Often when dealing in trigonometry, you have an algebraic equation such as:
sinΘ = √3 / 2
Where you want to find Θ.
Obviously:
Θ = sin-1(√2 / 2)
Θ = sin-1(0.70710678)
Θ = 45o or π/4c depending on your unit of angular measure
For now I will deal in radians (superscript c) not degrees (superscript o), but I will leave the information for degrees in for those of you who prefer them.:
Suppose we had:
sinΘ = √2 / 2
for 0 > Θ > 2π
We are presented with a problem, how do we find the other values of Θ for which sinΘ = 0?
Sine, Cosine and Tangent are repeating waves! All we need to know is the period, which is 2πc (360o) for sine and cosine, and πc (180o) for tangent and the shapes of the graphs. There is an excellent node in Trigonometry with the graphs.
A simplistic version for sine follows:
| _
| / \
|--------
| \_/
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The horizontal axis shows 2π, so we know that for sinΘ > 0 Θ = sin-1Θ and there will be two values of Θ, both before πc
There are similar rules for all the trigonometric functions, but there is also an easy way to remember them:
pi/2
sin \ | / all
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
pi / 4 rad _\ | /
/ \ | /_ pi / 4 rad
/ \ | / \
pi -----------------|-----\|/---|------------------- 0
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tan | cos
3pi/2
The angles marked by circle sections are equal to that found by the inverse trigonometric function, in this case sin-1. They drawn from the closest horizontal in the appropriate quadrant. In "All" all results are positive, so if you have a positive sine cos or tan, you draw a line in "All". The other quadrants work the same for specific functions.
This graph is between 0 and 2πc (360o), with each quadrant equal to π/2c (90o). To find the value of a line you've drawn, work out the angle from the starting position, (marked as 0) here. This is a simple matter of arithmatic. Say we have π/4 as our result (even though it would be exactly between the horizontal and the vertical, it is hard to draw), then we have found:
sin π/4 = sin ( π - π/4 ) = √3 / 2
Which is true!
Now, what if we have something nastier, like:
tanΘ = -0.5
Well, tan-1-0.5 = -0.463647609c
Remember, tan is NEGATIVE here
pi/2
sin \ | all
\ |
\ |
\ |
\ |
\ |
\ |
_\ |
0.463rad / \ |
/ \ |
pi -----------------|-----\|------------------------ 0
|\ |
| \ / 0.463rad
| \ /
| \/
| \
| \
| \
| \
| \
tan | \ cos
3pi/2
so, from the starting position again, we see that
tan(π - 0.463) = -0.5
tan(2π - 0.463) = -0.5
So once again, we have our results.
A final exercise!
cos2Θ = 0.5
(cosΘ)2 = 0.5
cosΘ = √(0.5)
Θ = cos-1( √0.5 )
Θ = cos-1( ±0.7071067812 )
Plus or Minus! This means, we have to draw lines in all 4 quadrants.
pi/2
sin \ | / all
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
pi / 4 rad _\ | /
/ \ | /_ pi / 4 rad
/ \ | / \
pi -----------------|-----\|/---|------------------- 0
| /|\ |
pi / 4 rad \/ | \ / pi / 4 rad
/ | \ _/
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
tan / | \ cos
3pi/4
so:
cos2(π / 4) = 0.5
cos2((&pi - (π / 4)) = 0.5
cos2((&pi + (π / 4)) = 0.5
cos2((2&pi - (π / 4)) = 0.5
One final thing I haven't addressed, how do you remember the order of the quadrants? Either use the graphs to work it out, memorise them, or use a mnemonic, such as:
The one I use: All Stations To Crew
for the less politically correct: All Sailors Take Cock
SamuraiSatan says The preferred mnemonic device used in my precalculus class is All students take calculus | 
