If $\sin A = \frac{3}{4}$, calculate $\cos A$ and $ an A$. Show all steps.

Question

If $\sin A = \frac{3}{4}$, calculate $\cos A$ and $	an A$. Show all steps.

Accepted Answer

Given: $\sin A = \frac{3}{4}$ 
We know that in a right-angled triangle, $\sin A = \frac{	ext{Opposite side}}{	ext{Hypotenuse}} = \frac{	ext{Perpendicular}}{	ext{Hypotenuse}}$. 
Let the perpendicular be $3k$ and the hypotenuse be $4k$, where $k$ is a positive real number.
Using Pythagoras theorem in right-angled triangle ABC: $(	ext{Hypotenuse})^2 = (	ext{Base})^2 + (	ext{Perpendicular})^2$ $(4k)^2 = (	ext{Base})^2 + (3k)^2$ $16k^2 = (	ext{Base})^2 + 9k^2$ $(	ext{Base})^2 = 16k^2 - 9k^2$ $(	ext{Base})^2 = 7k^2$ $	ext{Base} = \sqrt{7}k$ 
Now, calculating $\cos A$: $\cos A = \frac{	ext{Base}}{	ext{Hypotenuse}} = \frac{\sqrt{7}k}{4k} = \frac{\sqrt{7}}{4}$ 
Now, calculating $	an A$: $	an A = \frac{	ext{Perpendicular}}{	ext{Base}} = \frac{3k}{\sqrt{7}k} = \frac{3}{\sqrt{7}}$ 
Therefore, $\cos A = \frac{\sqrt{7}}{4}$ and $	an A = \frac{3}{\sqrt{7}}$.

Introduction to Trigonometry

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