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Math 103: Precalculus12 chapters | 92 lessons | 10 flashcard sets

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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.

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.

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.

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.

Let's look at some examples. If we had the function 2*cos(?)+1=0, how would we solve? First, we identify the variable, theta (?). This is what we want to isolate on one side of the equals sign. Since it is on the left side, we will start there. Let's do inverse operations to get theta by itself.

We start with the plus one. Since it is added, the inverse operation is to subtract one. We do this on both sides of the equation. This simplifies to 2*cos(?)=-1. The next thing we need to undo is the multiplication of two. The inverse operation is to divide by two, and we also do this to both sides of the equation. This simplifies to cos(?)=-1/2. The last thing we need to do to get theta by itself is do the inverse operation on the cosine function. We apply the inverse cosine function to both sides of the equation.

On the left side, the inverse cosine and cosine functions cancel each other, leaving us with theta by itself. To solve the right side of the equation we need to find the inverse cos of -1/2. Keep in mind the restriction that inverse cosine functions should be from zero to pi radians. The inverse cosine of -1/2 is 2pi/3 radians. Our answer is ?=2pi/3.

Solving trigonometric equations might look harder, but they are very similar to solving any equation for a variable. Find the variable you want to solve for and do inverse operations to simplify it. To undo trig functions, just use their respective inverse functions. When you use the inverse function, make sure you stay within the restricted domain so that you don't deal with repeating answers.

You should be able to solve equations containing trigonometric functions after watching this video lesson.

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Math 103: Precalculus12 chapters | 92 lessons | 10 flashcard sets

- Go to Functions

- Graphing Sine and Cosine 7:50
- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Unit Circle: Memorizing the First Quadrant 5:15
- Using Unit Circles to Relate Right Triangles to Sine & Cosine 5:46
- Special Right Triangles: Types and Properties 6:12
- Law of Sines: Definition and Application 6:04
- Law of Cosines: Definition and Application 8:16
- The Double Angle Formula 9:44
- Converting Between Radians and Degrees 7:15
- How to Solve Trigonometric Equations for X 4:57
- Go to Trigonometry

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