How do I compute sin or cos without a calculator?

Question

All I know is sin(90) = 1, sin(0) = 0, sin(30) = 0.5, cos(0) = 1, cos(90) = 0.

Yes, I could use a calculator now, but I need to know how to figure it out without aid, else I'll be helpless on my exams.

If I'm given something like sin(60), sin(50), cos(30), or cos(45), how could I figure these out without a calculator?

Answer 1

Is your exam room located in hell?
You will be provided a calculator.
They might even provide you with tables.

Answer 2

Draw two triangles:

The first one is an equilateral triangle. Drop a perpendicular from the top down to the base. Now you have 2 right triangles. Take the triangle on the left, and let the hypotenuse = 2. The base will be 1, since you split it in half with the perpendicular.
Using Pythagoras, the remaining side will be √3.

Now you can figure out all the trig functions for 30 degrees and 60 degrees. The angle at the lower left is 60 degrees. The one at the top is 30 degrees.

For 45 degrees, draw a right triangle that is isosceles, with the 2 arms coming out from the right angle set at 1 in length. Using Pythagoras the hypotenuse is √2. You can now figure out all the trig relationships for 45 degrees.

Don't try to remember all these relationships, when you are in an exam, just draw your triangles as described above.

By the way. I DON'T agree with others that say you should have a calculator. Congratulations to your teacher. I'm 74 years old, and I've used the above method all my life. It makes me sad when I go into a store to buy something these days, and young people are lost if they don't have a machine to do their thinking for them.

PS I'm a professional mathematician.

Good luck.

Answer 3

Calculating values for the trigonometric functions at specific angles is something like calculating the value for PI. There are several ways to estimate the value but no straightforward way to calculate it exactly. People who use trigonometry are often mostly interested in a few standard angles. (such as 30, 45, 60, and 90 degrees). For example, the sin of 30 degrees is 1/2, the cos is the square root of 3 all over 2.

Answer 4

I agree with all people who answer you on this. You have to memorize it. you already memorized some of it so here is another hint:
sin(60) = cos(30) = 0.866
sin(45) = cos(45) = 0.707
most exams are using the angles 60,30,90, and 45

Answer 5

You should memorize sin, cos, and tan 60 and also sin ,cos. tan 45. Then you should memorize the formulae for the sin,cos, tan of the sum and difference of two angles.

Do this and you will be fine.

Answer 6

I am shocked that you dont't have access to a calculator for an exam where trig ratios are required. It might be an idea to make sure about whether you can have one or not, or if log tables are available instead.

Answer 7

The best way is to memorize the Unit Circle:
http://www.hsu.edu/uploadedImages/Faculty/lloydm/classes/pt/UnitCircle.GIF

Answer 8

you wouldneed to laern your exact figures - many are surds

Answer 9

MEMORIZE THEM