Two straight roads and rise from the same point up a hill

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Trigonometry · Introduction to Trigonometry

Problem

Two straight roads

A

and

B

rise from the same point

P

up a hill. Road

A

makes an angle

\theta

with the horizontal, while road

B

makes an angle

(90^{\circ}-\theta)

with the horizontal, where

0<\theta<90^{\circ}

. A car travels a horizontal distance

x

metres up each road from

P

, reaching points

Q

and

R

respectively. (i) Express the vertical heights

PQ

and

PR

in terms of

x

and

\theta

. (ii) Obtain a simplified expression for

\dfrac{PQ+PR}{x}

in terms of

\theta

. (iii) Calculate the value of

\dfrac{PQ+PR}{x}

when

\theta=30^{\circ}

. (Use

\sqrt{3}=1.732

if required.)