How do you find the "exact value" of sin/cos/tan/csc, etc., just using the radian measure of the angle?

Question

Hi, I have a whole section of question to do which say simply: "Find the exact value without using a calculator."

For example, there is 150 degrees (or 5pi/6 radian). How do I find the exact value of something like that without using a calculator? THANKS!!!!!!!! I'll pick a best answer as soon as the 4 hour wait-requirement is up!

Answer 1

There are certain trigonometric constants that you are required to memorize, namely the values of sin and cos for 0, π/6, π/4, π/3, and π/2 (0°, 30°, 45°, 60°, and 90°, respectively). For other angles, recall from the unit circle that sin x = sin (π-x) (you can see this visually, by noting that π-x is the reflection across the y-axis of the angle x, and being a horizontal reflection, its height is unchanged). So in this case, sin (5π/6) = sin (π/6) = 1/2.

The "trick" here is to use the unit circle to find an angle in the first quadrant that has the same (or opposite) sine/cosine from the angle you are given, and whose sine you have memorized. If you can visualize the unit circle in your head, that's the best way to do it, but if not, you can memorize these algebraic relations and use them instead:

sin (π-x) = sin x
sin (-x) = - sin x
cos (π-x) = - cos x
cos (-x) = cos x
sin (x±2πk) = sin x (where k is any integer)
cos (x±2πk) = cos x (where k is any integer)

The first four identities are derived from considering reflections across the x and y-axes of the unit circle. The last two identities are derived from the fact that angles separated by an integral multiple of 2π are coterminal -- i.e. they end on the same ray, and thus have the same sines and cosines. On problems like these, the angle can always be reduced to one whose sine and cosine you have memorized by means of these transformations (well, unless you have a teacher who likes to give you stuff requiring extensive use of the half-angle identities to solve. But it doesn't sound like you do).

Answer 2

By the time you get too far in math, you will need to have them all memorized. For your example, sin of 150 degrees would be 1/2. Cos would be (sqrt3)/2 and tan would be sin/cosine. I am sure they are in your math book somewhere. Is there a big picture of a cirlce with different angles on it. It would take forever to explain them all here.

Answer 3

use sin(pi-x) = sinx, sin (pi+x) = -sinx etc to get them into first quadrant where you will spot things

then think that with a 45 degree triangle you have sides 1-1-sqrt(2)
and with a 30 degree you have 1- sqrt(3)-2

if you have pi/12 then use the half angle formulas
sin2x = 2 sinx cosx, cos2x = cos^2 - sin^2

Answer 4

minutes and seconds? does not make sense To convert a measure from radians to degrees, multiply by 180/pi: 2pi radians = 360 degrees pi radians = 180 degrees 1 radian = 180/pi degrees Therefore: If 1 radian = 180/pi degrees 0.5 radians = 0.5*(180/pi) degrees = 28.65 degrees radians is a measure of angular motion, nothing to do with time!