find the value of cospi/8 ,sinpi/8 ,tanpi/8 ,sinpi/24 ,cospi/24 ,tan(142x1/2)degree

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Solution

$(1) W e k n o w t h a t, cosθ = 2 \cos^{2} (\frac{θ}{2}) - 1 S o, \cos (\frac{π}{4}) = 2 \cos^{2} (\frac{π}{8}) - 1 \Rightarrow 2 \cos^{2} (\frac{π}{8}) = \cos (\frac{π}{4}) + 1 \Rightarrow 2 \cos^{2} (\frac{π}{8}) = \frac{1}{\sqrt{2}} + 1 \Rightarrow \cos^{2} (\frac{π}{8}) = \frac{1 + \sqrt{2}}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \Rightarrow \cos^{2} (\frac{π}{8}) = \frac{\sqrt{2} + 2}{4} \Rightarrow \cos (\frac{π}{8}) = \sqrt{\frac{\sqrt{2} + 2}{4}} \Rightarrow c o s (\frac{π}{8}) = \frac{1}{2} \sqrt{\sqrt{2} + 2}$
$(2) W e k n o w t h a t, \sin θ = 2 \sin (\frac{θ}{2}) \cos (\frac{θ}{2}) S o, \sin (\frac{π}{4}) = 2 \sin (\frac{π}{8}) \cos (\frac{π}{8}) \Rightarrow \frac{1}{\sqrt{2}} = 2 \sin (\frac{π}{8}) \times \frac{1}{2} \sqrt{\sqrt{2} + 2} \Rightarrow \frac{1}{\sqrt{2}} = \sin (\frac{π}{8}) \times \sqrt{\sqrt{2} + 2} \Rightarrow \sin (\frac{π}{8}) = \frac{1}{\sqrt{2} \sqrt{\sqrt{2} + 2}} \times \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2 - \sqrt{2}}} \Rightarrow \sin (\frac{π}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2} \sqrt{4 - 2}} \Rightarrow s i n (\frac{π}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{2}$

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