Questions: Given tan θ = -20/21 and cos θ < 0, find sin θ and cos θ.

Transcript text: Given $\tan \theta=-\frac{20}{21}$ and $\cos \theta<0$, find $\sin \theta$ and $\cos \theta$.

Solution

Solution Steps

To find $\sin \theta$ and $\cos \theta$ given $\tan \theta = -\frac{20}{21}$ and $\cos \theta < 0$, we can use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$. Since $\tan \theta$ is negative and $\cos \theta < 0$, $\theta$ is in the second quadrant. We can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find the values of $\sin \theta$ and $\cos \theta$.

Step 1: Establish Relationships

Given that $\tan \theta = -\frac{20}{21}$, we can express this as: \[ \frac{\sin \theta}{\cos \theta} = -\frac{20}{21} \] This implies: \[ \sin \theta = -\frac{20}{21} \cos \theta \]

Step 2: Use the Pythagorean Identity

Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we substitute $\sin \theta$: \[ \left(-\frac{20}{21} \cos \theta\right)^2 + \cos^2 \theta = 1 \] This simplifies to: \[ \frac{400}{441} \cos^2 \theta + \cos^2 \theta = 1 \] Combining the terms gives: \[ \left(\frac{400}{441} + 1\right) \cos^2 \theta = 1 \] \[ \frac{841}{441} \cos^2 \theta = 1 \]

Step 3: Solve for $\cos \theta$

Solving for $\cos^2 \theta$: \[ \cos^2 \theta = \frac{441}{841} \] Taking the square root, we find: \[ \cos \theta = -\frac{21}{29} \quad (\text{since } \cos \theta < 0) \]

Step 4: Solve for $\sin \theta$

Now substituting back to find $\sin \theta$: \[ \sin \theta = -\frac{20}{21} \left(-\frac{21}{29}\right) = \frac{20}{29} \]

Final Answer

Thus, the values are: \[ \sin \theta = \frac{20}{29}, \quad \cos \theta = -\frac{21}{29} \] The final boxed answers are: \[ \boxed{\sin \theta = \frac{20}{29}} \quad \text{and} \quad \boxed{\cos \theta = -\frac{21}{29}} \]

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