In Exercises 39–46, use a half-angle formula to find the exact va...

Question

In Exercises 39–46, use a half-angle formula to find the exact value of each expression.

sin 105°

Accepted Answer

Welcome back everyone in this video, we want to determine the exact value of the expression of sine 120 degrees using the half angle formula for answer choices A is a root of three divided by two B is a negative square root of three divided by two C is a negative or half and D is positive or half. Now, what do we already know? Well, it says that we're going to determine the exact value using the half angle formula and recall that the half angle formula states that the sign of a half angle is equal to the positive or negative square root of one minus the cosine of that angle all divided by two. So we need to figure out what that anger is and then use it to help us solve. We also need to figure out if we're going to use the positive or negative value for the half anger formula. How can we figure that out? Well, we're finding the sign of 120 degrees. So let's think about what that means. Let's draw a unit circle to help us. So if we draw our unit circle here what do we know? Well, we know that the sign of 120 would probably be somewhere over here. And if you notice, recall that we have our X axis and our Y axis recall that the Y value of a trigonometric function and the unit circle is equal to the sign of whatever that angle is. So in this case, the sign of 1 20 would be our Y value. And if you notice, if we come over here that Y value is positive. So that means we are going to use the positive version of our formula. So I can write that right here. So it's positive version of one minus the cosine of our anger all divided by two. So let's go ahead and figure out our angle and then solve. So what do we know? Well, since our angle which is theta divided by two is equal to 100 and 20 degrees, that means our anger must be theta, our angle must be 120 degrees multiplied by two, which is 240 degrees. So that's what we're going to use in our, in our formula. So this implies that the sign of degrees is going to be equal to the positive square root of one minus the core sign of 240 degrees divided by two. So for now, we need to figure out what the cosine of 240 degrees. Again, let's jump to our unit circle. Now, on our unit circle, the cosine of 240 degrees is probably gonna fall somewhere down here. And if you remember working with the unit circle, it's going to be our X value. So the X value is the cosine and that X value would work out to a half or negative a half rather. OK. So that would be the core sign of 240 degrees. So I can substitute that here. So that means a sign of 1 20 I can rewrite this and replace the core sign of 240 with negative a half. And then all of that is being divided by two. Now let's continue working to figure that out. No one minus negative or half, that would be 1.5 or three halves and that three half is gonna be divided by two, three houses divided by two would eventually were known to be 3/4. No, we're not finished yet because we can get well, two squared gives us four. So we can get a whole number in our denominator. So if we apply our law in this disease, then we can rewrite the square root of three quarters as a square of three divided by the square root of four, which the square of four is two. So therefore, the sign of 120 degrees would be the square root of three, all divided by two which when we look in our answer choices, that would be equal to a thanks for watching guys. I hope you found this video helpful.

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. sin 105°

Key Concepts

Half-Angle Formulas

Reference Angles

Exact Values of Trigonometric Functions

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