Question

If the secant of angle $\left(\theta \right)$ is $\frac{25}{7}$, what is sine of angle $\left(\theta \right)$.

role="math" localid="1648481459730" $\left(A\right)\frac{5}{25}\left(B\right)\frac{7}{25}\left(C\right)\frac{24}{25}\left(D\right)\frac{25}{7}$

Short answer

The value of $\sin e\left(\theta \right)=\frac{24}{25}$. So, option (C) is correct.

Step-by-step solution

Step 1. Write down the given information.

It is given that $\sec \left(\theta \right)=\frac{25}{7}$

Step 2. Calculation.

Since $\sec \left(\theta \right)=\frac{25}{7}$, so by reciprocal identity $\cos \left(\theta \right)=\frac{7}{25}$. Therefore, construct the triangle having angle $C=\theta$, base $BC=7$, hypotenuse $AC=25$and angle $B=90°$.

Now, by Pythagoras Theorem,

$\begin{array}{c}A{C}^2=A{B}^2+B{C}^2\\ {25}^2=A{B}^2+7^2\\ A{B}^2={25}^2-7^2\\ AB=\sqrt{{25}^2-7^2}\\ AB=24\end{array}$

Now in right triangle ABC by trigonometric ratios,

$\begin{array}{c}\sin \left(C\right)=\frac{AB}{AC}\\ \sin \left(\theta \right)=\frac{24}{25}\end{array}$

Since, the value of $\sin \left(\theta \right)=\frac{24}{25}$. Hence, option (C) is correct.

Step 3. Conclusion.

The value of $\sin \left(\theta \right)=\frac{24}{25}$. Hence, option (C) is correct.