Question

Express cos 20 theta in terms of sin 5 theta

Answer 1

you can use the formula cos twice thita is equal to 1-sin^2 thita

Answer 2

cos 20 $\begin{equation}\theta\end$  in terms of sin 5 $\begin{equation}\theta\end$ is expressed.

Given

To express cos 20 $\begin{equation}\theta\end$  in terms of sin 5 $\begin{equation}\theta\end$.

                             cos 20 $\begin{equation}\theta\end$ = cos (2 × 10) $\begin{equation}\theta\end$

                                            = cos 2 (10 $\begin{equation}\theta\end$)

                                            = 2 cos² 10 $\begin{equation}\theta\end$ -1        ( cos 2A = 2cos²A - 1)

                    Taking 2 from the above result, gives

                                            = 2 [cos 2(5$\begin{equation}\theta\end$)]² - 1

                                            = 2 [1 - 2sin²(5$\begin{equation}\theta\end$)]² - 1

                     [1 - 2sin²(5$\begin{equation}\theta\end$)]² is in (a-b)² form so, (a-b)² = a²+b²-2ab

                                            = 2 [1+4$\begin{equation}\sin ^{4} \end$ 5$\begin{equation}\theta\end$ - 4sin² 5$\begin{equation}\theta\end$] - 1

                                            = 2 + 8$sin^4$ 5$\begin{equation}\theta\end$ - 8sin² 5$\begin{equation}\theta\end$ -1

                                            = 8$sin^4$ 5$\begin{equation}\theta\end$ - 8$sin^2$ 5$\begin{equation}\theta\end$ + 1

Therefore, cos 20 $\begin{equation}\theta\end$  in terms of sin 5 $\begin{equation}\theta\end$ is expressed.

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