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Q1(ii):
Evaluate the following : (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$
Solution :
Given: An algebraic expression involving trigonometric ratios: $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$.
To Find: The numerical value of the given expression.
Step 1: Identification of Trigonometric Values
To evaluate the expression, we must recall the standard trigonometric ratios for the given angles from the trigonometric table:
- $\tan 45^\circ = 1$
- $\cos 30^\circ = \frac{\sqrt{3}}{2}$
- $\sin 60^\circ = \frac{\sqrt{3}}{2}$
Step 2: Substitution of Values into the Expression
Substitute the identified values into the expression $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$:
$= 2(1)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{\sqrt{3}}{2}\right)^2$
Step 3: Performing Arithmetic Operations
Now, we simplify each term step-by-step:
First, calculate the squares:
- $(1)^2 = 1$
- $\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{(\sqrt{3})^2}{(2)^2} = \frac{3}{4}$
Substitute these back into the expression:
$= 2(1) + \frac{3}{4} - \frac{3}{4}$
Step 4: Final Simplification
Perform the multiplication and addition/subtraction:
$= 2 + \frac{3}{4} - \frac{3}{4}$
[Since $\frac{3}{4} - \frac{3}{4} = 0$]
$= 2 + 0$
$= 2$
Final Answer: 2
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.2
- Q1(i): Evaluate the following : (i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$
- 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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