Solving trigonometric equations Solve the following equations....

Question

Solving trigonometric equations Solve the following equations.

tan x = 1

Accepted Answer

Hello. Today, we're going to be solving for a in the following trigonometric equation, and we'll be representing our answer in terms of pi. So the trigonometric equation given to us is tangent of a is equal to the square root of 3. Now in order to answer this question, we're going to use a unit circle. One way to think of the question is what angle can we plug into tangent that will give us the value of the square root of 3? Well on the unit circle, there are 2 angles where tangent is equal to the square root of 3. The first angle exists in the 1st quadrant of the unit circle at pi divided by 3, and the second angle exists in the third region of the unit circle at 4pi divided by 3. At both of these angles, tangent is equal to the square root of 3. Now notice that the original problem does not give us any restrictions for the angle a. And one thing to recall is that the period for tangent is given to us as pi. Notice that at the first angle where tangent is equal to the square root of 3, in order to travel from pi divided by 3 to 4 pi divided by 3, we will need to travel a total of 180 degrees. So what this means is that for every angle a where tangent is equal to the square root of 3, we need to travel one period of tangent. And since we have no restrictions, we need to figure out a way to represent the angle a with an unknown amount of rotations in both the positive and negative region. Well, we can start by allowing a to equal to the smallest angle possible where tangent is equal to the square root of 3 and that angle is going to be pi divided by 3 Now we need to ask ourselves What term can we add to pi divided by 3 that represents an unknown amount of periods being added to pithree. Well, we can write that quantity as k pi and this is where k is an unknown integer. What this means is that for any number k that we plug into the equation, that is the amount of rotations that we are traveling starting at pi divided by 3. Or in other words, whatever integer that we plug in for k, we're always going to end up an equivalent angle of pi divided by 3 where tangent is equal to the square root of 3. So this is going to be the answer to this problem. And with that being said, the overall answer is going to be a. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Solving trigonometric equations Solve the following equations.

tan x = 1

Key Concepts

Trigonometric Functions

Inverse Trigonometric Functions

Periodic Nature of Trigonometric Functions

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