Problem 925 Solve \(2 \sin ^{2} \theta+3 \co... [FREE SOLUTION]

Chapter 29: Problem 925

Solve $2 \sin ^{2} \theta+3 \cos \theta-3=0$ for $\theta$ if $0 \leq \theta<360^{\circ}$.

Short Answer

Expert verified

The possible values for $\theta$ that satisfy the given equation are: $\theta = 0^{\circ}, 60^{\circ}, 300^{\circ}$.

Step by step solution

Use the identity $\sin^2 \theta = 1 - \cos^2 \theta$

We are given the equation $2\sin^2 \theta + 3\cos\theta - 3 = 0$. To convert this equation to a single trigonometric function, we can use the identity $\sin^2 \theta = 1 - \cos^2 \theta$. Substituting this identity into the equation, we get: $$2(1 - \cos^2 \theta) + 3\cos\theta - 3 = 0.$$

Simplify the equation

Now, we need to simplify the equation. Expanding and simplifying, we have: $$2 - 2\cos^2 \theta + 3\cos\theta - 3 = 0.$$ This can be written as a quadratic equation in terms of the cosine function: $$2\cos^2 \theta - 3\cos\theta + 1 = 0.$$

Solve the quadratic equation

To find the values of $\cos\theta$, we need to solve the quadratic equation $2\cos^2 \theta - 3\cos\theta + 1 = 0$. Let's consider $x = \cos \theta$. The equation can be written as: $$2x^2 - 3x + 1 = 0.$$ By factoring the quadratic equation, we get: $$(2x - 1)(x - 1) = 0.$$ This equation has two possible solutions: $$x_1 = \frac{1}{2}, \quad x_2 = 1.$$

Find the values of $\theta$

Since $x = \cos \theta$, we have two possible values for the cosine function: $$\cos \theta_1 = \frac{1}{2}, \quad \cos \theta_2 = 1.$$ We need to find the angles $\theta$ that satisfy these values of cosine function within the domain $0 \leq \theta < 360^{\circ}$. For $\cos \theta_1 = \frac{1}{2}$: $$\theta_1 = 60^{\circ}, \quad \theta_2 = 300^{\circ}.$$ For $\cos \theta_2 = 1$: $$\theta_3 = 0^{\circ}.$$ Therefore, the possible values for $\theta$ that satisfy the given equation are: $$\theta = 0^{\circ}, 60^{\circ}, 300^{\circ}.$$

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91影视

Solve \(2 \sin ^{2} \theta+3 \cos \theta-3=0\) for \(\theta\) if \(0 \leq \theta<360^{\circ}\).

Short Answer

Step by step solution

Use the identity \(\sin^2 \theta = 1 - \cos^2 \theta\)

Simplify the equation

Solve the quadratic equation

Find the values of \(\theta\)

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