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How do you use the half angle formula to find sin75?

Answer
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Hint: Half-angle formulas allow the expression of trigonometric functions of angles equal to α2 in terms of α, which can simplify the functions and make it easier to perform more complex calculations, such as integration, on them. Half-angle formulas are especially useful in finding the values of unknown trigonometric functions.

Formula used: To find the value of sinθ by using half angle we will use the formula of sinθ=1cos2θ2.

Complete step-by-step solution:
We have to find the value of sin75 by using the half angle formula.
We know that,
sinθ=1cos2θ2 is the half angle formula of sinθ.
Substitute θ=75 we get,
sin75=1cos2×752
On multiply the terms and we get,
sin75=1cos1502
On splitting we get,
sin75=1cos(18030)2
Simplifying again we get,
sin75=1+cos302
On putting the values and again we get,
sin75=1+322
On simplifying again we get,
sin75=12+34

Hence, the value of sin75=12+34

Note: Cutting special angles on the unit circle in half gives you a variety of new angles that can’t be achieved by using the sum and difference formulas or the double-angle formulas. Although the half-angle formulas won’t give you all the angles of the unit circle, they certainly get you closer than you were before.
We also can find the values of trig functions for angles like pi/12, each of which are exactly half of angles on the unit circle. These angles aren’t the only types that the identities work for. we can continue to halve any angle on the unit circle and take the trig functions of those angles. For example, 15 degrees is half of 30 degrees, and 7.5 degrees is half of 15 degrees.
In the half-angle formula for sine and cosine, notice that a plus/minus sign appears in front of each radical (square root). Whether the answer is positive or negative depends on which quadrant the new angle (the half angle) is in. The half-angle formula for tangent doesn’t have a plus/minus sign in front, so the above doesn’t apply to tangent.
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