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Find all angles,
0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree.

6sin^(2)theta-5sin theta+4=3
Answer:
theta=

Find all angles, $0^{\circ} \leq \theta<360^{\circ}$ , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $6 \sin ^{2} \theta-5 \sin \theta+4=3$ $\newline$ Answer: $\theta=$

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Find the roots of factored polynomials

Full solution

Q. Find all angles,

0^{\circ} \leq \theta<360^{\circ}

, that satisfy the equation below, to the nearest tenth of a degree.

\newline

6 \sin ^{2} \theta-5 \sin \theta+4=3

\newline

Answer:

\theta=

Simplify Equation: First, we need to simplify the given equation by moving all terms to one side to set the equation to zero. $\newline$ $6\sin^2\theta - 5\sin \theta + 4 - 3 = 0$ $\newline$ $6\sin^2\theta - 5\sin \theta + 1 = 0$
Factor Quadratic: Next, we will factor the quadratic equation in terms of $\sin(\theta)$ . We are looking for two numbers that multiply to $6\times1=6$ and add up to $-5$ . The numbers that satisfy these conditions are $-3$ and $-2$ . So we can write the equation as $(3\sin(\theta) - 1)(2\sin(\theta) - 1) = 0$
Solve for $\sin(\theta)$ : Now, we will solve each factor for $\sin(\theta)$ . First, for $3\sin(\theta) - 1 = 0$ , we get $\sin(\theta) = \frac{1}{3}$ . Second, for $2\sin(\theta) - 1 = 0$ , we get $\sin(\theta) = \frac{1}{2}$ .
Find Angles ( $1$ / $3$ ): We will find the angles for $\sin(\theta) = \frac{1}{3}$ . Since the sine function is positive, the angles must be in the first or second quadrant. Using a calculator, we find $\theta \approx \arcsin(\frac{1}{3})$ . Theta $\approx 19.5$ degrees or $180 - 19.5 = 160.5$ degrees.
Find Angles $(1/2)$ : Next, we will find the angles for $\sin(\theta) = \frac{1}{2}$ . This is a known value on the unit circle, corresponding to $\theta = 30$ degrees and $\theta = 150$ degrees.
Final Angles: We have found all possible angles for the given equation within the range $0 \leq \theta < 360$ degrees. $\newline$ The angles are approximately $19.5$ degrees, $160.5$ degrees, $30$ degrees, and $150$ degrees.

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