5-1 Study Guide and Intervention - MRS. FRUGE

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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

Pythagorean Identities

sin = 1

csc

csc = 1

sin

cos = 1

sec

sec = 1

cos

tan = 1

cot

cot = 1

tan

sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2

Example:

If

sin

=

and

0?

<

<

90?,

find

tan

.

Use two identities to relate sin and tan .

sin2 + cos2 = 1

(3)2

5

+

cos2

=

1

cos2

=

16 25

Pythagorean Identity sin = 3

5

Simplify.

cos

=

?

16

25

or

?

4 5

Take the square root of each side.

Since 0? < < 90?, cos is positive. Thus, cos = 45.

Now find tan .

tan

=

sin cos

3

tan

=

5 4

5

tan

=

3 4

Quotient identity

sin = 3, cos = 4

5

5

Simplify.

Exercises

Find the value of each expression using the given information.

1. If cot = 12, find tan .

5

2. If sin = ? 1, find csc . ?4 4

3. If tan = 23, find cot .

4. If sec = ?2, find csc ( - 2). 2

5.

If

cot

=

?

4 3

and

sin

<

0,

find

cos

and

csc

.

cos = , and csc = ?

6. If sec = ?4 and csc > 0, find cos and tan .

cos

=

?

,

and

tan

=

?

Chapter 5

5

Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

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

sec

x

-

cos

x

=

1 cos

-

cos

x

= 1 - cos2

cos

= sin2

cos

=

sin

x

(sin

cos

)

= sin x tan x

b.

csc

x

x

+

csc

x

cot2

x

+

1 sin

=

csc

x

cot2

x

+

csc

x

= csc x (csc2 x ? 1) + csc x

= csc3 x ? csc x + csc x

= csc3 x

Reciprocal Identity Add. Pythagorean Identity Factor. Quotient Identity

Reciprocal Identity Pythagorean Identity Distributive Property Simplify.

Exercises Simplify each expression.

1. cos x (tan x + cot x) csc x

2. sin x + cos x cot x csc x

3.

1

csc2 + tan2

x

5. (cot2 x + 1)(sec2 x ? 1) x

7. csc x sin x + cot2 x x

4. (sec x ? tan x)(csc x + 1) cot x

6.

1

+

tan2 1 + sec

sec x

8. cos x (1 + tan2 x ) sec x

9. cos (2 ? ) x csc

10. cos (2 ? ) + cos2 x 1 csc

Chapter 5

6

Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

5-1 Practice

Trigonometric Identities

Find the value of each expression using the given information.

1. If cos

=

1 4

and

0?

<

<

90?,

find

tan

.

2.

If sin

=

2 3

and

0?

<

<

90?,

find cos

.

3.

If tan =

7 and 0? <

2

< 90?, find sin .

4. If tan = 2 and 0? < < 90?, find cot .

5. If sin =

?

4 5

and

cos

>

0,

find

tan

and

sec

.

tan = ? , sec =

6. If cot x =

?

3 2

and

sec

x

<

0,

find

sin

x

and

cos

x.

sin x = , cos x = ?

7. If cos = 0.54, find sin ( - 2). ?0.54

8. If cot x = ?0.18, find tan ( - 2). 0.18

Simplify each expression. 9. cos x + sin x tan x sec x

11. sin2 cos2 ? cos2 ?

10.

cot tan

? A

12.

csc2 - cot2 sin (-) cot

?sec x

13. KITE FLYING Brett and Tara are flying a kite. When the string is tied to the ground, the height of the kite can be determined by the formula = csc , where L is the length of the string and is the angle between the string and the

level ground. What formula could Brett and Tara use to find the height of the kite if they know the value of sin ?

H = L sin

Chapter 5

7

Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

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

?

=

x.

The left-hand side of this identity is more complicated, so start with that expression first.

sec2 sec2

?

1

=

(tan2 + sec2

1)

-

1

=

tan2 sec2

= (csoins22 ) 1 cos2

=

sin2 cos2

?

cos2

x

= sin2 x

Pythagorean Identity Simplify.

Quotient Identity and Reciprocal Identity Simplify. Multiply.

Notice that the verification ends with the expression on the other side of the identity.

Exercises Verify each identity.

1. sec ? cos = sin tan

sec - cos = ? cos = ? = = sin ( )

= sin tan

2. sec = sin (tan + cot )

sin (tan + cot ) = sin ( + ) = sin ( + )

= sin ( )= = sec

3. tan csc cos = 1 tan csc cos = ( ) ( ) cos = 1

4.

csc2 - cot2 1 - sin2

=

sec2

- ?

=

(

+ ) ?

=

=

Chapter 5

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Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

5-2 Study Guide and Intervention (continued)

Verifying Trigonometric Identities

Identifying Identities and Nonidentities You can use a graphing calculator to test whether an equation might be an identity by graphing the functions related to each side of the equation. If the graphs of the related functions do not coincide for all values of x for which both functions are defined, the equation is not an identity. If the graphs appear to coincide, you can verify that the equation is an identity by using trigonometric properties and algebraic techniques.

Example: Use a graphing calculator to test whether csc ? sin = cot cos is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.

The equation appears to be an identity because the graphs of the related functions coincide. Verify this algebraically.

csc

?

sin

=

1 sin

?

sin

=

1

- sin2 sin

Rewrite in terms of sine using a Reciprocal Identity. Rewrite using a common denominator.

=

cos2 sin

Pythagorean Identity

=

cos sin

?

cos

Factor cos2 .

