Problem 21 - Algebra 2 and Trigonometry

Question

In \(21-24,\) find, to the nearest tenth, the degree measures of all \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that make the equation true. $$ 8 \cos \theta=3-4 \cos \theta $$

Step-by-Step Solution

Answer

The angles are \(\theta \approx 75.5^{\circ}\) and \(\theta \approx 284.5^{\circ}\).

Step 1: Understand the Equation

Start with the given equation: \[ 8 \cos \theta = 3 - 4 \cos \theta \]Recognize that this is a trigonometric equation in terms of cos(\(\theta\)). We will first get all cos(\(\theta\)) terms on one side to solve for \(\theta\).

Step 2: Rearrange the Equation

Add \(4 \cos \theta\) to both sides to have all \(\cos \theta\) terms on one side of the equation:\[ 8 \cos \theta + 4 \cos \theta = 3 \]This simplifies to:\[ 12 \cos \theta = 3 \]

Step 3: Solve for \(\cos \theta\)

Divide both sides by 12 to isolate \(\cos \theta\):\[ \cos \theta = \frac{3}{12} = \frac{1}{4} \]

Step 4: Find \(\theta\) using Inverse Cosine

To find the principal value of \(\theta\), use the inverse cosine function:\[ \theta = \cos^{-1}\left( \frac{1}{4} \right) \]Calculate this using a calculator to find:\[ \theta \approx 75.5^{\circ} \]

Step 5: Find All Solutions in the Given Interval

Since cosine is positive in the first and fourth quadrants, we need to find another angle where cosine is \(\frac{1}{4}\). In the fourth quadrant, the angle will be:\[ \theta = 360^{\circ} - 75.5^{\circ} = 284.5^{\circ} \]

Step 6: Solution Summary

Thus, we find two angles within the interval \(0^{\circ} \leq \theta < 360^{\circ}\) that satisfy the equation:\[ \theta \approx 75.5^{\circ} \] and \[ \theta \approx 284.5^{\circ} \]

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry for finding the angles when the trigonometric ratios are known. When we know the value of a trigonometric function like cosine and want to find the corresponding angle, we use an inverse function.
For cosine, this is written as \( \cos^{-1} \) or arccos. It helps us find the angle \( \theta \) whose cosine is a specific value. For example, if \( \cos \theta = \frac{1}{4} \), then \( \theta = \cos^{-1} \left( \frac{1}{4} \right) \). This allows us to calculate the angle using a calculator or trigonometric tables.
Inverse trigonometric functions are crucial because they help us "undo" a trigonometric function. There are similar functions for sine and tangent: \( \sin^{-1} \) and \( \tan^{-1} \). Each of these functions has a specific range for which they return the principal value of the angle.

Cosine Function

The cosine function is one of the fundamental trigonometric functions used to relate angles to side lengths of a right triangle. It is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

• The cosine of an angle \( \theta \) can vary between -1 and 1.
• The cosine function is periodic, repeating its values every 360 degrees or \( 2\pi \) radians.
• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).

If we graph the cosine function, it appears as a wave oscillating between -1 and 1.
The function's positive values are found in the first and fourth quadrants of the unit circle, which is why, when solving for multiple solutions to a cosine equation within a 360-degree interval, we consider both these quadrants.
Understanding these properties also helps in solving equations and modeling periodic phenomena, such as sound waves or daylight patterns.

Angle Measurement

Angle measurement is crucial in understanding and solving trigonometric equations. An angle is measured in degrees or radians, which are related by the fact that 360 degrees is equal to \( 2\pi \) radians.
When solving trigonometric equations, it is important to consider the interval over which solutions are required. In many exercises, like the original exercise, we look for solutions within \(0^{\circ} \leq \theta < 360^{\circ}\). This ensures we account for one complete cycle of trigonometric function values.
Degrees divide a full circle into 360 parts, making it a convenient unit for most practical applications. Meanwhile, radians provide a direct relationship between arc length and radius, which is often more useful in calculus.

• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \).
• To convert from radians to degrees, multiply by \( \frac{180}{\pi} \).

Understanding these conversions and choosing the correct interval is key to accurately interpreting solutions to trigonometric problems.