Question

If $\tan \left(\mathrm{A}\right)=\sqrt{3}$ find values of all other trigonometric ratios.

Answer 1

Step 1: Solve for the value of $\cot \left(A\right)$

Given, $\tan \left(\mathrm{A}\right)=\sqrt{3}$

Since $\tan \left(\mathrm{A}\right)$ is positive in first or third quadrant so only $\cot \left(\mathrm{A}\right)$ will be positive rest all trigonometric ratios can have positive as well as negative values.

$\begin{array}{ccc}& \tan \left(\mathrm{A}\right)=\sqrt{3}& \\ \Rightarrow & \cot \left(\mathrm{A}\right)=\frac{1}{\tan \left(\mathrm{A}\right)}& \\ \Rightarrow & \cot \left(\mathrm{A}\right)=\frac{1}{\sqrt{3}}& \end{array}$

Step 2: Solve for the value of $\sec \left(A\right)$

$\begin{array}{ccc}\Rightarrow & \sec \left(\mathrm{A}\right)=\sqrt{1+{\tan }^2\left(\mathrm{A}\right)}& \left[\because {\sec }^2\left(\mathrm{A}\right)-{\tan }^2\left(\mathrm{A}\right)=1\right]\\ \Rightarrow & \sec \left(\mathrm{A}\right)=\pm 2& \end{array}$

Step 3: Solve for the value of $\cos \left(A\right)$

$\begin{array}{ccc}\Rightarrow & \cos \left(\mathrm{A}\right)=\frac{1}{\sec \left(\mathrm{A}\right)}& \\ \Rightarrow & \cos \left(\mathrm{A}\right)=\pm \frac{1}{2}& \end{array}$

Step 4: Solve for the value of $\sin \left(A\right)$

$\begin{array}{ccc}\Rightarrow & \sin \left(\mathrm{A}\right)=\sqrt{1-{\cos }^2\left(\mathrm{A}\right)}& \left[\because {\sin }^2\left(\mathrm{A}\right)+{\cos }^2\left(\mathrm{A}\right)=1\right]\\ \Rightarrow & \sin \left(\mathrm{A}\right)=\pm \frac{\sqrt{3}}{2}& \end{array}$

Step 5: Solve for the value of $\csc \left(A\right)$

$\begin{array}{ccc}\Rightarrow & cosec\left(\mathrm{A}\right)=\frac{1}{\sin \left(\mathrm{A}\right)}& \\ \Rightarrow & cosec\left(\mathrm{A}\right)=\pm \frac{2}{\sqrt{3}}& \end{array}$

Hence the values of all the trigonometric ratios are:

$$\begin{gathered} \sin \left(\mathrm{A}\right)=\pm \frac{\sqrt{3}}{2} \\ \cos \left(\mathrm{A}\right)=\pm \frac{1}{2} \\ \tan \left(\mathrm{A}\right)=\sqrt{3} \\ \csc \left(\mathrm{A}\right)=\pm \frac{2}{\sqrt{3}} \\ \sec \left(\mathrm{A}\right)=\pm 2 \\ \cot \left(\mathrm{A}\right)=\frac{1}{\sqrt{3}} \end{gathered}$$