Determine two angles between 0 and 360 that have a cosecant of -8. I don't understand what steps I should take in order to solve the question. Im not looking for the answer just a formula to do questions like this.

csc = 1 / sin = r / x

sin (and csc) are negative in quads III and IV

To determine the angles that have a cosecant of -8, you can follow these steps:

Step 1: Recall the definition of the cosecant function:

The cosecant (csc) of an angle in a right triangle is equal to the ratio of the length of the hypotenuse to the length of the side opposite that angle. Mathematically, csc(theta) = hypotenuse / opposite.

Step 2: Use the reciprocal property of the sine function:

Since the cosecant function is the reciprocal of the sine function, you can use the equation csc(theta) = 1/sin(theta) to find the angles.

Step 3: Rearrange the equation:

Rearrange the equation 1/sin(theta) = -8 to sin(theta) = -1/8.

Step 4: Find the reference angle:

Since the sine function is negative, the angle you're looking for lies in either the third or fourth quadrant. Find the reference angle in the first quadrant by taking the inverse sine of the absolute value of the ratio: sin^-1(1/8) ≈ 7.218 degrees.

Step 5: Determine the angles:

For angles in the third quadrant, subtract the reference angle from 180 degrees: 180 - 7.218 ≈ 172.782 degrees.
For angles in the fourth quadrant, consider that the sine function is positive in the fourth quadrant. Subtract the reference angle from 360 degrees: 360 - 7.218 ≈ 352.782 degrees.

Thus, the two angles that have a cosecant of -8 are approximately 172.782 degrees in the third quadrant and approximately 352.782 in the fourth quadrant.

To solve a trigonometric equation like this, you need to find the angles that have a particular trigonometric function value. In this case, we are given the cosecant of an angle, which is the reciprocal of the sine function.

The formula for solving an equation involving trigonometric functions is to take the inverse trigonometric function of both sides of the equation. Since we are dealing with the cosecant function, we take the inverse cosecant (or arcsin) function.

Here's the step-by-step process to find the two angles:

1. Start with the given equation: csc(θ) = -8.
2. Take the inverse cosecant (or arcsin) of both sides: arcsin(csc(θ)) = arcsin(-8).
- This step isolates the angle θ by applying the inverse cosecant function to both sides of the equation.
3. Simplify the left side: θ = arcsin(-8).
- Since the cosecant function is the reciprocal of the sine function, the inverse cosecant is equivalent to the inverse sine. Therefore, arcsin(csc(θ)) simplifies to just θ.
4. Use a calculator to find the inverse sine of -8.
- The inverse sine function is typically denoted as sin^(-1) or asin. Use your calculator's inverse sine function to find the angle whose sine is -8. Make sure your calculator is in the appropriate angle mode (degrees or radians).
5. The calculator will give you a value for θ, but keep in mind that the inverse sine function only gives you a single value between -90° and 90°. Since we want two angles between 0° and 360°, we need to find the second angle.
6. To find the second angle, subtract the first angle from 180°: 180° - θ.
- By subtracting the first angle from 180°, we get the second angle that is equivalent in sine value to the first angle. This is because the sine function is positive in both the first and second quadrants.
7. The two angles between 0° and 360° with a cosecant of -8 are θ and 180° - θ.

Remember to always verify your results and answers using the original equation and by checking the signs of the functions in the appropriate quadrants.