Give all six trigonometric function values for each angle θ. Rat...

Question

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. 
 
 cos θ = 1

Accepted Answer

Welcome back. I am so glad you're here. We're asked to enumerate the value of the other five trigonometric functions of H given the following trigonometric function, we are given the cosine of H which is equal to negative one. We recall from previous lessons and from the tables in the textbook that sometimes when we are given an exact value for a trigonometric function that indicates we have a special angle. And this is one of those cases, we have tables in the textbook that give us these exact values. It just so happens that when the cosine of an angle in this case, that angle theta is H is equal to negative one, our angle. So theta or H is equal to 180 degrees, it's a quadrant angle. So when we know this, those tables actually give us all of the other five trigonometric functions, so this one's quick to solve. But after I let you know what those are, I will show you how to solve it if we didn't have such an easily identifiable one. So we know from the tables in the book that the sign of this angle H is going to be zero, we know that the tangent of H is going to be zero. We know that the cotangent of H is going to be undefined. We know that the C can't of H is going to be negative one and we know that the Koi can of H is also going to be undefined. But how did they figure that out for those tables in the book? Let's go over that if we were to draw this angle and we have a vertical Y axis, a horizontal X axis, they come together at the origin in the middle. The initial side of our angle is along the positive part of the X axis. And we had counterclockwise 180 degrees and we are along the negative part of the X axis for our quadrant angle. If we overlay this with a unit circle, we know that where our 180 degrees intercepts the unit circle, we have an X value of negative one and A Y value of zero. So our X is negative one and our Y is zero. We recall from previous lessons that we can find our R and with Xy and R, we can determine all of our trigonometric functions. We can find our R by taking R squared equals X squared plus Y squared. Well, we know XX is ne negative one. We'll need to square that plus, we know our Y which is zero and we'll square that negative one. Squared is one plus zero, squared is zero. So one plus zero is one, we know that R squared equals one, we can take the square root of both sides, R equals plus or minus the square root of one, which is one. However, R is always greater than zero, R is a distance. And so it's always going to be positive. So our R value here, we can disregard the negative and it's just going to be positive one. So R equals one. Now with Xy and R, we just fill in for our sign tangent cotangent. And so on, our sign, we recall from previous lessons is going to be Y divided by RY is zero, divided by R is 10 divided by one is zero. There's our zero sign of H is zero. Now for the tangent tangent is found by taking Y divided by Xy again is zero divided by X is negative 10 divided by negative one is still zero. The tangent of H is zero for the cotangent cotangent is found by taking X divided by Y X is negative one divided by Y is zero, negative one divided by zero is undefined. And that's where we got the cotangent of H is undefined. The C can't, we know is R divided by X R is one divided by X is negative 11 divided by negative one is negative one. There's where we got the C can of H is negative one. And then finally, R Koi can is R divided by Y R is one, divided by Y is 01 divided by zero is undefined. And so our cosecant of H is undefined. That's where those tables got that, those five trigonometric function values. Well done. We'll catch you on the next one.

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7. cos θ = 1

Key Concepts

Trigonometric Functions

Unit Circle

Rationalizing Denominators

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