Question

Use the Half Angle Formulas to find the exact value of the following. You may have need of the Quotient, Reciprocal or Even/Odd Identities as well.

1/ cos(pi/8)

2/ sin(pi/8)

Answer 1

From the double/half angle formula for cosines:
\[\cos\theta = \pm\sqrt{\frac{\cos2\theta + 1}{2}}\]We know that the cosine of anything between 0 and π/2 is positive, so cos(π/8) will be positive and the plus/minus sign can be removed. So substituting in θ = π/8 gives:
\[\cos\frac{\pi}{8} = \sqrt{\frac{\cos\frac{\pi}{4} + 1}{2}}\]Since we know cos(π/4) = 1/√2, we can put that into the expression to get:
\[\cos\frac{\pi}{8} = \sqrt{\frac{\frac{1}{\sqrt{2}} + 1}{2}}=\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}\]

Similarly, the double/half angle formula for sines is:
\[\sin\theta = \pm\sqrt{\frac{1-\sin2\theta}{2}}\]Again, as π/8 is between 0 and π/2, we know that sin(π/8) is positive and the plus/minus sine can be removed. Substituting in θ = π/8:
\[\sin\frac{\pi}{8} = \sqrt{\frac{1-\sin\frac{\pi}{4}}{2}}\]As sin(π/4) = 1/√2:
\[\sin\frac{\pi}{8} = \sqrt{\frac{1-\frac{1}{\sqrt{2}}}{2}} = \sqrt{\frac{\sqrt{2}-1}{2\sqrt{2}}}\]

You can probably simplify the answers further but I'll leave that to you ;)