If sin θ = -  $\frac{3}{5}$ and θ lies in the third quadrant, find cos θ

Analysis

We are given the value of $\sin \theta$ and the specific quadrant where $\theta$ lies. We need to determine the value of $\cos \theta$.

Concept: Trigonometric Identities and Quadrants

1. Pythagorean Identity: The fundamental relationship between sine and cosine is: $$\sin^2 \theta + \cos^2 \theta = 1$$
2. Quadrant Rules (CAST Rule): • 1st Quadrant (All): All trigonometric ratios are positive. • 2nd Quadrant (Sine): Only $\sin \theta$ is positive. • 3rd Quadrant (Tangent): Only $\tan \theta$ is positive ($\sin \theta$ and $\cos \theta$ are negative). • 4th Quadrant (Cosine): Only $\cos \theta$ is positive.

Step-by-Step Solution

Step 1: Use the Pythagorean identity. Given $\sin \theta = -\frac{3}{5}$, substitute this into the identity:

Step 2: Solve for $\cos^2 \theta$.

Step 3: Find $\cos \theta$ by taking the square root.

Step 4: Determine the sign based on the quadrant. The question states that $\theta$ lies in the third quadrant. According to the CAST rule, only tangent is positive in the third quadrant, meaning both sine and cosine must be negative. Therefore, we choose the negative value:

Conclusion

Comparing our result to the given options, we find that our calculation matches option D.

Exam Tip: Always remember the CAST mnemonic (starting from the 4th quadrant or 1st quadrant counter-clockwise): Cosine (4th), All (1st), Sine (2nd), Tangent (3rd). It is a lifesaver for questions involving quadrants!