Need help with trigonometric function?

Question

i'm stuck finding the 6 trigonometric functions (sin,cos,tan,csc,sec,cot) for these two problems.

1. theta = tan^-1[-(5/12)]

i'm not sure for this one, i tried making a triangle and tried using the pythagorean theorem to figure out the last side, but i don't know which number is negative.

2. The point P (-3,4) is on the terminal side of theta

my teacher never gave an example on how to solve this kind of problem and neither does my book, i don't understand how to do it.

i'm just lost, 10 points best answer, any help is appreciated.

Answer 1

1) There are two ways you can draw a right triangle so that tan(θ) = -5/12.  The opposite over the adjacent is a negative number, so they have opposite signs.  This means you can draw the triangle in quadrant 2 or quadrant 4.  So first draw the triangle made up of points (0,0), (-12, 5) and (-12,0).  Then draw another triangle made up of points (0,0), (12,-5) and (12,0).  In both cases tan(θ) = -5/12, where θ is the triangle's angle near the origin.  From there you should be able to use the definitions of sin, cos etc. to find the possible values of sin(θ), cos(θ), etc.

2) This page gives a good explanation:

http://mathforum.org/library/drmath/view...

Answer 2

theta = tan^-1[-(5/12)] so tan(theta) = 5/12

tan^-1(-5/12) is -22.6 degrees. So 360-22.6 = 337.4. So the triangle lies in the 4th quadrant. This means x axis is positive, y is negative.

remember tan(theta) = opposite/adjacent. So for your triangle with angle theta, opposite is 5, adjacent is 12. Using pythagorean theorem we get

(adjacent)^2 + (opposite)^2 = (hypotenuse)^2. We plug in the numbers to figure out the hypotenuse.

(12)^2 + (5)^ = hypotenuse^2. ->

Solving we get hypotenuse = sqrt(144 + 25) = sqrt (169) = 13

So

cos(theta) = adj/hyp = 12/13

sin(theta) = opp/hyp = -5/13

tan(theta) = opp/adj = -5/12

sec(theta) = hyp/adj = 13/12

csc(theta) = hyp/opp = -13/5

cot(theta) = adj/opp = -12/5