In right-angled triangle with

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

Problem

In right-angled triangle

XYZ

with

\angle Y = 90^{\circ}

, the lengths of the two sides adjacent to the right angle are

XY = 3k + 2\,\text{cm}

and

YZ = 4k - 1\,\text{cm}

, where

k

is a real number greater than

1

. (i) Express the length of the hypotenuse

XZ

in terms of

k

. (ii) For

\angle X

, obtain simplified expressions for

\sin X

and

\cos X

in terms of

k

. (iii) Prove that

\sin^{2} X + \cos^{2} X = 1

for all permissible values of

k

. (iv) If

k = 3

, calculate the numerical values of

\tan X

and

\csc X

. (Leave answers as simplified fractions where possible.)