Question

How do you use the half angle formulas to determine the exact values of sine, cosine, and tangent of the angle `67^circ 30'`?

Answer 1

`sin t = sqrt(2 + sqrt2)/2`
`cos t = sqrt(2 - sqrt2)/2`
`tan t = sqrt(2 + sqrt2)/(sqrt(2 - sqrt2))`

Explanation:

Call `67^@30'= t` --> `2t = 135^@`
Use trig table, and trig identity:
`2sin^2 t = 1 - cos 2t`, and `2cos^2 t = 1 + cos 2t`
`2sin^2 t = 1 + sqrt2/2 = (2 + sqrt2)/2`
`sin^2 t = (2 + sqrt2)/4`
`sin t = sin (67.50) = +- sqrt(2 +- sqrt2)/2`
sin (67.50) is positive. Take the positive answer
`sin (67.50) = sqrt(2 + sqrt2)/2`
`2cos^2 t = 1 + cos 2t = 1 - sqrt2/2 = (2 - sqrt2)/2`
`cos t = cos (67.50) = sqrt(2 - sqrt2)/2`
`tan t = sin/(cos) = sqrt(2 + sqrt2)/(sqrt(2 - sqrt2))`