Introduction to Trigonometry

In $\Delta ABC$, right-angled at B, AB = 24 cm, BC = 7 cm. Determine: (i) sin A, cos A (ii) sin C, cos C

If sin A = $\frac{3}{4}$, calculate cos A and tan A.

Given 15 cot A = 8, find sin A and sec A.

Given sec $\theta = \frac{13}{12}$, calculate all other trigonometric ratios.

If $\angle A$ and $\angle B$ are acute angles such that cos A = cos B, then show that $\angle A = \angle B$.

If cot $\theta = \frac{7}{8}$, evaluate: (i) $\frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)}$ (ii) $\cot^2 \theta$

If 3 cot A = 4, check whether $\frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A$ or not.

In triangle ABC, right-angled at B, if tan A = $\frac{1}{\sqrt{3}}$, find the value of: (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

In $\Delta PQR$, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

State whether the following are true or false. Justify your answer. (i) The value of tan A is always less than 1. (ii) sec A = $\frac{12}{5}$ for some value of angle A. (iii) cos A is the abbreviation used for the cosecant of angle A. (iv) cot A is the product of cot and A. (v) sin $\theta = \frac{4}{3}$ for some angle $\theta$.

Evaluate the following: (i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$ (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$ (iii) $\frac{\cos 45^\circ}{\sec 30^\circ + \cosec 30^\circ}$ (iv) $\frac{\sin 30^\circ + \tan 45^\circ - \cosec 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$ (v) $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

Choose the correct option and justify your choice:

If $\tan(A + B) = \sqrt{3}$ and $\tan(A - B) = \frac{1}{\sqrt{3}}$; $0^\circ < A + B \leq 90^\circ$; $A > B$, find A and B.

State whether the following are true or false. Justify your answer. (i) $\sin(A + B) = \sin A + \sin B$. (ii) The value of $\sin \theta$ increases as $\theta$ increases. (iii) The value of $\cos \theta$ increases as $\theta$ increases. (iv) $\sin \theta = \cos \theta$ for all values of $\theta$. (v) $\cot A$ is not defined for $A = 0^\circ$.

Express the trigonometric ratios $\sin A$, $\sec A$ and $\tan A$ in terms of $\cot A$.

Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.

Choose the correct option. Justify your choice.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.