Question

If $\sin \left(\theta \right)=\frac{7}{25}$ then find the values of $\cos \left(\theta \right),\tan \left(\theta \right)$.

Answer 1

Using basic Trigonometric conversions

$$\begin{gathered} \cos \left(\theta \right)=\sqrt{1-{\sin }^2\left(\theta \right)}\left[\because si{n}^2\left(\theta \right)+co{s}^2\left(\theta \right)=1\right] \\ \Rightarrow \cos \left(\theta \right)=\sqrt{1-{\left(\frac{7}{25}\right)}^2} \\ \Rightarrow \cos \left(\theta \right)=\frac{24}{25} \end{gathered}$$

$\begin{array}{rcl}\mathbf{\because }\mathit{t}\mathit{a}\mathit{n}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}\mathbf{}& \mathbf{=}& \mathbf{}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}}\\ & \Rightarrow & \tan \left(\theta \right)=\frac{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$25$}\right.}}{{\displaystyle \raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$25$}\right.}}\\ & \Rightarrow & \tan \left(\theta \right)=\frac{7}{24}\end{array}$

Thus, value of $\cos \left(\theta \right)=\frac{24}{25}$ and $\tan \left(\theta \right)=\frac{7}{24}$.