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 θ and asked to find the possible values of cos θ. The question does not specify the quadrant in which θ lies.

Step 2: Key Formula or Approach (Alternate Method):
Use the Pythagorean identity directly: cos²θ = 1 - sin²θ, then take square root with ± sign since quadrant is unknown.

Step 3: Detailed Explanation:
Given: sin θ = 3/5. From sin²θ + cos²θ = 1: cos²θ = 1 - sin²θ = 1 - (3/5)² = 1 - 9/25 = 16/25. Taking square root: cos θ = ±√(16/25) = ±4/5. In QI, cos θ = 4/5. In QII, cos θ = -4/5. Since quadrant is not specified, both are valid.

Step 4: Final Answer:
The possible values for cos θ are 4/5 or -4/5.