Solve Trigonometric Equations with Identities & Inverses

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  • 0:01 Trigonometric Equations
  • 0:35 Identities and Inverses
  • 2:13 Example 1
  • 3:22 Example 2
  • 5:02 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to solve trigonometric equations by making use of trigonometric identities and inverses. Learn the steps to take to get to your solution.

Trigonometric Equations

You're studying for an upcoming math exam and you see that on the list is solving trigonometric equations, equations containing trig functions. You start thinking back to what you learned about solving trigonometric equations and you remember what the equations look like. You remember that the variable is part of the trig function's argument. You get equations like cos (x) = 1 and you need to find what x equals. You also remember that our answers repeat based on the trig function's period.

Identities and Inverses

You also think of the trigonometric identities, or true trig statements, that you learned such as the Pythagorean identity of sin^2 (x) + cos^2 (x) = 1 and others. You remember that there are many trigonometric identities that have been formally proven over the years that you can refer to, but you only had to remember the ones that are used more often than the others, such as angle sum and difference identities, double angle identities, sum to product identities, etc.

If you need to, take a moment to refresh yourself on these identities. (Check out our lesson 'Trigonometric Identities: Definition and Uses' here on this site or research them on the Internet.) You also remember that to get the variable by itself in an equation like cos (x) = 1, you need to apply the inverse function. So, cos (x) = 1 becomes x = cos^-1 (1). Each of our basic three trigonometric functions has an associated inverse function. The inverse of the sine function is called the arcsine, the inverse of the cosine function is called the arccosine, and the inverse of the tangent function is called the arctangent. In math, we note the inverse function with an exponent of -1.

All this you remember. So, now it's about putting it all together to help you solve more complicated trigonometric equations. Let's take a look at a couple.

Example 1

Solve 2 sin (x) - 1 = 0 for x between 0 and pi/2.

To solve this problem, we need to first get our trig function by itself. We add the 1 to both sides and then divide by 2. We get sin (x) = 1/2. We can now apply the inverse sine function to both sides to get the x by itself. We get x = sin^-1 (1/2). Looking at the unit circle, we have two possible answers that fit. We have pi/6 and 5pi/6 radians. Which one is the correct answer?

Unit Circle
unit circle

These answers do repeat every 2pi as well since our sine function has a period of 2pi. So, how do we choose the correct one? We look at what the problem tells us about the possible answers for x. It tells us that our x needs to be between 0 and pi/2. So, which one of pi/6 and 5pi/6 is within this range? Pi/6 is, so this is the correct answer.

Example 2

Let's look at one more:

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