Use a half-angle identity to find each exact value.
cos ...

Question

Use a half-angle identity to find each exact value.

cos 195°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given trigonometric expression using the half angle identity. And our given expression is the cosine of 202.5 degrees. Our answer choices are answer choice a negative. And then in the numerator, we have the square root of the quantity of two minus square root two. That's all divided by two. Answer to B negative. Again, a fraction in the numerator, the square root of the quantity of two plus square root two all divided by four answer choice C negative. In the numerator, the square root of the quantity of two plus square root two, all divided by two and answer choice D negative. In the numerator, the square root of the quantity of two minus square root two, all divided by four. All right. So we've got this cosine of an angle in degrees and we have to use the half angle identity to find an exact value. Well, we can set our 202.5 degrees to be equal to an angle divided by two. So considering that that would be half of an angle. If we solve for a here, our A would be equal to multiplying both sides by two, that would be equal to 405 degrees. Now, let's convert 405 degrees into radiance and 405 degrees multiplied by, we'll have pi radiance in the numerator divided by 180 degrees and then our degrees cancel out and we'd have 405 pi divided by 180 which would simplify to be nine pi divided by four. However, that is beyond one full rotation. So we'll subtract a full rotation, subtract two pi from nine pi divided by four. And we get uh an equivalent angle of pi divided by four. So we're looking at our co terminal angle pi divided by four for our A value. OK. So now we've got an A for our half angle identity. We recall from previous lessons, the half angle identity for cosine would be the cosine of a divided by two equals plus or minus the square root of. And inside of the radical is a fraction. In the numerator, we have one plus the cosine of A and that's divided by two. So we know our A is equal to pi divided by four. So let's plug that in, that's the cosine of instead of a, it's pi divided by four, divided by two. And then we have that equal to plus or minus the square root of. And inside of that radical is a fraction in the numerator, we have one plus the cosine of our A is pi divided by four. And then that's all divided by two. So now it's just a matter of identifying what our cosine of pi divided by four is exact value is, which is something that we can do because pi divided by four is on our unit circle and then simplifying. So let's figure out what the exact value of the cosine of pi divided by four is we recall from previous lessons, we've got our unit circle is going to be on our X horizontal and y vertical axes. Every point on the unit circle is one unit away from the origin, unlike what I drew. But you can imagine. And then the angle pi divided by four is in the first quadrant exactly in between the positive parts of the X and Y axes. And pi divided by four is xy coordinates are square root two divided by two square root two divided by two. And that's important because for an angle on the unit circle, the cosine is equal to the X value. So for the cosine of pi divided by four, that equals the X value of square two divided by two. Now we can fill that in, all right, if we're bringing down the left hand side, we've got the cosine of pi divided by four divided by two, which is what we started with. That's the cosine of 202.5 degrees. If you go back through the steps, we went through in reverse and then that's equal to plus or minus the square root of in the numerator of the fraction inside of the radical. We have one plus and the cosine of pi divided by four is equal to square root two, divided by two. That's all divided by two. OK. I said the only thing left was simplification but there are actually a lot of steps to simplification this time. Let's take a look at our numerator for a moment that inside of the radical, that's one plus square root two divided by two. We want to add fractions. We need a common denominator. That'll be two. We'll multiply one by two halves. So two halves plus square two divided by two is going to be in the numerator two plus square root two, all divided by two. So that's all in the radical. It's all plus or minus stone in the numerator of that radical is two plus square root two, all divided by two. And that's all divided by two. Again, let's think for a moment about simplifying if we took our numerator two plus square root two divided by two and divide that by the denominator two, that's the same as multiplying by the reciprocal of our second fraction, which is two. So this is the same as two plus square root two divided by two, multiplied by one half and then multiplying straight across, we can see that our numerator remains two plus square root two divided by, in the denominator two, multiplied by two is four. That's all still inside the radical. And it's still plus or minus. Now, we can simplify this further still because taking the square root of our numerator and our denominator, we can take the square root of our denominator which is four. So bringing over plus or minus, our numerator is just square root of two plus square root two. And that's divided by the square root of four is two. Are there enough twos and square roots and pluses and minuses for you. So now we only have one step left and that is taking a look at this plus or minus. Our original angle here is 202.5. That is going to be in the third quadrant 202.5 degrees is in the third quadrant and cosign is negative in the third quadrant. So we can disregard the positive that we've been carrying around. And this whole thing is just going to be a negative. And then in our numerator, it's the square root of two plus square root two. And that's all divided by two. And there we have it, we've got our cosine our exact value of the cosine of 202.5 degrees. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Use a half-angle identity to find each exact value.
cos 195°

Key Concepts

Half-Angle Identities

Reference Angles and Quadrants

Exact Values of Cosine for Common Angles

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