Examples for Homework 9

Half angle formulas:
cos½alpha = +/-sqrt((1+cosalpha)/2)

sin½alpha = +/-sqrt((1+cosalpha)/2)

The angle 315° is the same as (360-45)° or -45°, which means

sin315° = -sqrt(2)/2
cos315° = sqrt(2)/2

Half of 315 is 157.5°, which is in the second quadrant. Sine will be positive and cosine negative.

cos157.5° = -sqrt((1+sqrt(2)/2)/2

With a little algebraic manipulation, this becomes
-sqrt((2+sqrt(2))/4) and to have no radicals in the denominator the final answer is
cos157.5° = -sqrt(2+sqrt(2))/2

sin157.5° = sqrt((1-sqrt(2)/2)/2

With a little algebraic manipulation, this becomes
sqrt((2-sqrt(2))/4) and to have no radicals in the denominator the final answer is
sin157.5° = sqrt(2-sqrt(2))/2

157.5° in radians is (157.5/180)pi = (315/360)pi = (7/8)pi

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A start for cos3alpha

cos(alpha + 2alpha) = cosalpha×cos2alpha - sinalpha×sin2alpha
= cosalpha(cos²alpha - sin²alpha) - sinalpha(2cosalphasinalpha)

Continue the algebraic simplification.