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. -390°

Accepted Answer

Hello, today we're going to be finding the values of the six trigonometric functions for the following angle. We're going to provide the exact values and we'll be rationalizing the denominator when necessary. So the angle that is given to us is negative 420 degrees. Now, one thing to note about this angle is that it lies outside the first rotation of the unit circle. When we are working with problems like this, we normally want our angle theta to be within the first rotation of the unit circle which is between zero and 360 degrees. So let's go ahead and find an equivalent angle to negative degrees that lies between the region of zero and degrees. In order to get this angle, we can take our given angle which is negative 420 degrees and add 360 degrees to this value negative 420 plus 360 will give us the value of negative 60 degrees. This is still outside the first rotation of the unit circle. So we're going to add another 360 degrees to degrees. So negative 60 degrees plus 360 degrees will give us a final angle of positive 300 degrees. Now, what this means is that the angle 300 degrees is equivalent to the angle of negative 420 degrees. Let's go ahead and identify where this angle lies on the unit circle. Now, the angle 300 degrees can be found in the fourth quadrant of the unit circle. And the corresponding terminal point to this angle is given to us as positive one divided by two comma negative square root of three divided by two. This is where the terminal points are identified as cosine comma sign. This is where cosine is every X value on the terminal points and sine is every Y value on the terminal points. So now that we have the location of negative degrees and we have the corresponding terminal point, we can start identifying the values of the six trigonometric functions. Let's go ahead and start by finding sign cosine and tangent. Now again, recall that sign is every Y value on the terminal point. The Y value of the current terminal point for negative 420 degrees is negative square root three divided by two. So what this means is that sign of negative 420 degrees is equal to the negative square root three divided by two. This will be the value of our first trigonometric function. Now let's identify the value for cosine of negative 420 degrees. Now again, cosine is every X value on the unit circle or on the terminal points. And the terminal point or X value for negative 420 degrees is positive one divided by two. What this means is that cosine of negative 420 degrees is equal to positive one divided by two. This will be the value of our second trigonometric function. Now let's identify tangent. Now notice that the terminal points are identified using cosine and sine, they don't use tangent. So how can we find tangent of negative 420 degrees will recall that the identity for tangent of data is equal to the Y value divided by the X value. So if we want to identify tangent of negative 420 degrees, we're going to have 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 is negative square root three divided by two. We're going to divide this value by the X value of the terminal point which is positive one divided by two. This is going to leave us with a complex fraction. In order to simplify the complex fraction, we're going to multiply the denominator by its inverse, which will be two divided by one. And since we are multiplying the denominator by its inverse, we must multiply the numerator by the same quantity one divided by two multiplied by two, divided by one will reduce the denominator to just one. And in the numerator negative square over three divided by two multiplied by two divided by one. While we can use some cross cross reduction to reduce the twos to just one that will leave us with a negative square root three multiplied by one which will give us negative square root of three. So that means tangent of negative 420 degrees is equal to the negative square root of three. This is going to be the value of our third trigonometric function. Now let's go ahead and identify the values for co sequent se and cotangent. Now recall that the identity for co sequence of data is equal to the inverse of sine theta. So what this means is that if we want the value for code seeking of data, we're going to have to take the inverse of the current sine value. Now, earlier we identified that the sine value is equal to the negative squared three divided by two. So if we take the inverse of that value, that means that co sequent of negative 420 degrees is equal to the negative two divided by the square root of three. Since we have a square root in this value or in the denominator, we're going to need to rationalize the denominator in order to do this, we're going to multiply both the numerator and denominator by the square root of three. And this will allow us to simplify sequent of negative 420 degrees as negative two multiplied by the square root of three divided by three. This will be the value of the fourth trigonometric function. Now let's go ahead and identify second. Now recall that the identity for seeking data is equal to the inverse of cosine theta. So what this means is that if we want sequence of negative 420 degrees, we're going to have to take the inverse of the current cosine value. Now earlier we identified the cosine value as one divided by two. If we take the inverse of this value, that will mean that seeking of negative 420 degrees is equal to two divided by one and this value can be rewritten as just two. So seeking of negative 420 degrees is equal to two. Now, finally, we're going to identify the value for cotangent. Now recall that the identity for cotangent of data is equal to the inverse of tangent data. So if we want the cotangent data value, we're going to have to take the inverse of the current tangent value. Now the value for tangent is negative square root of three. If we take the inverse of this value, that means cotangent of theta is equal to negative one divided by the square root of three. And again, we have a square root in the denominator. So we're going to have to rationalize this fraction. So we're going to multiply both the numerator and denominator by the square root of three. Doing this will allow us to rewrite cotangent of negative 420 degrees as the negative square root of three divided by three. And this is going to be the final function that we are looking for. 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. -390°

Key Concepts

Reference Angles and Coterminal Angles

Signs of Trigonometric Functions in Different Quadrants

Exact Values and Rationalizing Denominators

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