Question
If $24\cos\theta = 7 \sin\theta ,$ find $\sin\theta + \cos\theta .$
✓
Answer
$24 \cos \theta=7 \sin \theta $
$\Rightarrow \frac{\sin \theta}{\cos \theta}=\frac{24}{7} $
$\Rightarrow \tan \theta=\frac{24}{7}=\frac{\text { Perpendicular }}{\text { Base }}$
Hypotenuse
$=\sqrt{(\text { Perpendicular })^2+(\text { Base })^2} $
$=\sqrt{(24)^2+(7)^2} $
$=\sqrt{576+49} $
$=\sqrt{625} $
$=25 $
$\sin \theta+\cos \theta $
$=\frac{\text { Perpendicular }}{\text { Hypotenuse }}+\frac{\text { Base }}{\text { Hypotenuse }} $
$=\frac{24}{25}+\frac{7}{25} $
$=\frac{24+7}{25} $
$=\frac{31}{25} .$
$\Rightarrow \frac{\sin \theta}{\cos \theta}=\frac{24}{7} $
$\Rightarrow \tan \theta=\frac{24}{7}=\frac{\text { Perpendicular }}{\text { Base }}$
Hypotenuse
$=\sqrt{(\text { Perpendicular })^2+(\text { Base })^2} $
$=\sqrt{(24)^2+(7)^2} $
$=\sqrt{576+49} $
$=\sqrt{625} $
$=25 $
$\sin \theta+\cos \theta $
$=\frac{\text { Perpendicular }}{\text { Hypotenuse }}+\frac{\text { Base }}{\text { Hypotenuse }} $
$=\frac{24}{25}+\frac{7}{25} $
$=\frac{24+7}{25} $
$=\frac{31}{25} .$
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