solve equation for solutionn over interval [0,360): cot(è)+4csc(è)=5
To solve the equation cot(θ) + 4csc(θ) = 5 over the interval [0, 360), you will need to follow these steps:
1. Rewrite cot(θ) and csc(θ) in terms of sine and cosine functions.
- cot(θ) = cos(θ) / sin(θ)
- csc(θ) = 1 / sin(θ)
2. Substitute the rewritten expressions into the original equation.
- cos(θ) / sin(θ) + 4 / sin(θ) = 5
3. Multiply the entire equation by sin(θ) to eliminate the denominators.
- cos(θ) + 4 = 5sin(θ)
4. Rearrange the equation in terms of sine and cosine.
- cos(θ) - 5sin(θ) = -4
5. Square both sides of the equation to eliminate the negative sign.
- (cos(θ) - 5sin(θ))^2 = (-4)^2
- cos^2(θ) - 10cos(θ)sin(θ) + 25sin^2(θ) = 16
6. Apply the trigonometric identity cos^2(θ) + sin^2(θ) = 1 to simplify the equation.
- 1 - 10cos(θ)sin(θ) + 25sin^2(θ) = 16
- 25sin^2(θ) - 10cos(θ)sin(θ) - 15 = 0
7. Rearrange the equation to make it a quadratic equation in terms of sin(θ).
- 25sin^2(θ) - 10cos(θ)sin(θ) - 15 = 0
- 25sin^2(θ) - 10sin(θ)cos(θ) - 15 = 0
8. Factorize or use the quadratic formula to find the values of sin(θ).
- Let sin(θ) = x: 25x^2 - 10xcos(θ) - 15 = 0
9. Solve the quadratic equation.
10. Once you've found the values of sin(θ), find the corresponding values of θ by using the inverse sine function (sin^(-1)).
11. Check if the solutions are within the desired interval [0, 360). If they are, those values of θ are the solutions to the original equation.
By following these steps, you should be able to solve the equation cot(θ) + 4csc(θ) = 5 over the interval [0, 360).