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Question

The value of sin16°+cos16° is


A

132cos1°+sin1°

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B

12cos1°+3sin1°

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C

13cos1°+2sin1°

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D

123cos1°+sin1°

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Solution

The correct option is D

123cos1°+sin1°


Explanation for the correct option:

Find the value of the expression:

sinA+B=sinAcosB+cosBsinAcosA+B=cosAcosB-sinAsinB

Use the above formula in the sin16°+cos16°

sin16°+cos16°=sin15°+1°+cos15°+1°=sin15°cos1°+cos15°sin1°+cos15°cos1°-sin15°sin1°=sin1°cos15°-sin15°+cos1°sin15°+cos15°...1

Now,

cos15°=cos45°-30°=cos45°cos30°+sin45°sin30°[cos(A-B)=cosAcosB-sinAsinB]=12·32+12·12[sin45°=cos45°=12,sin30°=12,cos30°=32]=3+122

And

sin15°=sin45°-30°=sin45°cos30°-cos45°sin30°[sinA-B=sinAcosB-cosAsinB]=12·32-12·12[sin45°=cos45°=12,sin30°=12,cos30°=32]=3-122

Substitute the value of cos15° and sin15° in the equation 1, we get

sin16°+cos16°=sin1°3+122-3-122+cos1°3-122+3+122=sin1°3+1-3+122+cos1°3-1+3+122=sin1°222+cos1°2322=12sin1°+32cos1°=12sin1°+3cos1°

Hence, the correct option is D.


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