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Question

Find the value of sin15using sin30.


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Solution

Step 1: Write the first trigonometric function in the form of the second trigonometric function :

Given trigonometric functions: sin15 and sin306282a90fcb5fe67a7b1b9ada 6282a90fcb5fe67a7b1b9ada

We know that for all values of the angle A,

sinA2+cosA22=sin2A2+cos2A2+2sinA2cosA2=1+sin(2×A2)[sin2θ+cos2θ=1andsin2θ=2sinθcosθ]=1+sinA

Therefore, sinA2+cosA2=±1+sinA2....(i)

Now, let A=30° then, A2=30°2=15° and from the above equation we get,
sin15+cos15=±1+sin30

Similarly, for all values of the angle A,

sinA2-cosA22=sin2A2+cos2A2-2sinA2cosA2=1-sin(2×A2)[sin2θ+cos2θ=1andsin2θ=2sinθcosθ]=1-sinA

Therefore, sinA2-cosA2=±1+sinA2...(ii)

Now, let A=30° then, A2=30°2=15° and from the above equation, we get
sin15-cos15=±1+sin30

Step 2: Find the exact values.

Clearly, sin15°>0and cos15˚>0

Therefore, sin15°+cos15°>0

Therefore, from equation (i) we get,
sin15+cos15=+1+sin30sin15+cos15=1+12...(iii)

Again,

sin15-cos15=212sin15-12cos15=2cos45°sin15-sin45°cos15=2sin(1545)=2sin-30=-2sin30°=-2×12=-12

Thus, sin15°-cos15°<0

Therefore, from equation (ii) we get,

sin15-cos15=-1+sin30sin15-cos15=-1+12...(iv)

Step 3: Find the value of the required trigonometric function.

Now, adding equation (iii) and equation (iv) we get,

2sin15=1+121122sin15=32122sin15=3-12sin15=3-122

Hence, the value of sin15is3122.


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