use the following to evaluate these inverse trigonometric functions on a calculator.

1sec x

can be evaluated as 1 1cos ;

x

1csc x

can be evaluated as 1 1sin ;

x

1cot x

can be evaluated as

1

1

1tan if 0

1180 tan if 0.

xx

xx

EXAMPLE 4 Finding Inverse Function Values with a Calculator

Use a calculator to give each value.

(a) Find y in radians if 1csc 3 .y (b) Find in degrees if arccot 0.3541 .

Be careful when using your calculator to evaluate the inverse cotangent of a negative quantity. To do this, we

must enter the inverse tangent of the _____________ of the negative quantity, which returns an angle in

quadrant _____________. Since inverse cotangent is _____________ in quadrant II, adjust your calculator

result by adding 180° or accordingly. Note that 1cot 0 .2

EXAMPLE 5 Finding Function Values Using Definitions of the Trigonometric Functions

Evaluate each expression without using a calculator.

(a) sin (sin−1 (0.7)) (b) sin−1 (sin (3π

4))

(c) 1 3sin tan

2

(d) 1 5tan cos

13

EXAMPLE 6 Finding Function Values Using Identities

Evaluate each expression without using a calculator.

(a) 1

cos arctan 3 arcsin3

(b)

2tan 2arcsin

5

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7.6 Solving Trigonometric Equations

■ Solving by Linear Methods ■ Solving by Factoring ■ Solving by Quadratic Methods

■ Solving by Using Trigonometric Identities ■ Equations with Half-Angles

■ Equations with Multiple Angles ■ Applications

Solving by Linear Methods

EXAMPLE 1 Solving a Trigonometric Equation by Linear Methods

Solve the equation 2sin 1 0

(a) over the interval [0°, 360°), and (b) for all solutions.

Solving by Factoring

EXAMPLE 2 Solving a Trigonometric Equation by Factoring

Solve sin tan sin over the interval [0°, 360°).

Solving by Quadratic Methods

EXAMPLE 3 Solving a Trigonometric Equation by Factoring

Solve 2tan tan 2 0x x over the interval 0, 2 .

EXAMPLE 4 Solving a Trigonometric Equation Using the Quadratic Formula

Find all solutions of cot cot 3 1.x x Write the solution set.

7-13

Solving a Trigonometric Equation

1. Decide whether the equation is ______________ or ______________ in form, so that you can determine

the solution method.

2. If only one trigonometric function is present, ______________ ______________ ______________ for

that function.

3. If more than one trigonometric function is present, rearrange the equation so that one side equals

______________. Then try to ______________and set each ______________equal to 0 to solve.

4. If the equation is quadratic in form, but not factorable, use the ______________ ______________. Check

that solutions are in the desired interval.

5. Try using ______________to change the form of the equation. It may be helpful to square each side of the

equation first. In this case, check for ______________solutions.

Equations with Half-Angles

EXAMPLE 6 Solving an Equation with a Half-Angle

Solve the equation 2sin 12

x

(a) over the interval 0, 2 , and (b) for all solutions.

Equations with Multiple Angles

EXAMPLE 7 Solving an Equation Using a Double Angle

Solve cos(2𝑥) =√2

2 over the interval 0, 2 .

EXAMPLE 8 Solving an Equation Using a Multiple-Angle

Solve sin(3𝑥) =−1

2

(a) over the interval [0°, 360°), and (b) for all solutions.