Given that $\cot(\theta) = -\frac{2}{5}$ and $\cos(\theta) > 0$, we need to find the values of the other trigonometric functions.

Step 1: Determine the quadrant

• $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = -\frac{2}{5}$. Since $\cot(\theta)$ is negative and $\cos(\theta) > 0$, $\theta$ must be in the fourth quadrant, where cosine is positive and sine is negative.

Step 2: Use the cotangent to find sine and cosine

Let's denote:

• $\cos(\theta) = x$
• $\sin(\theta) = y$

Given: $\frac{x}{y} = -\frac{2}{5}$

This implies: $x = -2y/5$

Step 3: Use the Pythagorean identity

We know from the identity: $x^2 + y^2 = 1$

Substitute $x = -\frac{2}{5}y$ into the identity: $\left(-\frac{2}{5}y\right)^2 + y^2 = 1$

Simplify: $\frac{4}{25}y^2 + y^2 = 1$ $\frac{4}{25}y^2 + \frac{25}{25}y^2 = 1$ $\frac{29}{25}y^2 = 1$ $y^2 = \frac{25}{29}$ $y = -\sqrt{\frac{25}{29}} = -\frac{5}{\sqrt{29}}$ (since $y = \sin(\theta)$ and $\sin(\theta)$ is negative in the fourth quadrant).

Now find $x$: $x = -\frac{2}{5}y = -\frac{2}{5} \times -\frac{5}{\sqrt{29}} = \frac{2}{\sqrt{29}}$

Step 4: Compute the trigonometric functions

• $\sin(\theta)$: $y = -\frac{5}{\sqrt{29}}$
• $\cos(\theta)$: $x = \frac{2}{\sqrt{29}}$
• $\tan(\theta)$: $\frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{5}{\sqrt{29}}}{\frac{2}{\sqrt{29}}} = -\frac{5}{2}$
• $\csc(\theta)$: $\frac{1}{\sin(\theta)} = -\frac{\sqrt{29}}{5}$
• $\sec(\theta)$: $\frac{1}{\cos(\theta)} = \frac{\sqrt{29}}{2}$
• $\cot(\theta)$: Given as $-\frac{2}{5}$

Final Answer:

$\sin(\theta) = -\frac{5}{\sqrt{29}}, \quad \cos(\theta) = \frac{2}{\sqrt{29}}, \quad \tan(\theta) = -\frac{5}{2}$ $\csc(\theta) = -\frac{\sqrt{29}}{5}, \quad \sec(\theta) = \frac{\sqrt{29}}{2}, \quad \cot(\theta) = -\frac{2}{5}$

Would you like more details on any step, or have any further questions?

---

Related Questions:

1. How do you determine the quadrant of an angle given one trigonometric function and its sign?
2. What is the relationship between $\sin(\theta)$ and $\cos(\theta)$ when given $\cot(\theta)$?
3. How do the signs of trigonometric functions vary across different quadrants?
4. What are the key identities used in trigonometry for solving such problems?
5. Can we solve for trigonometric functions if only $\tan(\theta)$ is given?

Tip: Always remember the ASTC rule (All Students Take Calculus) to determine the sign of trigonometric functions in different quadrants.