PreCalc Help! Unit Circle?

Question

Finals are in like 3 weeks, I'm so lost!

(a) List four angles (two positive, two negative) whose sine is -0.5 or (1/2) ---- I know this is easy to most but I really don't understand the unit circle!

(b) Suppose theta is an angle in standard position whose terminal edge meets the circle (centered at the origin) of radius 5 at the point ((3/2),(sq.root 91/2)) -- those are coordinates. Find sin, cos and tan theta.

I don't know what to do with the coordinates given, I don't know where they fit on the circle! I know you can't exactly draw one for me, but if anyone can attempt to explain what I'm required to do here, you'll be my hero!!

Love Tessa!

Thanks so much! That made heaps of sense.

If you could be bothered, I can't understand how to figure out where values go on the unit circle. Like if I'm asked to place -14pi / 5 on the unit circle, how do I know where this goes?

Thanks so much!! (L)

eek everyone has been so so helpful! Evan was so quick so thanks you but some of the others had stacks of detail.

Thanks everyone.. love tessa xx

Answer 1

Tessa
In 3 weeks you can put it all together.

for a, sin x = .5 at 30º and 150º, sin x = -.5 at -30º and -150º
those are equivalent to π/6, 5π/6, -π/6 and -5π/6 radians respectively.

for b, something is wrong. the point (1.5, √45.5) does not lie on a circle of radius 5. Neither does the point (1.5, √9.5). (these are my reads of what (3/2, 91/2) could mean). The problem HAS to be misstated somehow.

If you can check and post in additional details, I will go on and add some stuff about unit circles to this response.

Continued:
Part of the problem with understanding the unit circle is that both rectangular and polar coordinate systems can be used. r and theta are measured in the polar system. x,y are in the rectangular system. As you have seen, as the radius 1 moves around points on the circle, the x and y correspond to the cos and sin values of theta. If you look at http://www.analyzemath.com/unitcircle/unitcircle.html and run the applet, watch how the x and y values trace out a sin and cos wave. getting that picture in your mind is invaluable to your understanding. Watch it a few times.

Others have told you how to mark out angles on the unit circle. theta = 0 corresponds to the positive x axis. Increasing theta is counterclockwise. 0 and 180 degrees correspond with the + and - x axis. 90 and 270 degrees correspond to the + and - y axis.

There are 2π radians in 360 degrees. to convert degrees to radians, you multiply by 2π and divide by 360. ( that's the same as π/180. to convert radians to degrees you invert the multipliers. i.e 180/π

Try to understand what's happening with the unit circle and the sin and cos function -- the tan function is a little harder to follow, but you can view it if you want.

Ask more questions -- of Answers and of your teacher. in 3 weeks you can be the expert!

Continued2:
Northstar straightened out my concern with b. He got the number right. Please read the ref.

about your question, "Like if I'm asked to place -14pi / 5 on the unit circle, how do I know where this goes?"

check out http://www.hsu.edu/uploadedImages/Faculty/lloydm/classes/pt/UnitCircle.GIF

It's a drawing of the unit circle showing the relations between degrees, radians, x and y. It may be too big to print out (depending on your browser). If it is, send me the email address of an adult (for safety) before the end of the month and I'll send them a printable copy for you.

-14π/5 can be converted to a positive angle by adding 2π (whole revolutions) until it becomes positive. -14π/5 + 2π +2π = +6π/5, an angle in the 3rd quadrent. If you have trouble plotting that, convert to x and y coordinates. x=cos(-14π/5 ) = -0.81 and y = sin(-14π/5) = -0.59.

you should be able to plot that

Good Luck on your final Expert!

Answer 2

(a) List four angles (two positive, two negative) whose sine is -0.5 or (1/2) ---- I know this is easy to most but I really don't understand the unit circle!

This is a little unclear. Do you want sinθ to
equal +1/2 or -1/2?

sin(30°) = sin(150°) = 1/2
sin(-330°) = sin(-210°) = 1/2

sin(210°) = sin(330°) = -1/2
sin(-30°) = sin(-150°) = -1/2

(b) Suppose theta is an angle in standard position whose terminal edge meets the circle (centered at the origin) of radius 5 at the point ((3/2),(√91/2)) -- those are coordinates. Find sin, cos and tan theta.

