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10.3 Verify Trigonometric Identities

# 10.3 Verify Trigonometric Identities. Trigonometric Identities Trigonometric Identity: a trigonometric equation that is true for all values of the variable

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10.3 Verify Trigonometric Identities

Trigonometric Identities

• Trigonometric Identity: a trigonometric equation that is true for all values of the variable for which both sides of the equation are defined.

• There are 5 fundamental Trigonometric Identities. (See page 628 in your book.)

1+ tan 2 θ = sec2θ

22

2

cos

1

cos

sin1

2

2

2 cos

sin

cos

11

2

2

2

2

cos

cos

cos

sin1

22 cossin1

22 cossin1

1+ cot2 θ= csc2θ

22 sin

1

tan

11

2

2

2 sin

1

cossin

11

22

2

sin

1

sin

cos1

22

2

2

2

sin

1

sin

cos

sin

sin

1cossin 22

Section 9.3

Page 572

Given that sin q = and < q < π, find the values of the other five trigonometric functions of q .

45

π2

SOLUTION

STEP 1 Find cos q .

Write Pythagorean identity.

Substitute for sin .q

45( ) + cos q 4

52 2 1=

Subtract ( ) from each side.45

2cos q

2 245

1 – ( )=

Simplify.cos q 2 9

25=

Take square roots of each side.cosq 35

+–=

Because q is in Quadrant II, cos q is negative.

cosq 35–=

STEP 2 Find the values of the other four trigonometric functions of q using theknown values of sin q and cos q.

tan q sin qcos q= =

4535–

= 43–

cot q cos qsin q= =

45

35

–= 3

4–

csc q 1sin q= = 1

45

= 54

sec q 1cos q= = 3

5–

1 = 53–

Find the values of the other five trigonometric functions of q.

16

1. cos q , 0 < q < = π2

SOLUTION

STEP 1 Find sin q .

Write Pythagorean identity.sin q + cos q2 2 = 1

Substitute for cos q .16

Subtract ( ) from each side.16

2

sin q +2 = 1( )216

sin2 q = 1 – ( )216

Take square roots of each side.

Because q is in Quadrant I, sin q is positive.

Simplify.sin2 q = 3536

sin q = 356

sin q = 356

STEP 2 tan q sin qcos q= =

35 616

= 35

csc q 1sin q= = 1

35 6

= 6 35

cot q cos qsin q= =

16

35 6

= 135

sec q 1cos q= = 1

6

1 = 6

= 6 3535

= 3535

Find the values of the other five trigonometric functions of q.

2. sin q = , π < q<

3π 2

–3 7

SOLUTION

STEP 1 Find cos q .

Write Pythagorean identity.sin q + cos q2 2 = 1

Substitute for sin q .– 3 7

( )2 + cos2 q = 1 – 3 7

Simplify.cos q 2 40 49=

cos q 2 = 71 – ( )2– 3

Subtract from each side.

– 3 7

Take square roots of each side.

Because q is in Quadrant lII, cos q 18 is negative.

cos q +–= 2 10 7

cosq –= 2 10 7

cot q cos qsin q= =

37

72 10–

–= 2 10

3

csc q 1sin q= = 1

37

–= 7

3–

sec q 1cos q= =

2 10 7

1 = – 72010

10.3 Assignment Day 1

Page 631, 3-9 all

10.3 Verify Trigonometric Identities, day 2

2Simplify the expression csc q cot q + .

1sin q

Reciprocal identity2csc q cot q + 1

sin q csc q cot q + csc q2

=

Pythagorean identity= csc q (csc q – 1) + csc q2

Distributive property= csc q – csc q + csc q3

Simplify.= csc q3

Simplify the expression tan ( – q ) sin q.π2

Cofunction identitytan ( – q ) sin qπ2

cot q sin q=

Cotangent identity= ( ) ( sin q )cos qsin q

Simplify.= cos q

3. sin x cot x sec x

Simplify the expression.

sin xcos xsin x cos x

1· ·

Substitute identity functionsSimplify

Simplify the expression.

4. tan x csc xsec x

sin xcos x

cos x1

· sin x1

Substitute identity functions

Simplify

Simplify the expression.

cos –1 π2

– q

1 + sin (– q )5.

Substitute Cofunction identity;Substitute Negative Angle identity1 – sin ( θ )

sin ( θ ) – 1

– 1(sin ( θ ) – 1)sin ( θ ) – 1 Factor

Simplify

10.3 Assignment Day 2

Page 631, 10-19 all

10.3 Verify Trigonometric Identities, day 3

Verifying Trigonometric Identities

• When verifying an identity, begin with the expression on one side.

• Use algebra and trigonometric properties to manipulate the expression until it is identical to the other side.

Verify the identity = sin q .sec q – 12

sec q 22

Write as separate fractions.sec q – 12

sec q 2 = sec q2

sec q 2 – 1 sec q 2

Simplify.= 1 – ( )1 sec q

2

Reciprocal identity= 1 – cos q2

Pythagorean identity= sin q2

Verify the identity sec x + tan x = .cos x

1 – sin x

Reciprocal identitysec x + tan x 1cos x + tan x=

Tangent identity1

cos x + sin x cos x=

cos x =

Multiply by 1 – sin x 1 – sin x

1 + sin xcos x = 1 – sin x

1 – sin x

Simplify.cos x

1 – sin x=

Simplify numerator.1 – sin xcos x (1 – sin x)=

2

Pythagorean identitycos xcos x (1 – sin x)=

2

Shadow LengthA vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the sun above the horizon is q can be modeled by the equation below. Show that the equation is equivalent to s = h cot q .

sh sin (90° – q )

sin q=

SOLUTION

Simplify the equation.

Write original equation.sh sin (90° – q )

sin q=

Convert 90° to radians.h sin ( – q )

sin q

π2

=

Cofunction identityh cos q sin q=

Cotangent identity= h cot q

Verify the identity.

6. cot (– q ) = – cot q

SOLUTION

Reciprocal identity

Negative angle identity

cot (– q ) = tan (– θ )1

–tan ( θ )1

=

Reciprocal identity= – cot θ

= cot2 x

1sin2 x= cos2 x Reciprocal identity

Tangent identity and cotangent identities

7. csc2 x (1 – sin2 x) = cot2 x

SOLUTION

csc2 x (1 – sin2 x ) = csc2 x cos2x Pythagorean identity

Verify the identity.

Verify the identity.

8. cos x csc x tan x = 1

SOLUTION

cos x csc x tan x = cos x csc x sin xcos x Tangent identity and

cotangent identities

= cos x 1sin x

sin xcos x Reciprocal identity

= 1 Simplify

9. (tan2 x + 1)(cos2 x – 1) = – tan2 x

SOLUTION

(tan2 x + 1)(cos2 x – 1) = – sec2 x (–sin2x) Pythagorean identity

1cos2 x

(–sin2x)= Reciprocal identity

= –tan2 x Tangent identity and cotangent identities

Verify the Identity.

10.3 Assignment, day 3

Page 631, 25-32 all