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. 405°

Accepted Answer

Hello, today we're going to be finding the values of the six trigonometric functions using the given angle. We'll be providing the exact values and rationalizing the denominator when necessary. So the angle that is given to us is 870 degrees. Now, one thing to note is that this angle s lies outside the first rotation of the unit circle. Normally, we will want to have an angle. That was it, that's within the first rotation of the unit circle between zero and 360 degrees. So what we can go ahead and do is we can find a similar angle to degrees that lies within the first rotation of the unit circle. In order to find this angle, we can take our given angle 870 degrees and subtract 360 degrees from this spell minus 360 will give us the value of degrees. This is still outside the region of 360 degrees. So we're going to take 510 degrees and subtract another degrees from this value 510 minus 360 will give us the value of degrees. This is an angle that lies within the region between zero and degrees. And what this means is that the angle 870 degrees is equivalent to the angle of 150 degrees. So let's go ahead and identify the location of 150 degrees on the unit circle. Now, 150 degrees can be found in the second quadrant of the unit circle. And the corresponding terminal point is given to us as the negative square root three divided by two comma positive one divided by two. Now that we know the location of 150 degrees or degrees. And we have the corresponding terminal point, we can start to find the six trigonometric values. We'll go ahead and start by identifying sine, cosine and tangent. Now recall that any terminal point found on the unit circle is defined as cosine, comma sine. This is where cosine represents any X value on the terminal points and sine is any Y value on the terminal points. Now let's start by identifying sine of 870 degrees. Now again, sine is any Y value on the terminal point. So we'll just have to take a look at the Y value of the given terminal point. The Y value is positive one divided by two. And what that means is that sine of 870 degrees will equal to one divided by two. This will be the value of our first trigonometric function. Next, let's go ahead and identify cosine of 870 degrees. Now cosine is any X value on the terminal points. And for our terminal point, we have negative square root three divided by two. So what this means is that cosine of 870 degrees is equal to negative square root three divided by two. This will be the value of our second trigonometric function. Now let's go ahead and identify tangent of degrees. Now recall that any terminal point is identified as cosine comma sign but it does not use the tangent value. So how can we identify the value of the tangent function will recall that the identity for tangent of theta is given to us as the Y value divided by the X value. What this means is that we'll be taking the Y value of the terminal point and dividing it by the X value of the terminal point. So tangent of 870 degrees will equal to one divided by two, that quantity divided by negative square root three divided by two. So currently we have a complex fraction. In order to simplify this, we'll be multiplying the denominator by negative two divided by the square root of three And since we are multiplying the denominator by its reciprocal, we'll have to multiply the numerator by the reciprocal of the denominator. Doing so will allow us to reduce the denominator to just one. And so what we are left with is one divided by two, multiplied by negative two, divided by the square root of three, we can cross reduce to reduce the twos to just one. And that will leave us with one multiplied by negative one divided by the square root of three. So we have tangent of 870 degrees equal to negative one divided by the square root of three. Since we have a square root in the denominator, we'll need to rationalize this value. We're going to multiply the numerator and the denominator by the square root of three. And this will allow us to rewrite tangent of 870 degrees as negative square root three divided by three. This will be the value for a tangent of 870 degrees. Now that we have our sine cosine and tangent values. Let's start finding the values for co sequent sequent and cotangent. Now recall the co sequent of theta is equal to the inverse of the sine function. So what this means is that if we want to identify sequent of 870 degrees, we need to identify the inverse of the sine function. Well, currently we know that sine of 870 degrees is equal to one divided by two. And that means co sequent will equal to the inverse of this value, which will be two divided by one and two divided by one can be rewritten as just two. So cos of 870 degrees will equal to two. Now, let's identify the value of seek it. Now, second of theta is equal to the inverse of cosine theta. So that means second of 870 degrees will equal to the inverse of the current cosine value. Now again, cosine of 870 degrees is equal to negative square root three divided by two. That means seeking will equal to the inverse which is negative two divided by the square root of three. Again, we have a square root in the denominator. So we'll need to rationalize this value. We're going to multiply the numerator and denominator by the square root of three. And doing this will allow us to rewrite second of degrees as negative two multiplied by the square root of three divided by three. This will be the value of the seeking function for 870 degrees. Finally, let's identify our cotangent value. Now, cotangent of theta is equal to the inverse of tangent theta. So what this means is that cotangent of 870 degrees will equal to the inverse of the tangent function tangent of 870 degrees is equal to negative square root three divided by three. And that means that cotangent of 870 degrees will equal to negative three divided by the square root of three. We'll need to rationalize this, this value. So we're going to multiply the numerator and denominator by the square root of three, this will allow us to rewrite cotangent of 870 degrees as negative three multiplied by the square root of three divided by three. Since we have a three in both the numerator and denominator, we can reduce these values to just one. And that will leave us with cotangent of 870 degrees equal to the negative square root of three. And this is going to be the value for the 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. 405°

Key Concepts

Reference Angles and Angle Reduction

Signs of Trigonometric Functions in Quadrants

Exact Values and Rationalizing Denominators

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