(x,y) = (3/2, √91/2)

√(x² + y²) = √[(3/2)² + (√91/2)²] = √(9/4 + 91/4) = √25 = 5

sinθ = y/√(x² + y²) = (√91/2) / 5 = √91/10

cosθ = x/√(x² + y²) = (3/2) / 5 = 3/10

tanθ = y/x = (√91/2) / (3/2) = √91/3

Answer 3

It seems complicated but it really isn't. The unit circle is just a circle with radius=1 and an x and y axis passing through it. The origin (0,0) is in the exact middle of the circle, so if you draw a line in any direction when you hit the edge of the circle you will have a line that is the radius of the circle (half the diameter) and the radius will equal 1. You can use the line (radius) to form triangles. The whole sin and cos thing is not complicated either. cos(theta) is just the coordinate and sin(theta) is the y coordinate. With the line r you can draw a triangle and use the (cos,sin) system to find the angle closest to the origin. You just have to remember that:

sin(theta)=y/r
cos(theta)=x/r
tan(theta)=y/x

these are really basic trig identities. You have the worst teacher in the world, because by the end of the course you HAVE to know that..

there are also special points on the circle that represent different angles that you should know.
eg:
if theta is pi/6 (30 degrees) than sin(pi/6)=1/2

You just have to memorize these special points...the important ones in the first quadrant are:

0 (0 degrees, x=0), pi/6 (30 degrees), pi/4 (45 degrees), pi/3 (60 degrees), pi/2 (90 degrees). Anyways, I wish I could draw it and explain it, it's honestly just a matter of memorizing trig identities and special points. Go here, they have a cool picture of the special points:

http://en.wikipedia.org/wiki/Unit_circle

Anywho, you can find the points for sin(1/2) there...the unit circle is just a method for measuring angles.

for b)

to use the unit circle to solve b) you need to get the radius down to 1...so you have to divide it by five..but if you do that to the radius you have to divide the two other sides by 5 also. You will get a smaller triangle..but it will be a similar triangle, so luckily none of the angles will have changed. I guess you will need a calculator to find the sin cos and tan. Just remember that cos is the x coordinate and sin is the y coordinate and tan is just sin over cos.

so for our new triangle r=1 and the cos point is:
3/2 divided by five=3/10
the sin point is:
squareroot(9.5) divided by 5=(don't have a calculator)

so cos of theta will equal cos of theta=(3/10)
sin of theta is sin of theta=(sqrt9.5/5)
tan will be sin divided by the number you get for cos.

Anyways, good luck, hope this helped in some way..the best advice I can give is to go see the teacher after class if you don't understand something. Anyone on the planet can understand this, the only reason I can think of for you not knowing this by the end of the course is that your teacher totally sucks..

Answer 4

The unit circle is just a way of having the hypotenuse equal to one. That means that for any point (x,y) on the unit circle, the sine of the angle formed by connecting the point to the origin will always be y and the cosine will always be x (since opposite over hypotenuse is y/1 and adjacent over hypotenuse is x/1.

One angle, theta, at which sin(theta) = 1/2 is pi/6. Another will be symmetric to the y axis at 5pi/6. A negative angle for which this is true is -(2pi - p/6) or -11pi/6 and another is -(pi + pi/6) or -7pi/6.

b) by definition, the sin(theta) is opposite over hypotenuse. In this case, the hypotenuse is of length 5 and opposite is the y coordinate sqrt(91/2), so sin(theta) = sqrt(91/2)/5; cos(theta = adjacent/hypotenuse = x coordinate/5 = (3/2)/5 = 3/10, and tan(theta) = opposite/adjacent = y/x = sqrt(91/2)/(3/2)

Answer 5

I can't remember how to do this stuff since it has been so long since I've been in pre-calc. If you can, please explain how to do these step by step? Please explain the best you can in words since I know it's difficult without showing me in person. b. determine sin(7pi/6) = sin (pi + pi/6) = -- sin (pi/6) = -- 1/2 ANSWER and sin (11pi/6) = sin (2pi -- pi/6) = sin (--pi/6) = -- sin (pi/6) = -- 1/2 ANSWER. c. given that cos (pi/6)=3^(1/2)/2, determine cos (5pi/6) = cos (pi -- pi/6) = -- cos (pi/6) = -- 3^(1/2) / 2 ANSWER Ana is sitting in the bucket of a Ferris wheel. She is exactly 46.7 (radius) feet from the center and is at the 3 o'clock position as the Ferris wheel starts turning. a. If Ana swept out 2.58 radians on a circular arc as the Ferris wheel rotates, how many feet did she travel? Ana travelled 46.7*2.58 feet = 120.486 feet ANSWER Define a function to express the number of feet Ana travels on this Ferris wheel (with a 46.7 foot radius) as a function of the number of radians theta swept out by Ana = 46.7 theta feet ANSWER b. Ana's sister sweeps out 2.58 radians on a Ferris wheel with longer arms than Ana's. Did her sister travel the same number of feet as Ana? NO. ANSWER If not, who traveled farther and why?Sister travelled faster as her distance (radius ) was longer. ANSWER If I'm thinking about this correctly, would the answer to b be that her sister traveled further because of the formula s=r*theta saying that the radius times theta equals arc length (if I even remembered the formula right) so since the r is bigger, the radians of 2.58 is more of the sister's ferris wheel than of Ana's? YES. YOU ARE RIGHT. I hope that made sense. It jumbled in my head. At least tell me if I'm on the right track for that one.