Question

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possiblesolutions by including all answers in [0, 2pi) and indicating the remaining answers by using n to represent any integer. Round your answerto four decimal places, if necessary. If there is no solution, indicate "No Solution."cos^2(x) + 2cos(x) = 5cos(x) + 4

Answer 1

x = 0°, 180°

Step-by-step explanation:

cos²(x) + 2cos(x) = 5cos(x) + 4

let cos x = y

y² + 2y = 5y + 4

collect like terms:

y² + 2y - 5y - 4 = 0

y² - 3y - 4 = 0

Using factorisation method to solve the above equation:

The factors of -4 that we can sum up to get -3 are: -4 and +1

y² - 4y + y - 4 = 0

y(y +4) + 1(y - 4) = 0

(y + 1) (y - 4) = 0

y + 1 = 0 or y - 4 = 0

y = -1 or y = 4

Recall cosx = y

This means:

cos x = -1 or cos x = 4

We were given an interval of [0, 2π). This means the interval is from 0° ≤ x < 360°

Greater than 0° but less than 360°

cosx = -1

x = arc cos (-1)

x = 180 degrees

cos x = 4

x = arc cos (4)

x = 0 degrees

x = 0°, 180°