CBSE Class 10 Mathematics Introduction to Trigonometry Notes Set 19

Introduction to Trigonometry

In a right triangle, for an acute angle theta: \( \sin\theta=\frac{P}{H} \), \( \cos\theta=\frac{B}{H} \), \( \tan\theta=\frac{P}{B} \), and \( \operatorname{cosec}\theta,\ \sec\theta,\ \cot\theta \) are their reciprocals. Given one ratio, draw the right triangle with that ratio, find the third side by Pythagoras theorem and read off all the others.
Values at standard angles: sin 0, 30, 45, 60, 90 are \( 0,\ \frac12,\ \frac{1}{\sqrt2},\ \frac{\sqrt3}{2},\ 1 \); cos takes these in reverse order; tan 0, 30, 45, 60 are \( 0,\ \frac1{\sqrt3},\ 1,\ \sqrt3 \) and tan 90 is not defined.

Identities and Complementary Angles

\( \sin^2\theta+\cos^2\theta=1 \), \( 1+\tan^2\theta=\sec^2\theta \), \( 1+\cot^2\theta=\operatorname{cosec}^2\theta \). For complementary angles \( \sin(90^\circ-A)=\cos A \), \( \tan(90^\circ-A)=\cot A \), \( \sec(90^\circ-A)=\operatorname{cosec}A \). Pair tan x with tan(90 - x) so the product is 1; a product that contains cos 90 is zero.
To prove an identity, work on the more complicated side, convert everything to sines and cosines, and simplify using the identities. Typical results: \( \sqrt{\frac{1-\sin\theta}{1+\sin\theta}}=\sec\theta-\tan\theta \), \( \cot\theta-\tan\theta=\frac{2\cos^2\theta-1}{\sin\theta\cos\theta} \), and if \( \sec\theta+\tan\theta=p \) then \( \sin\theta=\frac{p^2-1}{p^2+1} \).

Solving Problems

If tan theta = m/n then \( \sin\theta=\frac{m}{\sqrt{m^2+n^2}} \). To use areas: area of a triangle with sides a, b and included angle C is \( \frac12ab\sin C \). A condition such as \( \tan\theta+\cot\theta=2 \) forces tan theta = 1, theta = 45 degrees, which makes any power-sum like \( \tan^9\theta+\cot^9\theta \) equal to 2.