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Q1(i):
Evaluate the following : (i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$
Solution :
Given: The trigonometric expression $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$.
To find: The numerical value of the given expression.
Visual Representation:
Step 1: Identify the values of the trigonometric ratios.
Based on the standard trigonometric table for specific angles, we have the following values:
- $\sin 60^\circ = \frac{\sqrt{3}}{2}$
- $\cos 30^\circ = \frac{\sqrt{3}}{2}$
- $\sin 30^\circ = \frac{1}{2}$
- $\cos 60^\circ = \frac{1}{2}$
Step 2: Substitute the values into the expression.
The given expression is $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$.
Substituting the values identified in Step 1:
$= \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) + \left( \frac{1}{2} \right) \left( \frac{1}{2} \right)$
Step 3: Perform the multiplication.
Multiply the fractions [Using the rule $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$]:
$= \left( \frac{\sqrt{3} \times \sqrt{3}}{2 \times 2} \right) + \left( \frac{1 \times 1}{2 \times 2} \right)$
$= \left( \frac{3}{4} \right) + \left( \frac{1}{4} \right)$
Step 4: Perform the addition.
Since the denominators are the same, add the numerators [Using the rule $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$]:
$= \frac{3 + 1}{4}$
$= \frac{4}{4}$
$= 1$
Final Answer: 1
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.2
- Q1(ii): Evaluate the following : (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$
- Q1(iii): Evaluate the following : (iii) $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$
- Q1(iv): Evaluate the following : (iv) $\frac{\sin 30^\circ + \tan 45^\circ – \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$
- Q1(v): Evaluate the following : (v) $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
- Q2(i): Choose the correct option and justify your choice : (i) $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$
- Q2(ii): Choose the correct option and justify your choice : (ii) $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =$
- Q2(iii): Choose the correct option and justify your choice : (iii) $\sin 2A = 2 \sin A$ is true when $A =$
- Q2(iv): Choose the correct option and justify your choice : (iv) $\frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =$
- Q3: If $\tan (A + B) = \sqrt{3}$ and $\tan (A – B) = \frac{1}{\sqrt{3}}$; $0^\circ < A + B \le 90^\circ$; $A > B$, find $A$ and $B$.
- Q4(i): State whether the following are true or false. Justify your answer. (i) $\sin (A + B) = \sin A + \sin B$.
- Q4(ii): State whether the following are true or false. Justify your answer. (ii) The value of $\sin \theta$ increases as $\theta$ increases.
- Q4(iii): State whether the following are true or false. Justify your answer. (iii) The value of $\cos \theta$ increases as $\theta$ increases.
- Q4(iv): State whether the following are true or false. Justify your answer. (iv) $\sin \theta = \cos \theta$ for all values of $\theta$.
- Q4(v): State whether the following are true or false. Justify your answer. (v) $\cot A$ is not defined for $A = 0^\circ$.
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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