Question

If $△ABC$ is right angled at $B$ and $BC=7cm,AC-AB=1$ cm, then find the value of $\cos A+\sin A$.

Answer 1

$\Delta ABC$ is right angled at B, so $AC$ is the hypotenuse.

$AC-AB=1$ ...Given ....(1)

By Pythagoras theorem,

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

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

$7^2=(AC-AB)(AC+AB)$

$49=1(AC+AB)$

So, $AC+AB=49$ ....(2)

Adding (1) and (2) we get,

$\Rightarrow 2AC=50$

$\Rightarrow AC=25$ cm

Sbstitute in eq (2), we get

$AB=24$ cm.

Now, $\cos A+\sin A=\frac{AB}{AC}+\frac{BC}{AC}$$=\frac{24}{25}+\frac{7}{25}=\frac{31}{25}$

$\Rightarrow \cos A+\sin A$$=\frac{31}{25}$