Question

If $\sin \theta =\frac{7}{25}$, find the values of cosθ and tan​θ.

Answer 1

We have,

$$\begin{gathered} {\sin }^2\theta +{\cos }^2\theta =1 \\ \Rightarrow {\left(\frac{7}{25}\right)}^2+{\cos }^2\theta =1 \\ \Rightarrow {\cos }^2\theta =1-\frac{49}{625}=\frac{625-49}{625}=\frac{576}{625} \\ \Rightarrow \cos \theta =\sqrt{\frac{576}{625}}=\frac{24}{25} \end{gathered}$$
Now,

$$\begin{gathered} \tan \theta =\frac{\sin \theta }{\cos \theta } \\ \Rightarrow \tan \theta =\frac{{\displaystyle \frac{7}{25}}}{{\displaystyle \frac{24}{25}}} \\ \Rightarrow \tan \theta =\frac{{\displaystyle 7}}{{\displaystyle 24}} \end{gathered}$$
Thus, the values of cosθ and tanθ are $\frac{24}{25}$ and $\frac{7}{24}$, respectively.