3.10 Trigonometric Equations and Inequalities

Solving Trigonometric Inequalities

The inverse trigonometric functions, such as arcsine and arccosine, are used to undo the actions of regular trigonometric functions, such as sine and cosine. This means that if you have an equation or inequality that contains one of these regular trigonometric functions, you can use the corresponding inverse trigonometric function to solve for the variable in that equation or inequality.

For example, let's say we have the equation sin(x) = 0.5. To solve for x, we can use the inverse sine function, or arcsin, which would give us the solution x = 30 degrees. This is because the sine of 30 degrees is 0.5. In this way, we can use inverse trigonometric functions to find the value of a variable that would produce a specific result when plugged into a regular trigonometric function.

When it comes to solving a trigonometric inequality, the process is similar. For example, let's say we have the inequality sin(x) > 0.5. To solve this inequality, we can use the inverse sine function to find the values of x that would produce a result greater than 0.5 when plugged into the sine function. In this case, the solution would be x > 30 degrees and x < 150 degrees.

You can use the unit circle or your calculator to solve trigonometric inequalities and equations. Be sure to pay attention to whether you need to use degree mode or radian mode to find the correct answer.

Solving Trigonometric Equations

It's important to note that when using inverse trigonometric functions, the solutions you find may need to be modified due to domain restrictions. This means that there are certain values for which the inverse trigonometric functions are not defined, and you will have to take these restrictions into account when interpreting and using the solutions you find. For example, the function arcsin(x) only outputs values between -90 and 90 degrees. Therefore, if we get a solution that is outside this range, we would need to adjust it accordingly.

Due to the periodic nature of sine, cosine, and tangent, there are infinite solutions to these equations. For example, the equation above, sin(x) = 0.5, has a solution at 30 degrees, but it also has a solution at 390 degrees, 750 degrees, and so on because the sine function has a period of 2𝛑. This means that every 2𝛑 radians, or 360 degrees, the sine of the angle repeats. This is why ranges of the inverse trigonometric functions are restricted to only include one period of their respective trigonometric functions.

Example

Let's solve the equation cos(x) = -0.2 by finding the value of x. To solve this equation, we can use the inverse cosine function, or arccosine, to find the value of x that would produce a result of -0.2 when plugged into the cosine function.

We first isolate the term cos(x) on one side of the equation: cos(x) = -0.2. In this case, it is already isolated. Next, we would take the arccosine of both sides: x = arccos(-0.2) to find the angle, x. Just like when you are solving the equation 3x = 6 by dividing both sides by 3 to find x, we are doing the same thing here because arccosine is the inverse of cosine, and division is the inverse of multiplication.

At this point, we would have a solution of x = arccos(-0.2). However, we must remember that the range of arccosine is between 0 and 180 degrees. Therefore, we need to restrict the domain of the solution in order to make sure that it falls within this valid range.

Image courtesy of Wiktionary.

In this case, the solution would be x = arccos(-0.2) = 163.4 degrees or x = 196.6 degrees. This means that the value of x that produces a cosine of -0.2 is 163.4 degrees and 196.6 degrees. We need to restrict our solution to x=163.4 degrees only because 196.6 degrees falls outside the range of the arccosine function. This example illustrates how it is important to be aware of the domain restrictions when working with inverse trigonometric functions and to adjust the solutions accordingly to ensure they fall within the valid range.

Domain and Range Restrictions on Trigonometric Functions

The same principles apply when solving equations and inequalities involving the sine and tangent functions. Below is a chart explaining the domain and range restrictions of the sine, cosine, and tangent functions, as well as their inverses. The square brackets indicate that the endpoints are included in the interval, and the open parentheses indicate that they are excluded.

Practice Problems

1. Which of the following is the correct solution to the equation sin(x) = 0.6, when restricted to the domain of 0 to 360 degrees?

A) 30 degrees

B) 150 degrees

C) 210 degrees

D) 330 degrees

Answer: B) 150 degrees

2. Which of the following is the correct solution to the inequality cos(x) > 0.5, when restricted to the domain of 0 to 360 degrees?

A) x < 30 degrees or x > 150 degrees

B) x > 30 degrees and x < 150 degrees

C) x < 30 degrees or x > 210 degrees

D) x > 30 degrees and x < 210 degrees

Answer: B) x > 30 degrees and x < 150 degrees

3. Which of the following is the correct solution to the equation tan(x) = -2, when restricted to the domain of -90 to 90 degrees?

A) -63.4 degrees

B) -116.6 degrees

C) 116.6 degrees

D) None of the above

Answer: B) -116.6 degrees