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Q2:
Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.
Solution :
Given: An angle $A$ in a right-angled triangle, where the trigonometric ratio is expressed in terms of $\sec A$.
To Find: Express $\sin A$, $\cos A$, $\tan A$, $\csc A$, and $\cot A$ in terms of $\sec A$.
Step 1: Expressing $\cos A$ in terms of $\sec A$
By the definition of reciprocal trigonometric ratios, we know that $\cos A$ is the reciprocal of $\sec A$.
$\cos A = \frac{1}{\sec A}$
Step 2: Expressing $\sin A$ in terms of $\sec A$
Using the Pythagorean identity: $\sin^2 A + \cos^2 A = 1$.
$\sin^2 A = 1 - \cos^2 A$ [Subtracting $\cos^2 A$ from both sides]
$\sin^2 A = 1 - \left(\frac{1}{\sec A}\right)^2$ [Substituting $\cos A = \frac{1}{\sec A}$]
$\sin^2 A = 1 - \frac{1}{\sec^2 A} = \frac{\sec^2 A - 1}{\sec^2 A}$
$\sin A = \sqrt{\frac{\sec^2 A - 1}{\sec^2 A}} = \frac{\sqrt{\sec^2 A - 1}}{\sec A}$
Step 3: Expressing $\tan A$ in terms of $\sec A$
Using the identity: $1 + \tan^2 A = \sec^2 A$.
$\tan^2 A = \sec^2 A - 1$ [Subtracting 1 from both sides]
$\tan A = \sqrt{\sec^2 A - 1}$
Step 4: Expressing $\csc A$ in terms of $\sec A$
By definition, $\csc A = \frac{1}{\sin A}$.
$\csc A = \frac{1}{\frac{\sqrt{\sec^2 A - 1}}{\sec A}}$ [Substituting the expression for $\sin A$ derived in Step 2]
$\csc A = \frac{\sec A}{\sqrt{\sec^2 A - 1}}$
Step 5: Expressing $\cot A$ in terms of $\sec A$
By definition, $\cot A = \frac{1}{\tan A}$.
$\cot A = \frac{1}{\sqrt{\sec^2 A - 1}}$ [Substituting the expression for $\tan A$ derived in Step 3]
Final Answer:
The trigonometric ratios in terms of $\sec A$ are:
$\sin A = \frac{\sqrt{\sec^2 A - 1}}{\sec A}$
$\cos A = \frac{1}{\sec A}$
$\tan A = \sqrt{\sec^2 A - 1}$
$\csc A = \frac{\sec A}{\sqrt{\sec^2 A - 1}}$
$\cot A = \frac{1}{\sqrt{\sec^2 A - 1}}$
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.3
- Q1: Express the trigonometric ratios $\sin A$, $\sec A$ and $\tan A$ in terms of $\cot A$.
- Q3(i): Choose the correct option. Justify your choice. (i) $9 \sec^2 A – 9 \tan^2 A =$
- Q3(ii): Choose the correct option. Justify your choice. (ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta – \text{cosec } \theta) =$
- Q3(iii): Choose the correct option. Justify your choice. (iii) $(\sec A + \tan A) (1 – \sin A) =$
- Q3(iv): Choose the correct option. Justify your choice. (iv) $\frac{1 + \tan^2 A}{1 + \cot^2 A} =$
- Q4(i): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) $(\text{cosec } \theta – \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$
- Q4(ii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) $\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A$
- Q4(iii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii) $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \text{cosec } \theta$ [Hint : Write the expression in terms of $\sin \theta$ and $\cos \theta$]
- Q4(iv): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iv) $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 – \cos A}$ [Hint : Simplify LHS and RHS separately]
- Q4(ix): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ix) $(\text{cosec } A – \sin A) (\sec A – \cos A) = \frac{1}{\tan A + \cot A}$ [Hint : Simplify LHS and RHS separately]
- Q4(v): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (v) $\frac{\cos A – \sin A + 1}{\cos A + \sin A – 1} = \text{cosec } A + \cot A$, using the identity $\text{cosec}^2 A = 1 + \cot^2 A$.
- Q4(vi): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vi) $\sqrt{\frac{1 + \sin A}{1 – \sin A}} = \sec A + \tan A$
- Q4(vii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vii) $\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$
- Q4(viii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii) $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$
- Q4(x): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (x) $(\frac{1 + \tan^2 A}{1 + \cot^2 A}) = (\frac{1 - \tan A}{1 - \cot A})^2 = \tan^2 A$
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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