Find all solutions in the interval 0 degrees<θ<360 degrees. If rounding necessary, round to the nearest tenth of a degree. 17sec2 θ − 15tanθsecθ − 15 = 0

17sec^2θ - 15secθ tanθ - 15 = 0

15secθ tanθ = 17sec^2θ - 15
225 sec^2θ tan^2θ = 289sec^4θ - 510sec^2θ + 225
225sec^4θ - 225sec^2θ = 289sec^4θ - 510sec^2θ + 225
64sec^4θ - 285sec^2θ + 225 = 0

That's just a quadratic in sec^2θ, so just solve it and you have your solution candidates.

However, because we squared things, there may be spurious solutions, so you have to check the values in the original equation.

In radians, I get .159,1.000,2.141,2.982

17sec2 θ − 15tanθsecθ − 15 = 0

17/cos^2 Ø - 15(sinØ/cosØ)(1/cosØ) - 15 = 0
times cos^2 Ø
17 - 15sinØ - 15cos^2 Ø = 0
17 - 15sinØ - 15(1 - sin^2 Ø) = 0
15sin^2 Ø - 15sinØ + 2 = 0
sinØ = (15 ± √105)/30
sinØ = .158435 or sinØ = .841565
Ø = 9.1° or 170.9° or Ø = 57.3° or 122.7°

To find all the solutions for the given equation 17sec^2θ - 15tanθsecθ - 15 = 0 in the interval 0 degrees ≤ θ ≤ 360 degrees, we can follow these steps:

Step 1: Simplify the equation if possible.
Since sec^2θ = 1 + tan^2θ, we can rewrite the equation as follows:
17(1 + tan^2θ) - 15tanθsecθ - 15 = 0

Step 2: Substitute secθ with 1/cosθ and simplify the equation further.
17(1 + tan^2θ) - 15tanθ(1/cosθ) - 15 = 0
17 + 17tan^2θ - 15tanθ/cosθ - 15 = 0
17 + 17tan^2θ - 15sinθ/cosθ - 15 = 0
17 + 17tan^2θ - 15sinθ/cos^2θ = 15

Step 3: Combine like terms on one side of the equation.
17tan^2θ - 15sinθ/cos^2θ = -2

Step 4: Multiply both sides of the equation by cos^2θ to eliminate the denominator.
17tan^2θ - 15sinθ = -2cos^2θ

Step 5: Use trigonometric identities to simplify the equation.
17sin^2θ/cos^2θ - 15sinθ = -2cos^2θ
(17sin^2θ - 15sinθcos^2θ)/cos^2θ = -2cos^2θ
17sin^2θ - 15sinθcos^2θ + 2cos^4θ = 0

Step 6: Factor the equation if possible.
Let's substitute sinθ with x for better readability.
17x^2 - 15x(1 - x^2) + 2(1 - x^2)^2 = 0
17x^2 - 15x + 15x^3 + 2 - 4x^2 + 2x^4 = 0

Step 7: Rearrange the equation in descending powers of x.
2x^4 - 4x^2 + 17x^2 + 15x^3 - 15x + 2 = 0
2x^4 + 13x^3 + 13x^2 - 15x + 2 = 0

Step 8: Solve the quartic equation.
Unfortunately, solving a general quartic equation can be quite complex and doesn't have a simple method. However, you can use numerical methods or graphing calculators to find the solutions. Additionally, using software like WolframAlpha can provide an approximate solution if you input the equation.

Once you have obtained the solutions for x, you can convert them to values for θ by using the inverse trigonometric functions.

Note: Since solving a quartic equation is quite sophisticated, using software or calculators is recommended to obtain accurate and approximate solutions.