How to Solve Trigonometric Equations for X

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  • 0:05 Are Trig Equations Different?
  • 0:44 Inverse Operations: Trig Style
  • 1:48 Trig Restrictions
  • 2:40 Example Problems
  • 4:24 Lesson Summary
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Lesson Transcript
Instructor: Tyler Cantway

Tyler has tutored math at two universities and has a master's degree in engineering.

Adding trigonometric functions into standard algebra equations adds complexity. Learn how to solve trigonometric equations for x and how to use inverse operations to simplify equations.

Are Trig Equations Different?

When I was younger, I wanted to learn how to cook. Since I had quite the sweet tooth, I thought the best way to learn was to start by baking cakes and cookies. I got good at it and wanted to make a huge cake for my sister's birthday. I thought there was no way I could learn it in time, but then someone told me a secret. All you have to do is make three normal cakes and stack them into a big cake. I didn't need special pans and utensils, I didn't need different ingredients. What I thought was a whole different task to master was actually just a small twist on something I had done before.

Inverse Operations: Trig Style

When we first learned to solve functions, we learned that the goal was to get the variable completely by itself on one side of the equation. After everything on the other side of the equals sign was simplified, you had your answer.

When we are dealing with trig functions, it's the same way. We still want to get the variable by itself on one side of the equation. For the numbers, we do this the exact same way we've learned. We simplify and perform inverse operations to slowly isolate the variable. But at some point, we'll run into a trig function. To get rid of that trig function, we have to do an inverse operation. The inverse operation of sin(x) is sin^-1(x). The inverse operation of cos(x) is cos^-1(x). The inverse operation of tan(x) is tan^-1(x). We will use these inverse operations to eliminate trig functions so we can continue and solve for our variable.

Trig Restrictions

Here's the catch. Trig functions are all repeating functions, which means they do the same thing over and over in both the positive and negative directions. Even though there are multiple answers, we want to choose the simplest angle possible. To do this, we only worry about answers that are in a specific domain. For sine and tangent, we let theta be between -pi/2 and pi/2. For cosine, we let theta be between zero and pi. This way, when we solve, we get the simplest answer and don't worry about all the repeated answers. To signal which answer we choose, we have to write the applicable restriction as part of our answer. This is useful because we choose the best answer and not all the repeated answers.

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