Question

If \(\sin\theta=\frac{24}{25}\) and \(0^\circ<\theta<90^\circ\), then what is the value of \(\cos\theta\)?

• A. \(\frac{12}{15}\)
• B. \(\frac{7}{25}\)
• C. \(\frac{3}{5}\)
• D. \(\frac{4}{5}\)

Hint:

If \(\sin\theta=\frac{24}{25}\), then use the \(7-24-25\) right triangle. So \(\cos\theta=\frac{7}{25}\).

Answer 1

The Correct Option is B

Concept:
We use the basic trigonometric identity: \[ \sin^2\theta+\cos^2\theta=1 \] Since: \[ 0^\circ<\theta<90^\circ \] the angle lies in the first quadrant. Therefore, both \(\sin\theta\) and \(\cos\theta\) are positive.

Step 1: Write the given value.
\[ \sin\theta=\frac{24}{25} \]

Step 2: Use the identity.
\[ \sin^2\theta+\cos^2\theta=1 \] Substitute: \[ \left(\frac{24}{25}\right)^2+\cos^2\theta=1 \] \[ \frac{576}{625}+\cos^2\theta=1 \]

Step 3: Find \(\cos^2\theta\).
\[ \cos^2\theta=1-\frac{576}{625} \] \[ \cos^2\theta=\frac{625-576}{625} \] \[ \cos^2\theta=\frac{49}{625} \]

Step 4: Find \(\cos\theta\).
\[ \cos\theta=\sqrt{\frac{49}{625}} \] \[ \cos\theta=\frac{7}{25} \] Since \(\theta\) is in the first quadrant, \(\cos\theta\) is positive. Hence, the correct answer is: \[ \boxed{(B)\ \frac{7}{25}} \]