Find all solutions in the inerval [0,360)

Question

cos^2(theta)+8cos(theta)+5=0

Answer in degrees

Answer 1

Cos²θ + 8Cosθ + 5 = 0 

 

Method 1

We will use "completing the square" method to factor this equation. Rewrite the equation as follows:

Cos²θ + 8Cosθ = -5

 

Add square of half the coefficient of Cosθ to both sides:

Cos²θ + 8Cosθ + 4²= -5 + 4²

 

Factoring, we have: (Cosθ + 4)² = -5 + 16

 

(Cosθ + 4)² = 11

 

Thus, Cosθ + 4 = ±√11 by taking square roots of both sides.

Cosθ = -4 ±√11

 

θ = Cos^-1(-4 +√11) and θ = Cos^-1(-4 -√11)

 

θ = Cos^-1(-0.6834) and θ = Cos^-1(-7.3166)

 

Since the range of the Cosine function is [-1,1], Cos^-1(-7.3166) cannot be a solution

 

Therefore Cos^-1(-4 +√11) = 133.1^o and 226.9^o - Remember that the Cosine function is negative in Quadrants II and III

.

Since we are restricted to [0,360) interval, the possible solutions are 133.1^o and 226.9^o, (133.1 + 360n)^o, (226 + 360n)^o, where n = 1, 2, 3, ...

 

OR

 

 

(133.1 + 360n)^o, (226.9 + 360n)^o, where n = 0, 1, 2, ...

 

Method 2

You can factor the given equation by using the quadratic formula. In this case, a = 1, b = 8, and c = 5.

Try this and compare your answers to the ones above.

 

Method 3

Let Cosθ = x. Then, rewrite the equation as x²+8x+5=0.

Solve and equate the x value to Cosθ, and find θ