Exercise 8.1

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

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

In $\triangle PQR$, 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) $\sec A = \dfrac{12}{5}$ 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 $\theta$.

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 $\dfrac{1 - \tan A}{1 + \tan A} = \cos^2 A - \sin^2 A$ or not.

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