# Thread: calculating trig. ratios w/ out a calculator

1. ## 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).

2. 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.

3. thank you.

but what if such a chart is not available?

4. 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

5. 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?

6. 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?

7. 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

8. Better yet, note that $\displaystyle \sin{\left(120^{\circ}\right)} = \sin{\left(180^{\circ} - 60^{\circ}\right)}$

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

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

9. Originally Posted by jonnygill
thank you.

but what if such a chart is not available?
live it ... learn it ... luv it.