A right-angled triangle has

MATHEMATICS · Multi-part · GCSE · Mathematics · Geometry and Measures · Geometry and Measures

Problem

A right-angled triangle

DEF

has

\angle E = 90^{\circ}

. The lengths of the sides are

DE = 2p

and

EF = (p + 3)

, where

p > 0

and

p \neq 3

. (i) Express the length of the hypotenuse

DF

in simplest form in terms of

p

. (ii) For the acute angle

\theta = \angle DFE

, obtain expressions for

\sin \theta

\cos \theta

and

\tan \theta

in terms of

p

. (iii) Hence, write algebraic expressions for

\csc \theta

\sec \theta

and

\cot \theta

. (iv) Prove that

1 + \cot^2 \theta = \csc^2 \theta

for this triangle by substituting your expressions from parts (ii) and (iii).