Math / Trigonometry

Question3) Find all solutions to the equations. List the solutions between $[0,2 \pi)$ .
24. $2 \sin ^{2} x-\cos x=1$

Studdy Solution

STEP 1

1. The equation $2 \sin^2 x - \cos x = 1$ involves trigonometric functions.
2. We will use trigonometric identities to simplify and solve the equation.
3. Solutions are required in the interval $[0, 2\pi)$ .

STEP 2

1. Use trigonometric identities to express the equation in terms of a single trigonometric function.
2. Solve the resulting trigonometric equation.
3. Find all solutions within the specified interval $[0, 2\pi)$ .

STEP 3

Use the Pythagorean identity $\sin^2 x = 1 - \cos^2 x$ to express $\sin^2 x$ in terms of $\cos x$ :
$2 \sin^2 x - \cos x = 1$ $2(1 - \cos^2 x) - \cos x = 1$

STEP 4

Simplify the equation:
$2 - 2 \cos^2 x - \cos x = 1$
Subtract 1 from both sides:
$2 - 1 - 2 \cos^2 x - \cos x = 0$ $1 - 2 \cos^2 x - \cos x = 0$

STEP 5

Rearrange the equation to form a quadratic in terms of $\cos x$ :
$-2 \cos^2 x - \cos x + 1 = 0$
Multiply the entire equation by $-1$ to make it easier to solve:
$2 \cos^2 x + \cos x - 1 = 0$

STEP 6

Factor the quadratic equation:
$2 \cos^2 x + \cos x - 1 = 0$
This can be factored as:
$(2 \cos x - 1)(\cos x + 1) = 0$

STEP 7

Solve each factor for $\cos x$ :
1. $2 \cos x - 1 = 0$ $2 \cos x = 1$ $\cos x = \frac{1}{2}$
2. $\cos x + 1 = 0$ $\cos x = -1$

STEP 8

Find all solutions for $\cos x = \frac{1}{2}$ within $[0, 2\pi)$ :
$x = \frac{\pi}{3}, \frac{5\pi}{3}$
Find all solutions for $\cos x = -1$ within $[0, 2\pi)$ :
$x = \pi$
The solutions within the interval $[0, 2\pi)$ are:
$x = \frac{\pi}{3}, \frac{5\pi}{3}, \pi$

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