Q11(v):

State whether the following are true or false. Justify your answer. (v) $\sin \theta = \frac{4}{3}$ for some angle $\theta$.

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.