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Find the exact value of sin75°cos15° - cos75°sin15°

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Answer:

√3/2

Explanation:

Easy Method

The equation above is in the forms of sin(a)cos(b) - cos(a)sin(b), which is sin(a-b) according to the trig identities. sin(75-15) = sin(60) = √3/2

Harder Method

Find sin75 with equation sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

sin75°

= sin(45° + 30°)

= [sin45°cos30° + cos45°sin30°]

= [√2/2 * √3/2 + √2/2 * 1/2] <-- unit circle known values

= [(√6 + √2)/4]

Find cos75 with the equation: cos(a+b) = sin(a)sin(b) - cos(a)cos(b)

cos75°

= cos(45° + 30°)

= [cos45°cos30° - sin45°sin30°]

= [√2/2 * √3/2 - √2/2 * 1/2] <-- unit circle known values

= [(√6 - √2)/4]

Find sin15 with the equation: sin(a-b) = sin(a)cos(b) - cos(a)sin(b)

sin15°

= sin(45° - 30°)

= [sin45°cos30° - cos45°sin30°]

= [√2/2 * √3/2 - √2/2 * 1/2] <-- unit circle known values

= [(√6 - √2)/4]

Find cos15 with the equation: cos(a-b) = sin(a)sin(b) + cos(a)cos(b)

cos15°

= cos(45° - 30°)

= [cos45°cos30° + sin45°sin30°]

= [√2/2 * √3/2 + √2/2 * 1/2] <-- unit circle known values

= [(√6 + √2)/4]

Now plug in all the solved values, we get: {[(√6 + √2)/4] * [(√6 + √2)/4]} - {[(√6 - √2)/4] * [(√6 - √2)/4]} = √3/2

Find the exact value of sin75°cos15° - cos75°sin15°-example-1
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