Problem 26

Question

Solve each equation for $0 \leq \theta<2 \pi$. $$ (\cos \theta)(\cos \theta+1)=0 $$

Step-by-Step Solution

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Answer

Solutions to the equation $(\cos \theta)(\cos \theta+1)=0$ are: $\theta=\pi/2$, $\theta=3\pi/2$, and $\theta=\pi$.

1Step 1: Set Each Factor to Zero

The equation can be broken down into two separate equations by setting each factor to zero: $cos \theta=0$ and $cos \theta+1=0$ which simplifies to $cos \theta=-1$.

2Step 2: Solve the First Equation

Solving $cos \theta=0$ for $\theta$ gives us two solutions in the range $0 \leq \theta<2 \pi$. By the unit circle or cos graph, these are $\theta=\pi/2$ and $\theta=3\pi/2$.

3Step 3: Solve the Second Equation

Next, we solve $cos \theta=-1$ for $\theta$. This gives us one solution in the given range, namely $\theta=\pi$.

4Step 4: Combine all the Solutions

Combining the solutions from Steps 2 and 3, the full solutions to the equation $(\cos \theta)(\cos \theta+1)=0$ in the range $0 \leq \theta<2 \pi$ are $\theta=\pi/2$, $\theta=3\pi/2$, and $\theta=\pi$.

Key Concepts

Unit CircleCosine FunctionSolving EquationsAngle Solutions

Unit Circle

The unit circle is a valuable tool in trigonometry, simplifying complex problems by providing a visual representation of angles and their corresponding trigonometric values. Imagine a circle with a radius of 1. The center is at the origin of a coordinate system. Each point on the circle represents an angle in radians measured from the positive x-axis. This allows us to easily understand the relationship between angles and the values of trigonometric functions like sine and cosine.

The x-coordinate of a point on the unit circle gives the cosine of the angle.
The y-coordinate represents the sine of the angle.

Understanding the unit circle is essential for solving trigonometric equations and finding angle solutions within a given range. When solving trigonometric equations, you can use the unit circle to pinpoint exact angle measures where specific trigonometric values occur.

Cosine Function

The cosine function is a fundamental part of trigonometry, and understanding it is key to solving trigonometric equations. This function associates an angle with a value between -1 and 1. The cosine of an angle ($\theta$) is the x-coordinate of the point where the corresponding angle intersects the unit circle.

When ($\theta$) is 0 or ($2\pi$), $\cos\theta$ equals 1.
At ($\pi$), $\cos\theta$ is -1.
For ($\pi/2$) and ($3\pi/2$), $\cos\theta$ is 0.

The cosine function is periodic with a period of ($2\pi$), meaning it repeats its values every ($2\pi$) radians. By understanding these key values and characteristics, you can use the cosine function effectively in solving equations.

Solving Equations

Solving trigonometric equations often involves transforming the equation into a simpler form, identifying where the function's value occurs, and then finding the corresponding angles. This process can be broken down into a few straightforward steps:

Break the equation into simpler parts. This involves setting each factor of the equation to zero.
Determine which trigonometric value is needed. In our example, we have ($\cos \theta = 0$) and ($\cos \theta = -1$).
Utilize tools like the unit circle or trigonometric graphs to find where these specific values occur within the specified range.

Once you complete these steps, you'll have found angle solutions that satisfy the original equation.

Angle Solutions

Finding the angle solutions for a trigonometric equation is the final step in the solving process. Once you've determined where a function equals a certain value, like ($\cos\theta = 0$) or ($\cos\theta = -1$), it's time to identify what those corresponding angles are within the required domain.

In our example, we use the unit circle and cosine graph to find:

For ($\cos \theta = 0$), the angles are ($\theta = \pi/2$) and ($\theta = 3\pi/2$).
For ($\cos \theta = -1$), the angle is ($\theta = \pi$).

By combining these results, we have all solutions for the equation within the range ($0 \leq \theta < 2\pi$). Remember, knowing how to use the unit circle and correcting interpreting trigonometric values is crucial to correctly finding the angle solutions.

Problem 26

Other exercises in this chapter

Problem 25

Given $\cos \theta=-\frac{15}{17}$ and \(180^{\circ}

View solution

Problem 26

Given $\cos \theta=-\frac{15}{17}$ and \(180^{\circ}

View solution

Problem 26

Find each exact value. Use a sum or difference identity. $$ \tan 75^{\circ} $$

View solution

Problem 26

Sketch a right triangle with $\theta$ as the measure of one acute angle. Find the other five trigonometric ratios of $\theta .$ $\cos \theta=\frac{7}{20}$

View solution