Solve the following equations.

Question

$an^22 heta=1,0\lt{ heta}\lt{\pi}$

Accepted Answer

Hello. Today, we're going to be solving the values of x for the following trigonometric equation, and we're going to be writing our answer in terms of pi. So the equation that is given to us is cotangent squared of x is equal to 1 and the value of x must be between 0 and pi. So, how do we go about solving for the value of x? Well, we can actually simplify the equation by taking the square root of both sides of the equation. This will allow us to simplify the equation to cotangent of x is equal to the square root of 1 which will give us the value of 1. Now we need to be careful though because whenever we take a square root of a number we end up with 2 solutions, a positive and negative solution. So currently we have cotangent of x is equal to plus and minus one. Now we can separate this equation into 2 separate equations. We can separate the equation as cotangent of x is equal to positive one and cotangent of x is equal to negative one and now we just need to solve the values of x individually for each equation. So in the first equation, the first equation is asking us what angle of x can we plug into cotangent that will give us the value of positive one? Well in order to answer this problem, we're going to need to utilize a unit circle. Now the angle that we can plug in a cotangent to give us the value of positive one is going to be the angle pi divided by 4. Now the terminal point at pi divided by 4 is the square root 2 divided by 2 comma square root 2 divided by 2 and recall that the cotangent of x is defined as the x value of the terminal point divided by the y value of the terminal point. So if you wanted to double check and make sure that this value of x works, we can divide the x value by the y value of the terminal point. So the x value at pi divided by 4 is the square root 2 divided by 2, and we'll be dividing that value by the square root of 2 divided by 2. So we have a complex fraction. In order to simplify this complex fraction, we can multiply the denominator by the reciprocal of the denominator which will be 2 divided by the square root of 2. And if we multiply the denominator by this value, we must also multiply the numerator by 2 divided by the square root of 2. The denominator will simplify to 1 and the numerator will simplify to 1. That leaves us with 1 divided by 1 which will simplify to just 1. So if we were to plug in cotangent or if we were to plug in pifour into cotangent, we'll get the value of 1. What this means is that the value of x for the first equation is going to be pifour. Next, we need to figure out an angle to plug in a cotangent that will give us the return value of negative one. Well, that angle in this case is going to be 3 pi divided by 4. The terminal point at 3pi divided by 4 is given to us as negative square root 2 divided by 2 comma positive square root 2 divided by 2. And and if we wanted to double check this angle, we can divide the x value of that terminal point by the y value of the terminal point. That will be negative square root 2 divided by 2 divided by positive square root 2 divided by 2. Again, since we have the complex fraction, we're going to multiply the numerator and denominator by the reciprocal of the denominator. Doing this will allow us to simplify the denominator to 1 and simplify the numerator to negative one. And negative one divided by positive one will give us the value of negative one. So what this means is that if we were to plug in the angle 3pi over 3pi divided by 4 into cotangent, we'll get the output of negative one and that means the value or the answer for x for the second equation will be 3pi divided by 4. So what this means is that we have 2 values of X that satisfy the original equation. Those values of X will be X is equal to pi divided by 4 and x is equal to 3 pi divided by 4. And with that being said, the answer to this problem is going to be d. 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.

Solve the following equations.
$\tan^22\theta=1,0\lt{\theta}\lt{\pi}$

Key Concepts

Trigonometric Functions

Inverse Trigonometric Functions

Periodic Nature of Trigonometric Functions

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