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Q11(v):
State whether the following are true or false. Justify your answer. (v) $\sin \theta = \frac{4}{3}$ for some angle $\theta$.
Solution :
Given: A trigonometric statement $\sin \theta = \frac{4}{3}$ for some angle $\theta$.
To Find: Determine whether the statement is true or false and provide a rigorous justification.
Step 1: Definition of the Sine Function
In a right-angled triangle, for an acute angle $\theta$, the sine function is defined as the ratio of the length of the side opposite to the angle $\theta$ to the length of the hypotenuse.
$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$
Step 2: Properties of a Right-Angled Triangle
In any right-angled triangle, the hypotenuse is the longest side. Let $a$ be the length of the opposite side and $c$ be the length of the hypotenuse. By the geometric property of triangles:
$c > a$ (The hypotenuse must be strictly greater than any other side of the triangle).
Step 3: Analyzing the given ratio
Given $\sin \theta = \frac{4}{3}$.
Comparing this to the definition $\sin \theta = \frac{a}{c}$, we have:
$a = 4$ units
$c = 3$ units
Step 4: Evaluating the validity
According to the property established in Step 2, the hypotenuse ($c$) must be greater than the opposite side ($a$).
Here, $c = 3$ and $a = 4$.
Since $3 < 4$, it implies that $c < a$.
This contradicts the fundamental property of a right-angled triangle where the hypotenuse is the longest side.
Step 5: Theoretical Justification
The range of the sine function for any real angle $\theta$ is restricted to the interval $[-1, 1]$.
Mathematically, $-1 \leq \sin \theta \leq 1$.
Since $\frac{4}{3} \approx 1.33$, and $1.33 > 1$, the value $\frac{4}{3}$ lies outside the possible range of the sine function.
Final Answer: False. The value of $\sin \theta$ cannot exceed 1 because the hypotenuse is always the longest side in a right-angled triangle, making the ratio $\frac{\text{Opposite}}{\text{Hypotenuse}}$ always less than or equal to 1.
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 $A$.
- 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) $\frac{(1 + \sin \theta) (1 - \sin \theta)}{(1 + \cos \theta) (1 - \cos \theta)}$
- 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) $\cos A \cos C – \sin A \sin C$
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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