If the terminal side of angle theta goes through the point (-3,-4), find cot()

To find \( \cot(\theta) \), we need to determine the values of \( \sin(\theta) \) and \( \cos(\theta) \).

Given that the terminal side of the angle \( \theta \) goes through the point (-3, -4), we can use this information to find the values of \( \sin(\theta) \) and \( \cos(\theta) \) by first calculating the hypotenuse \( r \).

### Step 1: Calculate the hypotenuse \( r \)
The hypotenuse \( r \) is the distance from the origin to the point (-3, -4), which can be found using the Pythagorean theorem:

\[
r = \sqrt{x^2 + y^2}
\]

Substituting the values \( x = -3 \) and \( y = -4 \):

\[
r = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

### Step 2: Find \( \sin(\theta) \) and \( \cos(\theta) \)
Now that we have \( r = 5 \), we can find:

\[
\sin(\theta) = \frac{y}{r} = \frac{-4}{5}
\]
\[
\cos(\theta) = \frac{x}{r} = \frac{-3}{5}
\]

### Step 3: Calculate \( \cot(\theta) \)
The cotangent of an angle is the ratio of \( \cos(\theta) \) to \( \sin(\theta) \):

\[
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{-3}{5}}{\frac{-4}{5}} = \frac{-3}{-4} = \frac{3}{4}
\]

Thus, \( \cot(\theta) = \frac{3}{4} \).

Would you like any further details or have any questions?

Here are 5 related questions to expand on this:

1. How do you find the other trigonometric functions from this point?
2. What is the significance of the hypotenuse in trigonometry?
3. How does the quadrant affect the signs of trigonometric functions?
4. What is the relationship between cotangent and tangent?
5. Can you calculate \( \tan(\theta) \) from the given information?

**Tip:** Remember that in trigonometry, the signs of the trigonometric functions depend on the quadrant where the terminal side of the angle lies.

Learn how to find the cotangent of an angle given a point on its terminal side. This problem covers key trigonometric concepts, including using the Pythagorean theorem to find the hypotenuse and calculating trigonometric ratios like sine, cosine, and cotangent. Suitable for students in Grades 9-11.

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