Question

If $\sin \theta = \frac{3}{5}$, then find $\cos \theta$. [$\sin^2\theta + \cos^2\theta = 1$]

• A. $\frac{4}{5}$
• B. $\frac{5}{4}$
• C. $\frac{3}{4}$
• D. $\frac{2}{5}$

Hint:

Recognize the Pythagorean Triple: (3, 4, 5). If $\sin\theta$ is $3/5$, then the sides are 3 (opposite) and 5 (hypotenuse). The adjacent side must be 4, so $\cos\theta$ is $4/5$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In trigonometry, the fundamental identity $\sin^2\theta + \cos^2\theta = 1$ relates the sine and cosine of the same angle. Alternatively, this can be solved using a right-angled triangle.

Step 2: Identifying the Formula and Values:
Given $\sin\theta = \frac{3}{5}$. We know that: \[ \cos^2\theta = 1 - \sin^2\theta \]

Step 3: Calculation:
\[ \cos^2\theta = 1 - \left(\frac{3}{5}\right)^2 \] \[ \cos^2\theta = 1 - \frac{9}{25} \] \[ \cos^2\theta = \frac{25 - 9}{25} = \frac{16}{25} \] Taking the square root: \[ \cos\theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

Step 4: Final Answer:
The value of $\cos \theta$ is $\frac{4}{5}$.