Solving Trigonometric Equations in the Unit Circle

Oct 13, 2025 by leolavanderia.com.br 125 views

Solving Trigonometric Equations in the Unit Circle

The unit circle is a fundamental concept in trigonometry, and it plays a crucial role in solving trigonometric equations. In this article, we will explore the values of $\theta$ that satisfy each of the given equations in the unit circle.

a. $\sin(\theta) = \frac{1}{2}$

To solve the equation $\sin(\theta) = \frac{1}{2}$ , we need to find the values of $\theta$ that lie on the unit circle and have a sine value of $\frac{1}{2}$ . The sine function is positive in the first and second quadrants, and the reference angle for $\sin(\theta) = \frac{1}{2}$ is $30^\circ$ or $\frac{\pi}{6}$ radians.

Using the unit circle, we can find the values of $\theta$ that satisfy the equation $\sin(\theta) = \frac{1}{2}$ .

In the first quadrant, the value of $\theta$ is $\frac{\pi}{6}$ .
In the second quadrant, the value of $\theta$ is $\frac{5\pi}{6}$ .

Therefore, the values of $\theta$ that satisfy the equation $\sin(\theta) = \frac{1}{2}$ are $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ .

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

To solve the equation $\cos(\theta) = -\frac{1}{2}$ , we need to find the values of $\theta$ that lie on the unit circle and have a cosine value of $-\frac{1}{2}$ . The cosine function is negative in the second and third quadrants, and the reference angle for $\cos(\theta) = -\frac{1}{2}$ is $60^\circ$ or $\frac{\pi}{3}$ radians.

Using the unit circle, we can find the values of $\theta$ that satisfy the equation $\cos(\theta) = -\frac{1}{2}$ .

In the second quadrant, the value of $\theta$ is $\frac{2\pi}{3}$ .
In the third quadrant, the value of $\theta$ is $\frac{4\pi}{3}$ .

Therefore, the values of $\theta$ that satisfy the equation $\cos(\theta) = -\frac{1}{2}$ are $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ .

c. $\sin(\theta) = -\frac{\sqrt{3}}{2}$

To solve the equation $\sin(\theta) = -\frac{\sqrt{3}}{2}$ , we need to find the values of $\theta$ that lie on the unit circle and have a sine value of $-\frac{\sqrt{3}}{2}$ . The sine function is negative in the third and fourth quadrants, and the reference angle for $\sin(\theta) = -\frac{\sqrt{3}}{2}$ is $60^\circ$ or $\frac{\pi}{3}$ radians.

Using the unit circle, we can find the values of $\theta$ that satisfy the equation $\sin(\theta) = -\frac{\sqrt{3}}{2}$ .

In the third quadrant, the value of $\theta$ is $\frac{4\pi}{3}$ .
In the fourth quadrant, the value of $\theta$ is $\frac{5\pi}{3}$ .

Therefore, the values of $\theta$ that satisfy the equation $\sin(\theta) = -\frac{\sqrt{3}}{2}$ are $\frac{4\pi}{3}$ and $\frac{5\pi}{3}$ .

d. $\cos(\theta) = \frac{\sqrt{3}}{2}$

To solve the equation $\cos(\theta) = \frac{\sqrt{3}}{2}$ , we need to find the values of $\theta$ that lie on the unit circle and have a cosine value of $\frac{\sqrt{3}}{2}$ . The cosine function is positive in the first and fourth quadrants, and the reference angle for $\cos(\theta) = \frac{\sqrt{3}}{2}$ is $30^\circ$ or $\frac{\pi}{6}$ radians.

Using the unit circle, we can find the values of $\theta$ that satisfy the equation $\cos(\theta) = \frac{\sqrt{3}}{2}$ .

In the first quadrant, the value of $\theta$ is $\frac{\pi}{6}$ .
In the fourth quadrant, the value of $\theta$ is $\frac{11\pi}{6}$ .

