Question

Solve the equation for exact solutions in the interval [0°,360°). Use an algebraic method.

6 sec 2θtan θ=8 tan θ
The solution set is___

Answer 1

To solve the equation 6 sec^2θtanθ = 8 tanθ for exact solutions in the interval [0°,360°), simplify the equation, find the square root of both sides, and use the inverse tangent function to find the exact solutions θ.

Step-by-step explanation:

To solve the equation 6 sec2θtanθ = 8 tanθ for exact solutions in the interval [0°,360°), we will first simplify the equation using trigonometric identities.

Let's start by dividing both sides of the equation by tanθ:

1. 6 sec2θtanθ / tanθ = 8 tanθ / tanθ
2. 6 sec2θ = 8

Next, we can rewrite sec2θ as 1 + tan2θ by using the Pythagorean identity:

1. 6 (1 + tan2θ) = 8
2. 6 + 6tan2θ = 8
3. 6tan2θ = 2
4. tan2θ = 2/6
5. tan2θ = 1/3

Now, take the square root of both sides:

1. tanθ = ±√(1/3)

The exact solutions for θ can be found by using the inverse tangent function:

1. θ = tan-1(±√(1/3))
2. θ ≈ 19.47° or θ ≈ 199.47°