Problem 12 - Algebra and Trigonometry

Question

\(5-16=\) Solve the given equation. $$ \cos \theta=0.32 $$

Step-by-Step Solution

Answer

\( \theta \approx 71.34^\circ \) or \( 288.66^\circ \).

Step 1: Understand the Problem

We need to find the angle \( \theta \) that satisfies the equation \( \cos \theta = 0.32 \). The goal is to determine the value of \( \theta \) such that when you take the cosine of it, the result is 0.32.

Step 2: Use Inverse Trigonometric Function

To find \( \theta \), use the inverse cosine function (also called arccosine). This is denoted as \( \theta = \cos^{-1}(0.32) \). This function will give us the angle whose cosine is 0.32.

Step 3: Calculate \( \theta \) Using a Calculator

Use a calculator to compute \( \theta = \cos^{-1}(0.32) \). Make sure the calculator is in degree mode if you want the answer in degrees. Inputting this will give you \( \theta \approx 71.34^\circ \).

Step 4: Consider the Range of Cosine

Since the cosine function is positive both in the first and the fourth quadrants, you should also consider the angle in the fourth quadrant, which would be \( 360^\circ - 71.34^\circ = 288.66^\circ \). Therefore, the possible angles \( \theta \) can be approximately \( 71.34^\circ \) and \( 288.66^\circ \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It describes the relationship between the angle of a right triangle and the length of the adjacent side to the hypotenuse. This relationship is defined by the cosine formula:

• \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)

The cosine of an angle can take values between -1 and 1, as it represents a projection of the unit circle radius on the horizontal axis.
When solving equations like \( \cos \theta = 0.32 \), you're essentially looking for the angle \( \theta \) that makes this horizontal projection equal 0.32.
Cosine values are periodic, repeating every 360 degrees or \( 2\pi \) radians. Therefore, when you find one angle solution, you can often derive additional solutions by considering symmetry and periodicity.

Arccosine

The arccosine function is the inverse of the cosine function. It allows us to find the angle when the cosine value is known. The notation typically appears as \( \cos^{-1}(x) \) or simply arccos(x).
Applying the arccosine function to \( 0.32 \) means you're finding an angle \( \theta \) such that \( \cos \theta = 0.32 \).

• The principal value of \( \theta \) is typically given in the range of \( [0, \pi] \) radians or \( [0, 180^\circ] \).
• For the equation \( \cos \theta = 0.32 \), the calculator will directly give \( \theta \approx 71.34^\circ \).

However, cosine is positive in both the first and fourth quadrants. This is why \( 288.66^\circ \) (or 360 minus the principal value) is also a solution. Understanding the output of arccosine is crucial for identifying all possible angle solutions for cosine equations.

Unit Circle

The unit circle is a valuable tool in trigonometry, as it sets the foundation for understanding the behavior of trigonometric functions such as cosine and sine. This circle is centered at the origin of a coordinate plane and has a radius of 1.

• Angles in the unit circle are measured from the positive x-axis, and each point on the circle can be expressed as \((\cos \theta, \sin \theta)\).
• For \( \cos \theta = 0.32 \), you're essentially determining the x-coordinate on the unit circle where the angle \( \theta \) corresponds to \( 0.32 \).

In practice, this means that two different points on the circle may yield the same cosine value due to symmetry.
By considering both the first quadrant (where angles increase from \( 0 \) to \( 90^\circ \)) and the fourth quadrant (where angles can be calculated as \( 360^\circ - \theta \)), you find multiple solutions to trigonometric equations. Effortlessly visualizing the relationships on the unit circle can enhance your understanding and ability to solve complex trigonometric problems.