Worksheet D: Trigonometric InequalitiesName:Directions: For Problems $1-3$, Indicate/highlight The Portion Of The Unit Circle That Satisfies The Given Inequality. Then, Write The Solution In Interval Notation Or As An Inequality.1. What Are

Introduction

Trigonometric inequalities are mathematical expressions that involve trigonometric functions, such as sine, cosine, and tangent, and are used to describe relationships between angles and side lengths of triangles. In this worksheet, we will explore how to solve trigonometric inequalities and graph the solutions on the unit circle.

Understanding Trigonometric Inequalities

A trigonometric inequality is an expression that involves a trigonometric function and is written in the form:

$$\sin x > \cos x$$

or

$$\tan x < \sin x$$

where $x$ is an angle measured in radians.

Solving Trigonometric Inequalities

To solve a trigonometric inequality, we need to find the values of $x$ that satisfy the inequality. We can do this by using the unit circle and graphing the solutions.

1. Solving $\sin x > \cos x$

To solve the inequality $\sin x > \cos x$, we need to find the values of $x$ for which the sine function is greater than the cosine function.

Step 1: Graph the Sine and Cosine Functions

The sine and cosine functions are periodic with a period of $2\pi$. We can graph the sine and cosine functions on the unit circle by plotting the points $(\cos x, \sin x)$ for $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.

Step 2: Identify the Regions Where the Sine Function is Greater Than the Cosine Function

From the graph, we can see that the sine function is greater than the cosine function in the regions where the sine function is positive and the cosine function is negative.

Step 3: Write the Solution in Interval Notation

The solution to the inequality $\sin x > \cos x$ is the set of all values of $x$ that satisfy the inequality. We can write this solution in interval notation as:

$$\left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \cup \left(\frac{5\pi}{4}, \frac{7\pi}{4}\right)$$

2. Solving $\tan x < \sin x$

To solve the inequality $\tan x < \sin x$, we need to find the values of $x$ for which the tangent function is less than the sine function.

Step 1: Graph the Tangent and Sine Functions

The tangent function is periodic with a period of $\pi$. We can graph the tangent and sine functions on the unit circle by plotting the points $(\tan x, \sin x)$ for $x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi$.

Step 2: Identify the Regions Where the Tangent Function is Less Than the Sine Function

From the graph, we can see that the tangent function is less than the sine function in the regions where the tangent function is negative and the sine function is positive.

Step 3: Write the Solution in Interval Notation

The solution to the inequality $\tan x < \sin x$ is the set of all values of $x$ that satisfy the inequality. We can write this solution in interval notation as:

$$\left(0, \frac{\pi}{4}\right) \cup \left(\frac{3\pi}{4}, \pi\right)$$

3. Solving $\sin x > \cos x$ and $\tan x < \sin x$

To solve the system of inequalities $\sin x > \cos x$ and $\tan x < \sin x$, we need to find the values of $x$ that satisfy both inequalities.

Step 1: Graph the Sine, Cosine, and Tangent Functions

We can graph the sine, cosine, and tangent functions on the unit circle by plotting the points $(\cos x, \sin x)$ and $(\tan x, \sin x)$ for $x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi$.

Step 2: Identify the Regions Where Both Inequalities are Satisfied

From the graph, we can see that both inequalities are satisfied in the regions where the sine function is positive and the cosine function is negative, and the tangent function is negative and the sine function is positive.

Step 3: Write the Solution in Interval Notation

The solution to the system of inequalities $\sin x > \cos x$ and $\tan x < \sin x$ is the set of all values of $x$ that satisfy both inequalities. We can write this solution in interval notation as:

$$\left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \cup \left(\frac{5\pi}{4}, \frac{7\pi}{4}\right)$$

Conclusion

In this worksheet, we have explored how to solve trigonometric inequalities and graph the solutions on the unit circle. We have seen how to use the unit circle to identify the regions where the trigonometric functions are positive or negative, and how to write the solutions in interval notation. We have also seen how to solve systems of trigonometric inequalities by graphing the functions and identifying the regions where both inequalities are satisfied.

Practice Problems

1. Solve the inequality $\cos x > \sin x$.
2. Solve the inequality $\tan x > \cos x$.
3. Solve the system of inequalities $\sin x > \cos x$ and $\tan x > \sin x$.

Answers

1. $\left(\frac{3\pi}{4}, \frac{7\pi}{4}\right)$
2. $\left(\frac{\pi}{4}, \frac{5\pi}{4}\right)$
3. $\left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \cup \left(\frac{5\pi}{4}, \frac{7\pi}{4}\right)$
   Q&A: Trigonometric Inequalities =====================================

Frequently Asked Questions

Q: What is a trigonometric inequality?

A: A trigonometric inequality is a mathematical expression that involves a trigonometric function, such as sine, cosine, or tangent, and is used to describe relationships between angles and side lengths of triangles.

Q: How do I solve a trigonometric inequality?

A: To solve a trigonometric inequality, you need to find the values of $x$ that satisfy the inequality. You can do this by using the unit circle and graphing the solutions.

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 used to graph trigonometric functions and to solve trigonometric inequalities.

Q: How do I graph a trigonometric function on the unit circle?

A: To graph a trigonometric function on the unit circle, you need to plot the points $(\cos x, \sin x)$ for $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.

Q: What is the difference between a trigonometric inequality and a trigonometric equation?

A: A trigonometric inequality is an expression that involves a trigonometric function and is written in the form of an inequality, such as $\sin x > \cos x$. A trigonometric equation is an expression that involves a trigonometric function and is written in the form of an equation, such as $\sin x = \cos x$.

Q: How do I solve a system of trigonometric inequalities?

A: To solve a system of trigonometric inequalities, you need to find the values of $x$ that satisfy both inequalities. You can do this by graphing the functions and identifying the regions where both inequalities are satisfied.

Q: What is the importance of trigonometric inequalities in real-life applications?

A: Trigonometric inequalities are used in a variety of real-life applications, including physics, engineering, and computer science. They are used to describe relationships between angles and side lengths of triangles, and to solve problems involving periodic functions.

Q: Can you provide some examples of trigonometric inequalities?

A: Yes, here are some examples of trigonometric inequalities:

• $\sin x > \cos x$
• $\tan x < \sin x$
• $\cos x > \sin x$
• $\tan x > \cos x$

Q: How do I write the solution to a trigonometric inequality in interval notation?

A: To write the solution to a trigonometric inequality in interval notation, you need to identify the regions where the inequality is satisfied and write the solution in the form of an interval, such as $\left(\frac{\pi}{4}, \frac{3\pi}{4}\right)$.

Q: Can you provide some practice problems for trigonometric inequalities?

A: Yes, here are some practice problems for trigonometric inequalities:

1. Solve the inequality $\cos x > \sin x$.
2. Solve the inequality $\tan x < \sin x$.
3. Solve the system of inequalities $\sin x > \cos x$ and $\tan x > \sin x$.

Answers

1. $\left(\frac{3\pi}{4}, \frac{7\pi}{4}\right)$
2. $\left(\frac{\pi}{4}, \frac{5\pi}{4}\right)$
3. $\left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \cup \left(\frac{5\pi}{4}, \frac{7\pi}{4}\right)$

Conclusion

In this Q&A article, we have explored some of the most frequently asked questions about trigonometric inequalities. We have seen how to solve trigonometric inequalities, graph the solutions on the unit circle, and write the solutions in interval notation. We have also seen how to solve systems of trigonometric inequalities and how to apply trigonometric inequalities in real-life applications.