Find all angles, 0^\circ\le \theta<360^\circ0 ∘ ≤θ<360 ∘ , that satisfy the equation below, to the nearest 10th of a degree. \tan^2\theta -5\tan\theta -6=0 tan 2 θ−5tanθ−6=0

Question

Quadratic Trig Equations

Answer 1

Find all the solutions to tan²(θ) - 5tan(θ) - 6 = 0 in the interval 0° ≤ θ ≤ 360^°

Notice the equation's similarity to x² - 5x - 6 = 0. The only difference is that x = tan(θ)

We can factor the left side of that quadratic equation so that

(x + 1)(x - 6) = 0

and therefore x = -1 or 6.

So it should be easier to see that tan²(θ) - 5tan(θ) - 6 factors to [tan(θ) + 1][tan(θ) - 6]

we can substitute into the trig equation to get

[tan(θ) + 1][tan(θ) - 6] = 0

which implies that

tan(θ) = -1 or 6

Using the calculator's inverse tangent on tan(θ) = -1 we get θ = -45^°

Using the calculator's inverse tangent on tan(θ) = 6 we get θ = 80.54^°

To finish, you need to find all the angles that are coterminal to -45^° and 80.54^° in the specified interval.