To solve the problem using the Pythagorean theorem, let's start with the given information:

• $\sin \theta = -\frac{4}{5}$
• $\theta$ is in the third quadrant.

Step 1: Interpret the given sine value

The sine of an angle in a right triangle is given by: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ Here, $\sin \theta = -\frac{4}{5}$, which means:

• Opposite side $= -4$ (negative because $\theta$ is in the third quadrant)
• Hypotenuse $= 5$ (always positive)

Step 2: Use the Pythagorean theorem to find the adjacent side

The Pythagorean theorem states: $\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$ Substitute the known values: $5^2 = (-4)^2 + \text{adjacent}^2$ $25 = 16 + \text{adjacent}^2$ $\text{adjacent}^2 = 25 - 16 = 9$ $\text{adjacent} = \pm 3$

Step 3: Determine the sign of the adjacent side

Since $\theta$ is in the third quadrant, both sine and cosine are negative. Therefore, the adjacent side (which corresponds to the cosine) is negative: $\text{adjacent} = -3$

Step 4: Find $\cos \theta$

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-3}{5}$

Step 5: Find $\tan \theta$

Tangent is given by: $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}$

Final Answers

• $\cos \theta = -\frac{3}{5}$
• $\tan \theta = \frac{4}{3}$

Would you like further details or have any other questions?

Here are some related questions:

1. How would the process change if $\theta$ were in the second quadrant?
2. What is the geometric interpretation of the Pythagorean theorem in trigonometry?
3. How do you find the cotangent using sine and cosine?
4. What are the steps to find the secant and cosecant of $\theta$ using this method?
5. Can you apply the Pythagorean theorem in non-right triangles?

Tip: When using the Pythagorean theorem, always consider the signs of the sides based on the quadrant where the angle lies.