The diagram below shows the unit circle labeled at sixteen...

The diagram below shows the unit circle labeled at sixteen different locations (note these are THE important locations on the unit circle). Identify the location number, 1 through 16, that is the correct answer to each question below. For example: Where does the ordered pair (1, 0) lie? 16 • Where is the 45 degree angle? 2 Where does cosine equal $\frac{\sqrt{3}}{2}$ and sine < 0? Where is the angle $-\frac{\pi}{6}$? Where does cosine equal -1? Where is the angle $-\pi$? Where is the angle $-64x$? Where does tangent equal 1 and sine > 0? Where does tangent equal $-\frac{\sqrt{3}}{3}$ and cosine > 0? Where does cotangent equal $-\frac{\sqrt{3}}{3}$ and cosecant > 0? Where does secant equal 1? Where does cosecant equal 2 and tangent < 0?

Transcript

00:01 So the first thing we're going to do here is find theta when cosine theta is 1 half.

00:07 Looking at the unit circle, cosine of theta is 1 half when theta is pi over 3 degrees.

00:19 So cosine of theta 1 half means that theta is equal to the inverse cosine of 1 half, which is the angle pi over 3.

00:38 Or if you want it in degrees, that is equal to 60 degrees.

00:43 Then for part 6, we want to find a reference angle for 240.

00:49 So reference angle for 240 degrees.

00:58 If you look at your unit circle, 240 degrees will be between 180 and 270, so it will be in the third quadrant over here, 240 degrees.

01:16 So then our reference angle is going to be the angle in the first quadrant that corresponds to that.

01:25 So that will be 240 degrees minus 180 degrees, or 60 degrees.

01:33 So this angle is 60 degrees, and that is 240 minus 180 degrees.

01:43 That will give you an angle in the first quadrant that is the reference for 240.

01:51 Then for part 8, we want to evaluate the cosecant of negative 3 pi.

02:08 Nope, that's not the one we want to do.

02:10 We want to do part 7.

02:12 So part 7, we want to evaluate sine, cosine, and tangent of 150 degrees without a calculator.

02:40 Well, again, we're going to use the unit circle.

02:44 When you have a unit circle, negative 150 degrees, this is negative 90, this is negative 180, negative 150 is in the third quadrant, has a reference angle of 30, 80 minus 150 is 30 degrees.

03:16 Now here you have your y -coordinate will be negative y of the y of your reference angle, and your x will be negative x compared to the x of your reference angle, because negative 150 is in quadrant 3, where both y and x are negative, and 30 is in quadrant 1, where both x and y are positive.

03:47 So this then will be negative sine of 30 degrees, this will be negative cosine of 30 degrees, and this will be sine over cosine, both of which are negative, so we get a positive tangent of 30 degrees.

04:11 And then sine of 30 degrees is 1 half, so we have a negative 1 half, cosine of 30 degrees is square root of 3 over 2, so we have negative square root of 3 over 2, and the tangent is the sine over the cosine, which will be 1 half over square root of 3 over 2, both negative, so the negative cancels, you get 1 over square root of 3, which is equal to square root of 3 over 3, because we don't want a square root in the denominator...

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