Find exact values of the six trigonometric functions of each a...

Question

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -510°

Accepted Answer

Hello, today we're going to be listing the six trigonometric functions using the following angle. We're going to provide exact values and we're going to rationalize the denominator when necessary. So the angle given to us is negative degrees. One thing to note about this angle is that it lies outside the region of one rotation of the unit circle. Normally, when we're working with an angle, theta we would like for our angle theta to be within the first rotation of the unit circle which is between zero and 360 degrees. So what we can go ahead and do is we can find an equivalent angle to negative 480 degrees. In order to do that, we're going to take our given angle which is negative 480. And we're going to add 360 degrees to this value negative 480 plus 360 will give us the value of negative 120 degrees. Now, this value still lies outside the first rotation of the unit circle. So we're going to take negative 120 degrees and add another 360 degrees to this value. Negative 120 plus 360 will give us positive 240 degrees. Now, 240 degrees lies between the region of zero and 360 degrees. And what this means is that negative 480 degrees is equivalent to 240 degrees. Now, what we want to do is we want to locate the angle degrees on a unit circle. So let's go ahead and draw that value. So where is 240 degrees located on the unit circle? While this angle can be found in the third quadrant of the unit circle. And the terminal point at this given angle is negative one divided by two comma negative square root three divided by two. So now that we have the location of negative 480 degrees and we have the following terminal point, we want to use this information to find the six trigonometric functions. Let's go ahead and start by finding the values of sine, cosine and tangent. Now recall that every terminal point found on the unit circle is defined as cosine comma sine. This means that cosine represents every X value on the terminal points and sine represents every Y value on the terminal points. So if we wanted to find the value of sine of negative 480 degrees, all we would need to do is we need to identify the Y value of the corresponding terminal point. Well, in this case here, the Y value of the terminal point is going to be negative square root three divided by two. And what this means is that sign of negative 480 degrees is equal to negative square root three divided by two. This is going to be our first trigonometric value. Next, let's go ahead and identify cosine of negative 480 degrees. Now again, cosine represents every X value on the terminal points. So we'll just need to identify the corresponding X value to the given angle. And that XX value for the terminal point is going to be negative one divided by two. So what this means is that cosine of negative 480 degrees is equal to negative one divided by two. This will be our second trigonometric value. Now, what we want to do is we want to identify the value of tangent. Well, notice that every terminal point is defined as cosine comma sign, it doesn't use the value of tangent. So how can we identify the values of tangent? Will recall that the identity for tangent data is given to us as the Y value divided by the X value of the terminal point. So in order to identify tangent of negative 480 degrees, what we'll need to do is we'll need to take the Y value of the terminal point and divide it by the X value of the terminal point. Now again, the Y value of our terminal point is negative square root three divided by two. We're going to want to divide this value by the X value of the terminal point which is negative one divided by two. So what we are left with is a complex fraction. In order to simplify the complex fraction, we're going to multiply the denominator by the reciprocal of the denominator that reciprocal will be negative two divided by one. And since we are multiplying the denominator by this value, we must also multiply the numerator by the same value negative one divided by two multiplied by negative two divided by one will reduce the denominator to just one. So what we are left with in the numerator is negative square root three divided by two, multiplied by negative two divided by one, we can cross reduce the twos and reduce them to one that will essentially leave us with negative square root of three multiplied by negative one. And that is going to give us a final value of positive square root of three. So that means tangent of negative degrees is equal to positive square root of three. This will be our third trigonometric value. Now, what we want to do is we want to identify the values for co sequent sequent and cotangent. Now recall that the identity for sequent is that cos is the inverse of sign data. So what this means is that in order to get the coin value, we're going to take the inverse of the current sine value. Now, the current sine value we identified is negative square root three divided by two. And so that means sequent of negative degrees is equal to the inverse which is negative two divided by the square root of three. Now, since we have a square root in the denominator, we're going to need to rationalize this value in order to rationalize we multiply the numerator and denominator by the square root of three doing so will allow us to simplify sequent of negative 480 degrees as negative two multiplied by the square root of three, that quantity divided by three. This will be the fourth value that we are looking for. Now, we need to identify the second value. Now recall that the identity for second is that sin is the inverse of cosine. So what we'll need to do is we'll need to take the inverse of the current cosine value in order to identify sequent. Now again, cosine of negative 480 degrees is equal to negative one divided by two. And that means that the inverse for sequence of theta or sequent of negative 480 degrees is equal to negative two divided by one and negative two divided by one can just be rewritten as negative two. So this is going to be the value for C and finally, we need to identify the cotangent value. Now just like the other values cotangent of theta is the inverse of tangent theta. So what this means is that we'll need to take the inverse of the current tangent value. Now, the current tangent value is the square root of three. And that means the inverse for cotangent of negative 480 degrees is going to equal to one divided by the square root of three. Now, just like cos we have a square root in the denominator. So we're going to need to rationalize this value. We can rationalize by multiplying both the numerator and denominator by the square root of three. And this will allow us to rewrite the cotangent of negative 480 degrees equal to the square root of three divided by three. And this is going to be the final trigonometric value for the given angle of four hun negative 480 degrees. 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.

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -510°

Key Concepts

Reference Angles and Coterminal Angles

Signs of Trigonometric Functions in Different Quadrants

Rationalizing Denominators

Watch next