How Do You Find Sin, Csc, Cos, Sec, Tan and Cot USING THE UNIT CIRCLE?!?

Question

I have a trigonometry quiz tomorrow. We aren't allowed to use a calculator, just our unit circle which has degrees, coordinates, the location and numbers such as 5pi/6.

How do you find Sin, Csc, Cos, Sec, Tan and Cot using this unit circle? Thanks!!

Answer 1

If the give you the coordinates of a point on the circle such as (a,b), then the angle x is formed by the positive x-axis and the line drawn from the origin to the point (a,b).

Then sin x = b csc x = 1/b cos x = a sec x = 1/a tan x = b/a and cot x = a/b

Special angles are 0, 30, 45,60,90, 180, 270 which in radians are 0, pi/6, pi/4, pi/3, pi/2, pi. 3pi/2. To change from radians to degrees multiply by 180/pi so 5pi/6 = 150 degrees.

sin 0 = 0, sin 30 = 1/2, sin 45 = sqrt(20/2, sin 60= sqrt(3)/2 sin 90 = 1, sin 180 =0 sin 270 = -1

cos 0 =1, cos30 = sqrt(3)/2, cos45= sqrt(2)/2, cos 60 = 1/2, cos 90 = 0, cos 180 = -1 cos 270 = 0

tan 0 = 0, tan 30 = sqrt(3)/3, tan45 =1, tan 60 =sqrt(3) tann90 = undefined, tan 180 = 0 tan 270 = undefined

Answer 2

Well, you definitely want to draw these things out. The numbers given can be drawn as sides of a right triangle. Just remember: sin = opposite/hypotenuse cos = adjacent/hypotenuse tan = opposite/adjacent sec = reciprocal of cos (just flip the number) csc = reciprocal of sin cot = reciprocal of tan For example, on the first one, draw an x/y axis and use the values of cot to draw the sides. Since cot= reciprocal of tan, then the length of adjacent side (the side adjacent to theta) is 2 units on the +x axis and the opposite side is 3 units on the +y axis (you know this is correct because all of the functions are positive in the 1st quadrant). So you have the opposite and adjacent side, use the pythagorean theorem to find the hypotenuse. Sec is hypotenuse/adjacent so the value of number 1 is the value of the hypotenuse over 2. Hope that helps For number 4, plot the point. You'll find that the adjacent side is -7 and the opposite side is 3. Find the hypotenuse. From that you can get the cosine. 5 is basically the same. If you map it out on the axes like the other ones, the fact that angle C isn't in the 2nd quadrant will be apparent.