Question

Find the function value of cos^2 (7pi/8)

Answer 1

Mhm, absolutely. Notice that we do not have a common unit circle value for any division of 8. But we do have a value for is 7pi/4. Now in order to actually get the value of 7pi/8, we use a half angle formula. So what we do is we will write out the formula and plug 7pi/4 into that formula, this will give us the value of 7pi/8. So the formula we need to plug 7pi/4 into is this: ±1+cos2θ2−−−−−−−−−√SO if you plug in 7pi/4 for theta in that formula, you think you could simplify it to find 7pi/8?

Answer 2

Cos²(7pi/8)

cos²(7π/8)

But π radians = 180°

7π/8 = 7*180/8 = 157.5°

cos²(7π/8) = cos²(157.5°) = (cos(157.5))²

cos 157.5 = - cos(180 - 157.5)            Second quadrant angle

  cos 157.5     =  -cos22.5

  cos²157.5     =  cos²22.5.....................(a)


Note:  cos2θ= 2cos²θ - 1              Trigonometric Identity

cos45 = 2cos²(45/2) - 1

cos45 = 2cos²22.5 - 1            cos 45 = √2/2

√2/2 =  2cos²22.5 - 1

 √2/2 + 1 = 2cos²22.5

 2cos²22.5 = √2/2 + 1

  cos²22.5 = (√2/2 + 1)/2           

Recall equation (a)

cos²157.5     =  cos²22.5.....................(a)

cos²157.5     =  cos²22.5 = (√2/2 + 1)/2

Therefore cos²(7pi/8) = cos²157.5     =  cos²22.5 = (√2/2 + 1)/2
 
Hope this explains it.