In Exercises 61–86, use reference angles to find the exact value ...

Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.

sec 495°

Accepted Answer

Welcome back. I am so glad you're here. We're asked to evaluate the following expression manually by using reference angles. Our expression is the sea can of 855 degrees. Our answer choices are answer choice. A, the Squire of three divided by two answer choice B the square of three answer trace C negative square root of two divided by two and answer choice D the negative square root of two. All right. Well, 855 degrees that's already in degrees, not in radiance. So that's nice, but it's well, more than one full rotation around. So it's more than one full circle. So we can take our 855 degrees and subtract a circle. We can subtract 360 degrees that gets us down to 495 degrees. But that's still more than a circle. So we can subtract another full rotation, another 360 degrees. And that will put us right at 135 degrees which we can place. If we draw a rough sketch of a coordinate plane, we have a vertical Y axis and a horizontal X axis. We draw our initial side of our angle vertex at the origin and initial side heading toward positive infinity. For the X values along the X axis, our terminal side, well, 135 degrees is going to be halfway in between degrees and 180 degrees. So it's halfway between the positive part of the Y axis and the negative part of the X axis, there's our 135 degrees. Now, what's our reference angle? If we're taking the C can't of degrees, what's our reference angle for this? Well, for the reference angle, normally, we would take in the second quadrant, we would be taking pi minus our angle, but this time we're in degrees. So instead of taking pie, we're going to take 180 degrees and subtract 135 degrees. And that's going to give us a 45 degree reference angle. And for a 40 45 degree angle, we can draw a 45 45 90 triangle. Take note that mine looks exactly like my 30 60 triangle. You just have to label it and then it makes it 45 45 right. So with the right angle is the 90 degrees here and the other two angles are both 45 degrees when you mark them that way. Mine's just a rough sketch. And so what we know about the 45 45 90 is that the hypotenuse is the squared of two and the opposite and the adjacent are both one. So we're taking the sea can of our degrees. Well, C can is going to be taking the hypotenuse and dividing that by the adjacent. And so we take the hypotenuse here. That's the square root of two, divide that by the adjacent, that's one, the square root of two divided by one is the square root of two. But we have one important question. Still here. We're in the second quadrant is C can positive or negative in the second quadrant C can is only positive in the 1st and 4th quadrants. So this time it's negative. So we've got the negative squarer of two. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sec 495°

Key Concepts

Reference Angles

Secant Function

Unit Circle

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