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Q4(ii):
State whether the following are true or false. Justify your answer. (ii) The value of $\sin \theta$ increases as $\theta$ increases.
Solution :
Given: A trigonometric function $f(\theta) = \sin \theta$, where $\theta$ represents an angle in a right-angled triangle.
To Determine: Whether the statement "The value of $\sin \theta$ increases as $\theta$ increases" is True or False, and provide a justification.
Step 1: Definition of Sine Function
In a right-angled triangle, the sine of an angle $\theta$ is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse:
$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
Step 2: Evaluating $\sin \theta$ at specific angles in the interval $0^\circ \le \theta \le 90^\circ$
We use the standard trigonometric values for common angles:
| $\theta$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin \theta$ | $0$ | $0.5$ | $\frac{1}{\sqrt{2}} \approx 0.707$ | $\frac{\sqrt{3}}{2} \approx 0.866$ | $1$ |
Step 3: Analyzing the Trend
By observing the values in the table above:
- At $\theta = 0^\circ$, $\sin \theta = 0$
- At $\theta = 30^\circ$, $\sin \theta = 0.5$
- At $\theta = 45^\circ$, $\sin \theta \approx 0.707$
- At $\theta = 60^\circ$, $\sin \theta \approx 0.866$
- At $\theta = 90^\circ$, $\sin \theta = 1$
As $\theta$ increases from $0^\circ$ to $90^\circ$, the value of $\sin \theta$ consistently increases from $0$ to $1$.
Step 4: Geometric Justification
As the angle $\theta$ increases in a right-angled triangle (while keeping the hypotenuse constant), the length of the side opposite to $\theta$ increases. Since $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, an increase in the numerator (Opposite side) while the denominator (Hypotenuse) remains fixed results in an increase in the overall value of the fraction.
Conclusion:
Since the value of $\sin \theta$ rises as $\theta$ increases from $0^\circ$ to $90^\circ$, the statement is mathematically correct.
Final Answer: True. The value of $\sin \theta$ increases as $\theta$ increases from $0^\circ$ to $90^\circ$.
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(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(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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