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Trigonometry Week 12 Test Grade Report

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Trigonometry

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Trigonometry Week 12 Test - Grade Report

Score:

92% (92 of 100 pts)

Submitted:

Jul 24 at 10:03am

Question: 1

Grade: 1.0 / 4.0

Given that tanθ=−tanθ=−7676 and π2<θ<ππ2<θ<π, find the exact values of the

trigonometric functions.

sin2θ=sin2θ= -84/85 (17%)

csc2θ=csc2θ= -85/84 (17%)

cos2θ=cos2θ= -13/85 (17%)

sec2θ=sec2θ= -85/13 (17%)

tan2θ=tan2θ= 84/13 (17%)

cot2θ=cot2θ= 13/84 (17%)

Solution

Draw the angle in quadrant 2 since π2<θ<ππ2<θ<π.

Use the Pythagorean Theorem to find the unknown side of the triangle.

In this case, the hypotenuse has length 85−−√85.

Substitute the lengths of the three sides of the right triangle into the double-angle formulas

to find each exact value.

sin2θsin2θ

=2sinθcosθ=2sinθcosθ

=2⋅785−−√⋅−685−−√=2⋅785⋅−685

=−8485=−8485

cos2θcos2θ

=cos2θ−sin2θ=cos2θ−sin2θ

=(−685−−√)2−(785−−√)2=−6852−7852

=3685−4985=3685−4985

=−1385=−1385

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tan2θtan2θ

=2tanθ1−tan2θ=2tanθ1−tan2θ

=2(−76)1−(−76)2=2−761−−762

=−1461−4936=−1461−4936

=−146−1336=−146−1336

=8413=8413

Use the reciprocals of sin2θsin2θ, cos2θcos2θ and tan2θtan2θ to find the vlues

of csc2θcsc2θ, sec2θsec2θ, and cot2θcot2θ.

Therefore, csc2θ=−8584csc2θ=−8584, sec2θ=−8513sec2θ=−8513,

and cot2θ=1384cot2θ=1384.

Question: 2

Grade: 1.0 / 4.0

What is the value of cos(−22.5°)cos−22.5°?

2+2–√−−−−−−√22+22 (100%)

Solution

cos(−22.5°)=cos(−45°2)cos−22.5°=cos−45°2

Write −22.5 as a

fraction with a

denominator 2.

=1+cos(−45°)2−−−−−−−−−−−√=1+cos−45°2

Half-Angle

Formula

=1+2–√22−−−−−−− ⎷ =1+222

Evaluate

cos(−45°).

=2+2–√22−−−−−−− ⎷ =2+222

Common

denominator.

=2+2–√4−−−−−−√=2+24

Simplify.

=2+2–√−−−−−−√2=2+22

Simplify the

denominator.

Therefore, cos−22.5∘=2+2–√−−−−−√2.cos−22.5∘=2+22.

Question: 3

Grade: 1.0 / 4.0

Use the product-to-sum formula to write cos6xsin9xcos6xsin9x as a sum or difference.

12sin15x+12sin 3x12sin15x+12sin 3x (100%)

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Trigonometry Week 12 Test - Grade Report Score: 92% (92 of 100 pts) Submitted: Jul 24 at 10:03am Question: 1 Grade: 1.0 / 4.0 Given that tanθ=−tanθ=−7676 and π2 ...
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