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Trigonometry Problems

Trigonometry is a branch of mathematics which studies about triangles and relationships between their sides and angles. It is all about right triangle and trigonometric functions (sine, cosine, tangent, cosecent, secent and cotangent). Let us consider a right triangle as shown below:

Then, we can define trigonometric ratios as follows:

In trigonometry, we learn different important problems which are utilized not only in pure mathematics, but also in real life circumstances.

Trigonometry is the foundation for many other higher and college level courses.

Here, we are going to discuss few basic problems related to trigonometry.

Problem 1:

Given that $\sin \theta$ = $\frac{3}{5}$, find all the remaining five trigonometric ratios, where $\theta$

Solution:

$\sin \theta$ = $\frac{3}{5}$

Using the identity $\sin ^{2}\theta +\cos ^{2}\theta =1$

$\cos ^{2}\theta=1-\sin ^{2}\theta$

$\cos ^{2}\theta$ =1 - $(\frac{3}{5})^{2}$

$\cos ^{2}\theta$ = 1 - $\frac{9}{25}$

$\cos ^{2}\theta$ = $\frac{16}{25}$

$\cos \theta$ = $\frac{4}{5}$

$\csc \theta$ = $\frac{1}{\sin \theta }$

$\csc \theta$ = $\frac{1}{\frac{3}{5}}$

$\csc \theta$ = $\frac{5}{3}$

$\sec \theta$ = $\frac{1}{\cos \theta }$

$\sec \theta$ = $\frac{1}{\frac{4}{5}}$

$\sec \theta$ = $\frac{5}{4}$

$\tan \theta$ = $\frac{\sin \theta }{\cos \theta }$

$\tan \theta$ = $\frac{\frac{3}{5}}{\frac{4}{5}}$

$\tan \theta$ = $\frac{3}{4}$

Problem 2:

Find the value of $\sin ^{-1}$$(\frac{\sqrt{3}}{2}). Solution: \sin ^{-1}$$(\frac{\sqrt{3}}{2})$

= $\sin ^{-1}\sin $$(\frac{\pi }{3}) We know that \sin ^{-1}\sin \theta =\theta Therefore, \sin ^{-1}\sin$$(\frac{\pi }{3})$ = $\frac{\pi }{3}$

Problem 3:

Simplify $(\sin A+\cos A)^{2}$.

Solution:

Using the identity $(a+b)^{2}=a^{2}+b^{2}+2ab$, we get
$(\sin A+\cos A)^{2}$
= $\sin ^{2}A+\cos ^{2}A+2\sin A \cos A$
We have $\sin ^{2}\theta +\cos ^{2}\theta =1$
and $2\sin \theta \cos \theta =\sin 2\theta$
Using them, we obtain
$(\sin A+\cos A)^{2}=1+\sin 2A$

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