Question

How do you use a double angle formula to find the exact value of $\cos 2u$ when $\sin u = \dfrac{7}{{25}}$, where $ - \dfrac{\pi }{2} \leqslant u \leqslant \dfrac{\pi }{2}$ ?

Answer 1

Hint: The given problem can be solved by using the double angle formula of cosine. Use of the double angle formula of cosine helps us to convert $\cos \left( {2u} \right)$ to an expression consisting of $\sin \left( u \right)$ . There are multiple double formulae for cosine in terms of sine, cosine and tangent. But we have to take into consideration the one related to sine.

Complete step by step answer:
The given problem requires us to find the exact value of $\cos 2u$ using the double angle formula of cosine. The double angle formula for tangent is: \[\cos \left( {2x} \right) = 1 - 2{\sin ^2}x\].
Now, we are given the value of \[\sin u\] as \[\dfrac{7}{{25}}\]. So, we can easily find out the value of $\cos 2u$ by substituting the value of \[\sin u\]in the double angle formula for cosine.
So, we have, \[\cos \left( {2u} \right) = 1 - 2{\sin ^2}u\]
\[ \Rightarrow \cos \left( {2u} \right) = 1 - 2{\left( {\dfrac{7}{{25}}} \right)^2}\]
Evaluating the square term so as to simplify the expression, we get,
\[ \Rightarrow \cos \left( {2u} \right) = 1 - 2\left( {\dfrac{{49}}{{625}}} \right)\]
\[ \Rightarrow \cos \left( {2u} \right) = 1 - \dfrac{{98}}{{625}}\]
Simplifying the calculations further, we get,
\[ \Rightarrow \cos \left( {2u} \right) = \dfrac{{625 - 98}}{{625}}\]
\[ \Rightarrow \cos \left( {2u} \right) = \dfrac{{527}}{{625}}\]
So, we get the value of $\cos 2u$ as \[\dfrac{{527}}{{625}}\] when we are given that value of $\sin u$ is $\dfrac{7}{{25}}$ where u lies in the interval $ - \dfrac{\pi }{2} \leqslant u \leqslant \dfrac{\pi }{2}$.

Note: The above question can also be solved by using compound angle formulae instead of double angle formulae such as $\cos (A + B) = \cos A\cos B - \sin A\sin B$ . This method can also be used to get to the correct answer of the given problem but we would have to first evaluate the values of sine and cosine so as to apply this method and get to the final answer.