If tan θ=247, find that sin θ + cos θ.

Solution:

Given:

$\tan \theta=\frac{24}{7}$....(1)

To find:

$\sin \theta+\cos \theta$

Now we know $\tan \theta$ is defined as follows

$\tan \theta=\frac{\text { Perpendicular side opposite to } \angle \theta}{\text { Base side adjacent to } \angle \theta}$....(2)

Now by comparing equation (1) and (2)

We get

Perpendicular side opposite to $\angle \theta=24$

Base side adjacent to $\angle \theta=7$

Therefore triangle representing angle $\theta$ is as shown below

Side AC is unknown and can be found using Pythagoras theorem

Therefore,

$A C^{2}=A B^{2}+B C^{2}$

Now by substituting the value of known sides from figure

We get,

$A C^{2}=24^{2}+7^{2}$

$=576+49$

$=625$

Now by taking square root on both sides

We get,

$A C=\sqrt{625}$

$=25$

Therefore Hypotenuse side AC = 25 …… (3)

Now we know, $\sin \theta$ is defined as follows

$\sin \theta=\frac{\text { Perpendicular side opposite to } \angle \theta}{\text { Hypotenuse }}$

Therefore from figure (a) and equation (3)

We get,

$\sin \theta=\frac{A B}{A C}$

$=\frac{24}{25}$

$\sin \theta=\frac{24}{25}$....(4)

Now we know, $\cos \theta$ is defined as follows

$\cos \theta=\frac{\text { Base side adjacent to } \angle \theta}{\text { Hypotenuse }}$

Therefore from figure (a) and equation (3)

We get,

$\cos \theta=\frac{B C}{A C}$

$=\frac{7}{25}$

$\cos \theta=\frac{7}{25}$.....(5)

Now we need to find the value of expression $\sin \theta+\cos \theta$

Therefore by substituting the value of $\sin \theta$ and $\cos \theta$ from equation (4) and (5) respectively, we get,

$\sin \theta+\cos \theta=\frac{24}{25}+\frac{7}{25}$

$=\frac{24+7}{25}$

$=\frac{31}{25}$

Hence $\sin \theta+\cos \theta=\frac{31}{25}$