How do you use the sum or difference identities to find the exact value of cos105^circcos105∘?

1 Answer
Nghi N
Dec 25, 2016

- (sqrt(2 - sqrt3))/2−√2−√32

Explanation:

Use trig identity:
2cos^2 a = 1 + cos 2a2cos2a=1+cos2a
2cos^2 (105) = 1 + cos (210)2cos2(105)=1+cos(210). Find cos (210)
Use trig table and unit circle
cos (210) = cos (30 + 180) = - cos 30 = - sqrt3/2cos(210)=cos(30+180)=−cos30=−√32
2cos^2 (105) = 1 - sqrt3/2 = (2 - sqrt3)/22cos2(105)=1−√32=2−√32
cos^2 (105) = (2 - sqrt3)/4cos2(105)=2−√34
cos (105) = +- sqrt(2 - sqrt3)/2cos(105)=±√2−√32
Since cos 105 (Quadrant II) is negative, take the negative value as answer.

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