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Q2:
In Fig. 8.13, find $\tan P – \cot R$.
In Fig. 8.13, find $\tan P – \cot R$.
Solution :
Given: A right-angled triangle $PQR$ where $\angle Q = 90^\circ$. The length of side $PQ = 12\text{ cm}$ and the length of the hypotenuse $PR = 13\text{ cm}$.
To Find: The value of the expression $\tan P - \cot R$.
Step 1: Determine the length of side $QR$ using the Pythagoras Theorem.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. [Pythagoras Theorem: $PR^2 = PQ^2 + QR^2$]
Substituting the given values:
$13^2 = 12^2 + QR^2$
$169 = 144 + QR^2$
$QR^2 = 169 - 144$
$QR^2 = 25$
$QR = \sqrt{25} = 5\text{ cm}$
Step 2: Calculate $\tan P$.
For $\angle P$, the side opposite is $QR$ and the side adjacent is $PQ$.
The trigonometric ratio for tangent is defined as: $\tan P = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{QR}{PQ}$
$\tan P = \frac{5}{12}$
Step 3: Calculate $\cot R$.
For $\angle R$, the side opposite is $PQ$ and the side adjacent is $QR$.
The trigonometric ratio for cotangent is defined as: $\cot R = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{QR}{PQ}$
$\cot R = \frac{5}{12}$
Step 4: Evaluate the expression $\tan P - \cot R$.
Substitute the values obtained in Step 2 and Step 3:
$\tan P - \cot R = \frac{5}{12} - \frac{5}{12}$
$\tan P - \cot R = 0$
Final Answer: 0
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) $\sin C, \cos C$
- 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) $\frac{(1 + \sin \theta) (1 - \sin \theta)}{(1 + \cos \theta) (1 - \cos \theta)}$
- Q7(ii): If $\cot \theta = \frac{7}{8}$, evaluate : (ii) $\cot^2 \theta$
- 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) $\cos A \cos C – \sin A \sin C$
- 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 $\theta$.
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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