To find the values of the trigonometric functions of $\theta$ given that $\cos(\theta) = -\frac{2}{5}$ and $\tan(\theta) < 0$, we can follow these steps:

Step 1: Determine the Quadrant

• Since $\cos(\theta) = -\frac{2}{5}$, the cosine function is negative.
• $\tan(\theta) < 0$ indicates that tangent is negative.

Cosine is negative in the second and third quadrants, but tangent is only negative in the second quadrant. Therefore, $\theta$ lies in the second quadrant.

Step 2: Find $\sin(\theta)$

We know that: $\sin^2(\theta) + \cos^2(\theta) = 1$ Substitute $\cos(\theta) = -\frac{2}{5}$: $\sin^2(\theta) + \left(-\frac{2}{5}\right)^2 = 1$ $\sin^2(\theta) + \frac{4}{25} = 1$ $\sin^2(\theta) = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}$ $\sin(\theta) = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}$ Since $\theta$ is in the second quadrant, where sine is positive: $\sin(\theta) = \frac{\sqrt{21}}{5}$

Step 3: Find $\tan(\theta)$

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{21}}{5}}{-\frac{2}{5}} = -\frac{\sqrt{21}}{2}$

Step 4: Find the Remaining Trigonometric Functions

1. Cosecant ($\csc(\theta)$): $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{\sqrt{21}} = \frac{5\sqrt{21}}{21}$

2. Secant ($\sec(\theta)$): $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{2}{5}} = -\frac{5}{2}$

3. Cotangent ($\cot(\theta)$): $\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{2}{\sqrt{21}} = -\frac{2\sqrt{21}}{21}$

Summary of Trigonometric Function Values:

• $\sin(\theta) = \frac{\sqrt{21}}{5}$
• $\cos(\theta) = -\frac{2}{5}$
• $\tan(\theta) = -\frac{\sqrt{21}}{2}$
• $\csc(\theta) = \frac{5\sqrt{21}}{21}$
• $\sec(\theta) = -\frac{5}{2}$
• $\cot(\theta) = -\frac{2\sqrt{21}}{21}$

Would you like more details or have any questions?

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Related Questions:

1. How do you determine the quadrant in which an angle lies?
2. What are the signs of trigonometric functions in different quadrants?
3. How do you derive the Pythagorean identities in trigonometry?
4. What is the significance of the tangent function being negative?
5. How do you calculate trigonometric functions for angles in standard position?

Tip: Always check the quadrant when determining the sign of a trigonometric function.