Question

If $\sin \left(\mathrm{A}\right)=\frac{3}{4}$, calculate $\cos \left(\mathrm{A}\right)$ and $\tan \left(\mathrm{A}\right)$.

Answer 1

Step 1: Find the value of $\cos \left(\mathrm{A}\right)$

Given: $\sin \left(\mathrm{A}\right)=\frac{3}{4}$

Since, $\mathrm{sine}$ is the ratio of perpendicular and hypotenuse.

So, $\sin \left(\mathrm{A}\right)=\frac{\mathrm{BC}}{\mathrm{AB}}$

$\Rightarrow$$\mathrm{BC}=3x$ and $\mathrm{AB}=4x$ where $x$ is arbitary constant

Using Pythagoras Theorem in triangle $\mathrm{ABC}$, sum of the square of hypotenuse is equal to the sum of square of other sides.

$\Rightarrow$${\mathrm{AB}}^2={\mathrm{BC}}^2+{\mathrm{AC}}^2$

$\Rightarrow$ ${\left(4x\right)}^2={\left(3x\right)}^2+{\mathrm{AC}}^2$

$\Rightarrow$ $16x^2=9x^2+{\mathrm{AC}}^2$

$\Rightarrow$ ${\mathrm{AC}}^2=7x^2$

$\Rightarrow$ $\mathrm{AC}=x\sqrt{7}$

Now, $\cos \left(\mathrm{A}\right)=\frac{\mathrm{AC}}{\mathrm{AB}}$

$\Rightarrow$ $\cos \left(\mathrm{A}\right)=\frac{x\sqrt{7}}{4x}$

$=\frac{\sqrt{7}}{4}$

Step 2: Find the value of $\tan \left(\mathrm{A}\right)$

As $\mathrm{tangent}$ is the ratio of $\mathrm{sine}$ and $\mathrm{cosine}$.

So, $\tan \left(\mathrm{A}\right)=\frac{\sin \left(\mathrm{A}\right)}{\cos \left(\mathrm{A}\right)}$

$\Rightarrow$ $\tan \left(\mathrm{A}\right)=\frac{{\displaystyle \frac{3}{4}}}{{\displaystyle \frac{\sqrt{7}}{4}}}$

$\Rightarrow$ $\tan \left(\mathrm{A}\right)=\frac{{\displaystyle 3}}{{\displaystyle \sqrt{7}}}$

Hence, the value of $\cos \left(\mathrm{A}\right)=\frac{\sqrt{7}}{4}$ and $\tan \left(\mathrm{A}\right)=\frac{{\displaystyle 3}}{{\displaystyle \sqrt{7}}}$