# calculating trig. ratios w/ out a calculator

• Apr 12th 2011, 02:05 PM
jonnygill
calculating trig. ratios w/ out a calculator
Hello,

So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.

Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
• Apr 12th 2011, 02:07 PM
TheChaz
Quote:

Originally Posted by jonnygill
Hello,

So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.

Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).

Yes. Lookup "unit circle".
The format of the points around the circle is (cosx, sinx). So find the point corresponding to 120 degrees, find the sin value (√3/2) and flip it.
• Apr 12th 2011, 02:14 PM
jonnygill
thank you.

but what if such a chart is not available?
• Apr 12th 2011, 02:15 PM
topsquark
Quote:

Originally Posted by jonnygill
Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).

$\displaystyle csc(120) = \frac{1}{sin(120)} = \frac{1}{sin(2 \cdot 60 )} = \frac{1}{2~sin(60)~cos(60)}$

$\displaystyle = \frac{1}{2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}$

$\displaystyle = \frac{2}{\sqrt{3}}$

-Dan
• Apr 12th 2011, 02:18 PM
pickslides
You have to commit to memory the special triangles for 30,45 and 60 and then know the sign of each function corresponding to each quadrant. Have you seen CAST?
• Apr 12th 2011, 02:25 PM
jonnygill
Quote:

Originally Posted by topsquark
$\displaystyle csc(120) = \frac{1}{sin(120)} = \frac{1}{sin(2 \cdot 60 )} = \frac{1}{2~sin(60)~cos(60)}$

$\displaystyle = \frac{1}{2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}$

$\displaystyle = \frac{2}{\sqrt{3}}$

-Dan

Where did cos come from? how does one know when to use cos instead of sin in problems like this?
• Apr 12th 2011, 02:33 PM
topsquark
Quote:

Originally Posted by jonnygill
Where did cos come from? how does one know when to use cos instead of sin in problems like this?

I used the identity $sin(2 \theta) = 2~sin( \theta )~cos( \theta )$. It's one of the double angle formulas.

-Dan
• Apr 12th 2011, 06:15 PM
Prove It
Better yet, note that $\displaystyle \sin{\left(120^{\circ}\right)} = \sin{\left(180^{\circ} - 60^{\circ}\right)}$

$\displaystyle = \sin{\left(60^{\circ}\right)}$

$\displaystyle = \frac{\sqrt{3}}{2}$.
• Apr 12th 2011, 06:35 PM
skeeter
Quote:

Originally Posted by jonnygill
thank you.

but what if such a chart is not available?

live it ... learn it ... luv it.

http://t3.gstatic.com/images?q=tbn:A...edargny2VL&t=1