If $sin\ A = \frac{3}{4}$, calculate $cos\ A$ and $tan\ A$.

Given:

$sin\ A = \frac{3}{4}$.

To do:

We have to find $cos\ A$ and $tan\ A$.

Solution:  

Let, in a triangle $ABC$ right-angled at $B$, $sin\ A = \frac{3}{4}$.

We know that,

In a right-angled triangle $ABC$ with a right angle at $B$,

By Pythagoras theorem,

$AC^2=AB^2+BC^2$

By trigonometric ratios definitions,

$sin\ A=\frac{Opposite}{Hypotenuse}=\frac{BC}{AC}$

$cos\ A=\frac{Adjacent}{Hypotenuse}=\frac{AB}{AC}$

$tan\ A=\frac{Opposite}{Adjacent}=\frac{BC}{AB}$

Here,

$AC^2=AB^2+BC^2$

$\Rightarrow (4)^2=(AB)^2+(3)^2$

$\Rightarrow 16=(AB)^2+9$

$\Rightarrow AB=\sqrt{16-9}=\sqrt7$

Therefore,

$cos\ A=\frac{AB}{AC}$

$=\frac{\sqrt7}{4}$

$tan\ A=\frac{BC}{AB}$

$=\frac{3}{\sqrt7}$  

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Updated on: 10-Oct-2022

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