Find All Angles { \theta$}$ Between ${ 0^{\circ}\$} And ${ 180^{\circ}\$} Satisfying The Given Equation. Round Your Answer To One Decimal Place. (Enter Your Answers As A Comma-separated List.)$[ \cos (\theta) =

Introduction

Trigonometric equations are a fundamental part of mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of trigonometric equation involving the cosine function. We will find all angles $\theta$ between $0^{\circ}$ and $180^{\circ}$ that satisfy the given equation.

The Equation

The equation we will be solving is:

$$\cos (\theta) = \frac{1}{2}$$

This equation involves the cosine function, which is a periodic function that oscillates between -1 and 1. The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Understanding the Cosine Function

Before we proceed to solve the equation, let's take a closer look at the cosine function. The cosine function has a period of $360^{\circ}$, which means that it repeats itself every $360^{\circ}$. The cosine function is also an even function, which means that $\cos (-\theta) = \cos (\theta)$.

Solving the Equation

To solve the equation $\cos (\theta) = \frac{1}{2}$, we need to find all angles $\theta$ between $0^{\circ}$ and $180^{\circ}$ that satisfy this equation. We can start by finding the reference angle, which is the acute angle between the terminal side of the angle and the x-axis.

Finding the Reference Angle

The reference angle for $\cos (\theta) = \frac{1}{2}$ is $60^{\circ}$, since $\cos (60^{\circ}) = \frac{1}{2}$. This means that the angle $\theta$ is either $60^{\circ}$ or $120^{\circ}$.

Finding the Angles in the Given Range

Since we are looking for angles between $0^{\circ}$ and $180^{\circ}$, we can conclude that the only angle that satisfies the equation is $\theta = 60^{\circ}$.

Rounding the Answer

We are asked to round our answer to one decimal place. Since $60^{\circ}$ is already an integer, we can conclude that the rounded answer is still $60^{\circ}$.

Conclusion

In this article, we solved a trigonometric equation involving the cosine function. We found that the only angle between $0^{\circ}$ and $180^{\circ}$ that satisfies the equation is $\theta = 60^{\circ}$. We also rounded our answer to one decimal place, which is still $60^{\circ}$.

Final Answer

The final answer is $\boxed{60.0}$.

Additional Information

If you want to find the angles in the range $0^{\circ}$ to $360^{\circ}$, you can use the following formula:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$$

where $k$ is an integer. This will give you all the angles in the range $0^{\circ}$ to $360^{\circ}$ that satisfy the equation.

Example Use Case

Suppose you want to find the angles in the range $0^{\circ}$ to $360^{\circ}$ that satisfy the equation $\cos (\theta) = \frac{1}{2}$. You can use the formula above to find the angles:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$$

where $k$ is an integer. Plugging in $k = 0$, you get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right)$$

which gives you the angle $\theta = 60^{\circ}$. Plugging in $k = 1$, you get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ}$$

which gives you the angle $\theta = 120^{\circ}$. Plugging in $k = 2$, you get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 720^{\circ}$$

which gives you the angle $\theta = 180^{\circ}$. Plugging in $k = 3$, you get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 1080^{\circ}$$

which gives you the angle $\theta = 240^{\circ}$. Plugging in $k = 4$, you get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 1440^{\circ}$$

which gives you the angle $\theta = 300^{\circ}$. Plugging in $k = 5$, you get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 1800^{\circ}$$

which gives you the angle $\theta = 0^{\circ}$.

Code

Here is some sample code in Python to find the angles in the range $0^{\circ}$ to $360^{\circ}$ that satisfy the equation $\cos (\theta) = \frac{1}{2}$:

import math
def find_angles():
angles = []
for k in range(6):
theta = math.acos(0.5) + 2 * math.pi * k
angles.append(math.degrees(theta))
return angles
angles = find_angles()
print(angles)

$$$$

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These functions relate the angles of a triangle to the ratios of the lengths of its sides.

Q: What is the cosine function?

A: The cosine function is a trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is denoted by the symbol $\cos (\theta)$, where $\theta$ is the angle.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the trigonometric function and then use the inverse function to find the angle. For example, if you have the equation $\cos (\theta) = \frac{1}{2}$, you can use the inverse cosine function to find the angle $\theta$.

Q: What is the inverse cosine function?

A: The inverse cosine function is a function that takes a value between -1 and 1 and returns the angle whose cosine is that value. It is denoted by the symbol $\cos^{-1} (x)$, where $x$ is the value.

Q: How do I use the inverse cosine function to solve a trigonometric equation?

A: To use the inverse cosine function to solve a trigonometric equation, you need to isolate the trigonometric function and then take the inverse cosine of both sides of the equation. For example, if you have the equation $\cos (\theta) = \frac{1}{2}$, you can take the inverse cosine of both sides to get:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right)$$

Q: What is the reference angle?

A: The reference angle is the acute angle between the terminal side of the angle and the x-axis. It is used to find the angle whose cosine is a given value.

Q: How do I find the reference angle?

A: To find the reference angle, you need to use the inverse cosine function. For example, if you have the equation $\cos (\theta) = \frac{1}{2}$, you can use the inverse cosine function to find the reference angle:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right)$$

Q: What is the period of the cosine function?

A: The period of the cosine function is the distance between two consecutive points on the graph of the function that have the same y-coordinate. It is equal to $360^{\circ}$.

Q: How do I find the angles in a given range that satisfy a trigonometric equation?

A: To find the angles in a given range that satisfy a trigonometric equation, you need to use the inverse cosine function and the formula for the period of the cosine function. For example, if you have the equation $\cos (\theta) = \frac{1}{2}$ and you want to find the angles in the range $0^{\circ}$ to $360^{\circ}$ that satisfy the equation, you can use the following formula:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$$

where $k$ is an integer.

Q: What is the final answer to the trigonometric equation $\cos (\theta) = \frac{1}{2}$?

A: The final answer to the trigonometric equation $\cos (\theta) = \frac{1}{2}$ is $\boxed{60.0}$.

Q: How do I use the code to find the angles in a given range that satisfy a trigonometric equation?

A: To use the code to find the angles in a given range that satisfy a trigonometric equation, you need to modify the code to use the inverse cosine function and the formula for the period of the cosine function. For example, if you have the equation $\cos (\theta) = \frac{1}{2}$ and you want to find the angles in the range $0^{\circ}$ to $360^{\circ}$ that satisfy the equation, you can use the following code:

import math
def find_angles():
angles = []
for k in range(6):
theta = math.acos(0.5) + 2 * math.pi * k
angles.append(math.degrees(theta))
return angles
angles = find_angles()
print(angles)

This code uses the math.acos function to find the reference angle, and then uses the formula above to find the angles in the range $0^{\circ}$ to $360^{\circ}$. The math.degrees function is used to convert the angles from radians to degrees. The print statement is used to print the angles.