Question

If \(\sin \theta = \frac{3}{5}\), then \(\cos \theta =\)

• A. \(\frac{4}{5}\) but not \(-\frac{4}{5}\)
• B. \(\frac{4}{5}\) or \(-\frac{4}{5}\)
• C. \(-\frac{4}{5}\) but not \(\frac{4}{5}\)
• D. \(\frac{3}{5}\) but not \(-\frac{3}{5}\)

Hint:

Always be careful when taking the square root in trigonometric problems. Unless the quadrant of the angle is specified, you must consider both the positive and negative roots. A positive \(\sin \theta\) value means the angle can be in Quadrant I (where cosine is positive) or Quadrant II (where cosine is negative).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the value of \(\sin \theta\) and asked to find the possible values of \(\cos \theta\). The question does not specify the quadrant in which \(\theta\) lies.

Step 2: Key Formula or Approach:
The fundamental trigonometric identity relating sine and cosine is:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
We will use this identity to solve for \(\cos \theta\).

Step 3: Detailed Explanation:
We are given \(\sin \theta = \frac{3}{5}\).
Substitute this value into the Pythagorean identity:
\[ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \]
\[ \frac{9}{25} + \cos^2 \theta = 1 \]
Now, solve for \(\cos^2 \theta\):
\[ \cos^2 \theta = 1 - \frac{9}{25} \]
\[ \cos^2 \theta = \frac{25 - 9}{25} \]
\[ \cos^2 \theta = \frac{16}{25} \]
Take the square root of both sides to find \(\cos \theta\):
\[ \cos \theta = \pm \sqrt{\frac{16}{25}} \]
\[ \cos \theta = \pm \frac{4}{5} \]
Since the quadrant of \(\theta\) is not specified, both positive and negative values are possible.
If \(\theta\) is in the first quadrant, \(\cos \theta = \frac{4}{5}\).
If \(\theta\) is in the second quadrant, \(\cos \theta = -\frac{4}{5}\).
Therefore, \(\cos \theta\) can be either \(\frac{4}{5}\) or \(-\frac{4}{5}\).

Step 4: Final Answer:
The possible values for \(\cos \theta\) are \(\frac{4}{5}\) or \(-\frac{4}{5}\).