= cot cos Rewrite in terms of cot using a Quotient Identity.

Exercises

Test whether each equation is an identity by graphing. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.

1. sin x + cos x cot x = csc x

sin x + cos x cot x

= sin x + cos x ( )

= sin x + = +

= = csc x

2. 2 ? cos2 x = sin2 x

When

x

=

,

=

1.5

and = 0.5;

therefore, the equation is not an identity.

Chapter 5

11

Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

5-2 Practice

Verifying Trigonometric Identities

Verify each identity.

1.

cot

csc +

tan

=

cos

x

2.

sin

1

-

1

-

sin

1

+

1

=

-2

sec2

y

=

+

?

+

=

+

-

?

+

=

+ ? + ?

=

?

= =

3. sin3 x ? cos3 x = (1 + sin x cos x)(sin x ? cos x) x ? x = (sin x ? cos x)

( x + sin x cos x + x)

= ?2 y

4.

tan

+

cos 1 + sin

=

sec

tan + = +

+ +

= + +

( )(+ )

= (sin x ? cos x)(1 + sin x cos x)

=

+ ( )(+

)

=

sec

= (1 + sin x cos x)(sin x ? cos x)

5. (sec

-

tan )2 =

1 - sin 1 + sin

6.

1

sin + cos

+

1

+ cos sin

=

2

csc

( ? ) = ? 2 sec tan +

=

?

2(

)

(

)

+

+ + = ++ +

+

( + )

=

?

+

= +

( + )

=

( ? ) ?

=

( ? ) ( + )( ?

)

=

? +

= ( + )

( + )

= = 2 csc

Test whether each equation is an identity by graphing. If it appears to be an identity, verify it.

If not, find an x-value for which both sides are defined but not equal.

7.

cos 1- sin

=

1 + sin cos

= ? +

? ? +

=

( + ?

)

=

( +

)

= +

8. sin x(sec x + cot x) = cos x

When

x

=

,

=

1.7

and

=

0.7;

therefore, the equation is not an identity.

9. PHYSICS The work done in moving an object is given by the formula W = Fd cos , where d is the displacement, F is

the

force

exerted,

and

is

the

angle

between

the

displacement

and

the

force.

Verify

that

W

=

cot csc

is

an

equivalent

formula. W =

=

( )

=

?

sin

=

Fd

cos

Chapter 5

12

Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

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, 2).

tan x cos x ? cos x = 0

Original equation

cos x (tan x ? 1) = 0

Factor.

cos x = 0 or

tan x ? 1 = 0

Set each factor equal to 0.

x

=

2

or

3 2

tan x = 1

x

=

4

or

5 4

When

x

=

2

or

32,

tan

x

is

undefined,

so

the

solutions

of

the

original

equation

are

4

or

54.

When

you

solve

for

all

values

of

x, the solution should be represented as x + 2n for sin x and cos x and x + n for tan x, where n is any integer. The

solutions

are

4

+

n

or

5 4

+

n.

Example 2: Find all solutions of sin x + = ?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.

x

=

4 3

or

5 3

Solve for x.

The

solutions

are

4 3

+

2n

or

5 3

+

2n.

Exercises Solve each equation for all values of x. 1. cos x = ?1 + 2n

2. sin3 x ? 4 sin x = 0 n

3. sin x cos x ?3 cos x = 0 + n

4. 2 sin3 x = sin x n, + n, + n

Find all solutions of each equation on the interval [0, 2).

5. 2 cos x = 1 ,

6. 5 + 2 sin x ? 7 = 0

7. 4 sin2 x tan x = tan x

8. 2 cos x ? 3 = 0

0, , , , ,

Chapter 5

,

16

Glencoe Precalculus

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

5-3 Study Guide and Intervention (continued)

Solving Trigonometric Equations

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 x ? x + 3 = 1 ? 2 tan x on the interval [0, 2).

2 tan2 x ? sec2 x + 3 = 1 ? 2 tan x

Original equation

2 tan2 x ? (tan2 x + 1) + 3 = 1 ? 2 tan x tan2 x + 2 = 1 ? 2 tan x

sec2 x = tan2 x + 1 Simplify.

tan2 x + 2 tan x + 1 = 0

Simplify.

(tan + 1)2 = 0

Factor.

tan x = ?1

Take the square root of each side.

x

=

3 4

or

7 4

Solve for x on [0, 2).

Example 2: Find all solutions of 1 + cos x = sin x on the interval [0, 2).

1 + cos x = sin x

Original equation

(1 + cos )2 = (sin x)2 1 + 2 cos x + cos2 x = sin2 x 1 + 2 cos x + cos2 x = 1 ? cos2 x

Square each side. Multiply. Pythagorean Identity

2 cos2 x + 2 cos x = 0

Simplify.

2 cos x (cos x + 1) = 0

Factor.

cos x = 0 or cos x = ?1

x

=

2,

,

3 2

Zero Product Property Solve for x on [0, 2).

Exercises

Solve each equation for all values of x.

1. tan2 x = 1 +

3. sin x cos x ?3 cos x = 0 + n

2. 2 sin2 x ? cos x = 1

+ 2n, + 2n, + 2n

4. cos2 x + sin x + 1 = 0 + 2n

Find all solutions of each equation on the interval [0, 2).

5. cos x = sin x ,

6. 3 cos x tan x ? cos x = 0

,

7. tan2 x + sec x ? 1 = 0

0, ,

8. 1 + cos x = 3 sin x

,

Chapter 5

17

Glencoe Precalculus

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