Therefore, the values of $\theta$ that satisfy the equation $\cos(\theta) = \frac{\sqrt{3}}{2}$ are $\frac{\pi}{6}$ and $\frac{11\pi}{6}$ .

Conclusion

In this article, we have solved four trigonometric equations in the unit circle. We have found the values of $\theta$ that satisfy each of the given equations and have used the unit circle to visualize the solutions. The values of $\theta$ that satisfy each equation are:

$\sin(\theta) = \frac{1}{2}$ : $\frac{\pi}{6}$ and $\frac{5\pi}{6}$
$\cos(\theta) = -\frac{1}{2}$ : $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$
$\sin(\theta) = -\frac{\sqrt{3}}{2}$ : $\frac{4\pi}{3}$ and $\frac{5\pi}{3}$
$\cos(\theta) = \frac{\sqrt{3}}{2}$ : $\frac{\pi}{6}$ and $\frac{11\pi}{6}$

These values of $\theta$ can be used to solve a variety of trigonometric problems and can be applied to real-world situations where trigonometry is used.
Frequently Asked Questions (FAQs) about Trigonometric Equations in the Unit Circle

In this article, we will answer some of the most frequently asked questions about trigonometric equations in the unit circle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. It is a fundamental concept in trigonometry and is used to define the trigonometric functions sine, cosine, and tangent.

Q: How do I use the unit circle to solve trigonometric equations?

A: To use the unit circle to solve trigonometric equations, you need to identify the quadrant in which the angle lies and then use the reference angle to find the value of the trigonometric function.

Q: What is a reference angle?

A: A reference angle is an angle that has the same value as the trigonometric function as the original angle, but is in a different quadrant. For example, if the original angle is in the second quadrant, the reference angle is in the first quadrant and has the same value as the trigonometric function.

Q: How do I find the reference angle?

A: To find the reference angle, you need to subtract the angle from 360 degrees (or 2π radians) and then take the absolute value of the result.

Q: What are the values of θ that satisfy the equation sin(θ) = 1/2?

A: The values of θ that satisfy the equation sin(θ) = 1/2 are π/6 and 5π/6.

Q: What are the values of θ that satisfy the equation cos(θ) = -1/2?

A: The values of θ that satisfy the equation cos(θ) = -1/2 are 2π/3 and 4π/3.

Q: What are the values of θ that satisfy the equation sin(θ) = -√3/2?

A: The values of θ that satisfy the equation sin(θ) = -√3/2 are 4π/3 and 5π/3.

Q: What are the values of θ that satisfy the equation cos(θ) = √3/2?

A: The values of θ that satisfy the equation cos(θ) = √3/2 are π/6 and 11π/6.

Q: How do I use the unit circle to find the values of θ that satisfy a trigonometric equation?

A: To use the unit circle to find the values of θ that satisfy a trigonometric equation, you need to identify the quadrant in which the angle lies and then use the reference angle to find the value of the trigonometric function.

Q: What are some common trigonometric equations that can be solved using the unit circle?

A: Some common trigonometric equations that can be solved using the unit circle include:

sin(θ) = 1/2
cos(θ) = -1/2
sin(θ) = -√3/2
cos(θ) = √3/2

Q: How do I apply the unit circle to real-world problems?

A: The unit circle can be applied to a variety of real-world problems, including:

Calculating the height of a building or a mountain
Determining the distance between two points on a map
Finding the angle of elevation or depression of a line
Calculating the area or perimeter of a triangle or other polygon

Q: What are some common mistakes to avoid when using the unit circle?

A: Some common mistakes to avoid when using the unit circle include:

Failing to identify the quadrant in which the angle lies
Using the wrong reference angle
Failing to take the absolute value of the result
Failing to consider the periodic nature of the trigonometric functions

By following these tips and avoiding common mistakes, you can use the unit circle to solve a variety of trigonometric equations and apply trigonometry to real-world problems.