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. 495°

Accepted Answer

Hello, today we're going to be identifying the six trigonometric function values for the given angle. And we're going to be providing the exact values for each function. When necessary, we're going to be rationalizing the denominator. So the angle that is given to us is 840 degrees. Now, one thing to note about this angle is that it lies outside the first rotation of the unit circle. Normally, we would like 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 identify an equivalent angle to 840 degrees that lies within the first rotation of the unit circle. In order to get this angle, we can take our given angle which is 840 degrees and subtract 360 degrees from this value 840 degrees minus 360 degrees will leave us with 480 degrees. This is still outside the first rotation of the unit circle. So we're going to take 480 degrees and subtract another 360 degrees from this value. 480 minus 360 will leave us with 120 degrees. This angle lies within the first rotation of the unit circle. So what this means is that the angle 840 degrees is equivalent to the angle at 120 degrees. So let's go ahead and identify the location of 120 degrees on the unit circle. Now, the angle of 120 degrees can be found in the second quadrant of the unit circle. And the terminal point at this angle is given to us as negative one divided by two comma positive square root three divided by two. So now that we have the location of the angle and the terminal point, we can start identifying the values of 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 on the unit circle is defined as cosine comma sign. This is where cosine represents every X value on the unit circle and sine represents every Y value on the unit circle. So let's start by identifying sine of 840 degrees. Now again, sinus every Y value of the terminal points and the current Y value at our terminal point is positive squared three divided by two. What this means is that sine of 840 degrees is equal to the positive square root three divided by two. This is going to be the value of our first trigonometric function. Now let's go ahead and identify cosine of degrees. Now again, cosine is every X value on the terminal point. And the current X value on our terminal point is negative one divided by two. What this means is that cosine of 840 degrees is equal to negative one divided by two. This will be the value of our second trigonometric function. Now let's go ahead and identify tangent of 840 degrees. Now recall that every terminal point is defined as cosine comma sign, but it does not use the tangent function. So how can we identify tangent of 840 degrees will recall that tangent of theta is equal to the Y value divided by the X value. What this means is that if we want to identify tangent of degrees, we can take the Y value of the terminal point and divide it by the X value of the terminal point. So again, the Y value of our terminal point is the square three divided by two. We're going to take this value and divide it by the X value which is negative one divided by two. This is going to leave us with the complex fraction. Now, in order to simplify the complex fraction, we're going to multiply the denominator by its inverse which is negative two divided by one. And since we are multiplying this denominator by the value, we must also multiply the numerator by negative two divided by one. Doing so will allow us to reduce the denominator to just one. And what we are left with is the square three divided by two multiplied by negative two divided by one in the numerator. We can use cross reduction to reduce both of the twos to just one. This will leave us with a squared of three multiplied by negative one, which will leave us with the negative squared of three. So what this means is that tangent of 840 degrees is equal to the negative square root of three. This will be the value of our third trigonometric function. Now that we have the values of sine cosine and tangent, let's identify the values for co sequent sequent and cotangent. Now recall that cos seeking of data is equal to the inverse of sine theta. So if we want to identify co sequent of 840 degrees, we're going to be taking the inverse of the current sign value. Now, we identified the current sine value to be the square root of three divided by two. So if we take the inverse, that means coking of 840 degrees will equal to two divided by the square root of three. Since we have a square root value in the denominator. We're going to need to rationalize this value. We're going to multiply the numerator and denominator by the square root of three. This will allow us to rewrite the value of co sequent of degrees as two multiplied by the square root of three divided by three. This will be the value of our fourth trigonometric function. Now let's go ahead and identify the value for sequent. Now, recall that seeking of data is equal to the inverse of cosine data. What this means is that if we want the value for second of 840 degrees, we're going to have to take the inverse of the current cosine value. Now the current cosine value is negative one divided by two. That means the inverse of this value will equal to negative two divided by one and two divided by one can be rewritten as just two. So that means that seeking of 840 degrees is equal to negative two, that will be the value of our fifth trigonometric function. Finally, let's go ahead and identify the value for cotangent. Now recall that co tension of the is equal to the inverse of tangent data. What this means is that if we want cotangent of 840 degrees, we're going to have to take the inverse of the current tangent value. Now again, tangent is equal to the negative square root of three that means the inverse of that value will be negative one divided by the square root of three. We're going to have to rationalize this value since we have a square root in the denominator. So we're going to multiply both the numerator and denominator by the square root of three. Doing so will allow us to rewrite cotangent of degrees as the negative square root of three divided by three. And this is going to be the value of our final trigonometric function. 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. 495°

Key Concepts

Reference Angles and Coterminal Angles

Signs of Trigonometric Functions in Different Quadrants

Rationalizing Denominators

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