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Q7(ii):
If $\cot \theta = \frac{7}{8}$, evaluate : (ii) $\cot^2 \theta$
Solution :
Given: The trigonometric ratio $\cot \theta = \frac{7}{8}$.
To Find: The value of $\cot^2 \theta$.
Visual Representation:
Step 1: Understanding the definition of $\cot \theta$
In a right-angled triangle, for an angle $\theta$, the cotangent ratio is defined as the ratio of the length of the adjacent side to the length of the opposite side:
$\cot \theta = \frac{\text{Adjacent side}}{\text{Opposite side}}$
Step 2: Formulating the expression for $\cot^2 \theta$
The expression $\cot^2 \theta$ is mathematically equivalent to $(\cot \theta)^2$. This notation indicates that the entire value of the cotangent of angle $\theta$ must be raised to the power of 2.
Step 3: Substitution and Calculation
Given that $\cot \theta = \frac{7}{8}$, we substitute this value into the expression:
$\cot^2 \theta = (\cot \theta)^2$
$\cot^2 \theta = \left( \frac{7}{8} \right)^2$
Applying the exponent rule $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$:
$\cot^2 \theta = \frac{7^2}{8^2}$
Calculating the squares of the numerator and the denominator:
$7^2 = 7 \times 7 = 49$
$8^2 = 8 \times 8 = 64$
Therefore:
$\cot^2 \theta = \frac{49}{64}$
Final Answer: Final Answer: \frac{49}{64}
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.1
- Q1(i): In $\triangle ABC$, right-angled at $B$, $AB = 24$ cm, $BC = 7$ cm. Determine : (i) $\sin A, \cos A$
- Q1(ii): In $\triangle ABC$, right-angled at $B$, $AB = 24$ cm, $BC = 7$ cm. Determine : (ii) $\sin C, \cos C$
- Q10: In $\triangle PQR$, right-angled at $Q$, $PR + QR = 25$ cm and $PQ = 5$ cm. Determine the values of $\sin P, \cos P$ and $\tan P$.
- Q11(i): State whether the following are true or false. Justify your answer. (i) The value of $\tan A$ is always less than 1.
- Q11(ii): State whether the following are true or false. Justify your answer. (ii) $\sec A = \frac{12}{5}$ for some value of angle $A$.
- Q11(iii): State whether the following are true or false. Justify your answer. (iii) $\cos A$ is the abbreviation used for the cosecant of angle $A$.
- Q11(iv): State whether the following are true or false. Justify your answer. (iv) $\cot A$ is the product of cot and $A$.
- Q11(v): State whether the following are true or false. Justify your answer. (v) $\sin \theta = \frac{4}{3}$ for some angle $\theta$.
- Q2: In Fig. 8.13, find $\tan P – \cot R$.
- Q3: If $\sin A = \frac{3}{4}$, calculate $\cos A$ and $\tan A$.
- Q4: Given $15 \cot A = 8$, find $\sin A$ and $\sec A$.
- Q5: Given $\sec \theta = \frac{13}{12}$, calculate all other trigonometric ratios.
- Q6: If $\angle A$ and $\angle B$ are acute angles such that $\cos A = \cos B$, then show that $\angle A = \angle B$.
- Q7(i): If $\cot \theta = \frac{7}{8}$, evaluate : (i) $\frac{(1 + \sin \theta) (1 - \sin \theta)}{(1 + \cos \theta) (1 - \cos \theta)}$
- Q8: If $3 \cot A = 4$, check whether $\frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A – \sin^2 A$ or not.
- Q9(i): In triangle ABC, right-angled at B, if $\tan A = \frac{1}{\sqrt{3}}$, find the value of: (i) $\sin A \cos C + \cos A \sin C$
- Q9(ii): In triangle ABC, right-angled at B, if $\tan A = \frac{1}{\sqrt{3}}$, find the value of: (ii) $\cos A \cos C – \sin A \sin C$
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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