How do you find the exact values of sin 75 degrees using the half angle formula?

Question

Daniel Acosta · Accepted Answer

<p><img alt="enter image source here" src="https://tutor.hix.ai Find: sin 75  deg<br>
Answer: <mathjax>#sin 75 = +- sqrt(2 + sqrt3)/2# Call sin 75 = sin t --> cos 150 = cos 2t 
On the trig unit circle, 
cos (150) = cos (180 - 30) = - cos 30 =# -(sqrt3)/2#
#cos 150 = (-sqrt3)/2 = 1 - 2sin^2 t#
#2sin^2 t = (2 + sqrt3)/2# #sin (75) = sin t = +- (sqrt(2 + sqrt3))/2#

Caroline Adams · Answer

undefined To find the exact value of sin 75 degrees using the half-angle formula, follow these steps:

1. Start with the half-angle formula for sine: sin(θ/2) = ± √[(1 - cosθ) / 2]
2. Rewrite 75 degrees as the sum of two angles whose sines are known. We can express 75 degrees as 30 degrees + 45 degrees.
3. Calculate the values of sin(30 degrees) and sin(45 degrees). These values are known to be 1/2 and √2/2 respectively.
4. Use these values to find the exact value of sin(75 degrees) using the half-angle formula.sin(75°) = sin(45° + 30°) = sin[(45° + 30°)/2]

= sin(75°/2)

= ±√[(1 - cos(75°)) / 2]

Now, we need to find the value of cos(75°).

cos(75°) = cos(45° + 30°) = cos(45°)cos(30°) - sin(45°)sin(30°)

= (√2/2)(√3/2) - (√2/2)(1/2)

= (√6 - √2) / 4

Substitute cos(75°) into the half-angle formula:

sin(75°) = ±√[(1 - (√6 - √2)/4) / 2]

= ±√[(4 - (√6 - √2)) / 8]

= ±√[(4 + √6 - √2) / 8]

So, the exact value of sin(75°) using the half-angle formula is ±√[(4 + √6 - √2) / 8].