Introduction to Trigonometry

In $\Delta \text{ABC}$, right-angled at B, $\text{AB} = 24\text{ cm}$, $\text{BC} = 7\text{ cm}$. Determine :

(i) $\sin A$, $\cos A$

(ii) $\sin C$, $\cos C$

In Fig. 8.13, find $\tan P - \cot R$.

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

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

Given $\sec \theta = \dfrac{13}{12}$, 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 $\cot \theta = \dfrac{7}{8}$, evaluate :

(i) $\dfrac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)}$

(ii) $\cot^2 \theta$

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

In triangle ABC, right-angled at B, if $\tan A = \dfrac{1}{\sqrt{3}}$, find the value of:

(i) $\sin A \cos C + \cos A \sin C$

(ii) $\cos A \cos C - \sin A \sin C$

In $\Delta \text{PQR}$, right-angled at Q, $\text{PR} + \text{QR} = 25\text{ cm}$ and $\text{PQ} = 5\text{ cm}$. 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) $\sec A = \dfrac{12}{5}$ 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) $\sin \theta = \dfrac{4}{3}$ 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) $\dfrac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}$

(iv) $\dfrac{\sin 30^\circ + \tan 45^\circ - \csc 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$

(v) $\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

Choose the correct option and justify your choice :

(i) $\dfrac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$ (A) $\sin 60^\circ$ (B) $\cos 60^\circ$ (C) $\tan 60^\circ$ (D) $\sin 30^\circ$

(ii) $\dfrac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =$ (A) $\tan 90^\circ$ (B) $1$ (C) $\sin 45^\circ$ (D) $0$

(iii) $\sin 2A = 2 \sin A$ is true when $A =$ (A) $0^\circ$ (B) $30^\circ$ (C) $45^\circ$ (D) $60^\circ$

(iv) $\dfrac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =$ (A) $\cos 60^\circ$ (B) $\sin 60^\circ$ (C) $\tan 60^\circ$ (D) $\sin 30^\circ$

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

(i) $9 \sec^2 A - 9 \tan^2 A =$

(A) $1$ (B) $9$ (C) $8$ (D) $0$

(ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta - \csc \theta) =$

(A) $0$ (B) $1$ (C) $2$ (D) $-1$

(iii) $(\sec A + \tan A) (1 - \sin A) =$

(A) $\sec A$ (B) $\sin A$ (C) $\csc A$ (D) $\cos A$

(iv) $\dfrac{1 + \tan^2 A}{1 + \cot^2 A} =$

(A) $\sec^2 A$ (B) $-1$ (C) $\cot^2 A$ (D) $\tan^2 A$

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

(i) $(\text{cosec } \theta - \cot \theta)^2 = \dfrac{1 - \cos \theta}{1 + \cos \theta}$

(ii) $\dfrac{\cos A}{1 + \sin A} + \dfrac{1 + \sin A}{\cos A} = 2 \sec A$

(iii) $\dfrac{\tan \theta}{1 - \cot \theta} + \dfrac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \text{ cosec } \theta$ [Hint : Write the expression in terms of $\sin \theta$ and $\cos \theta$]

(iv) $\dfrac{1 + \sec A}{\sec A} = \dfrac{\sin^2 A}{1 - \cos A}$ [Hint : Simplify LHS and RHS separately]

(v) $\dfrac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \text{cosec } A + \cot A$, using the identity $\text{cosec}^2 A = 1 + \cot^2 A$.

(vi) $\sqrt{\dfrac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$

(vii) $\dfrac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$

(viii) $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$

(ix) $(\text{cosec } A - \sin A)(\sec A - \cos A) = \dfrac{1}{\tan A + \cot A}$ [Hint : Simplify LHS and RHS separately]

(x) $\left(\dfrac{1 + \tan^2 A}{1 + \cot^2 A}\right) = \left(\dfrac{1 - \tan A}{1 - \cot A}\right)^2 = \tan^2 A$