the signs of trig ratios vary in different quadrants right? could you pls give me the proof of it?

Question

i am able to prove it only for acute angles in various quadrants. not obtuse ie. the ones starting all the way from the 1st quadrant.

plas help:)

thanks

and also, could you give me the proof for sin(-theta) = -sin theta and cos(-theta) = cos theta for angles that are not acute?

thanks:)

Answer 1

http://i20.photobucket.com/albums/b249/LindySoul/mathematics/sinabove90comparative.gif

http://i20.photobucket.com/albums/b249/LindySoul/mathematics/negativesine.gif

Answer 2

Trig functions are based on rations of the lengths of the sides of a triangle formed as a point sweeps a circle. The radius of the circle is always positive. However the x and y coordinates change from quadrant to quadrant.

In the first quadrant they're both positive So in the first quadrant, all functions are positive

In the second quadrant, the x coordinate is negative. So any functions invloving x will be negative.

In the third quadrant, the x and y coordinates are both negative. So any function that is a ratio between one of them and the radius will be negative. But any function that involves a ratio between x and y will be positive.

In the fourth quadrant, the y coordinate is negative, So, any ratio that contains y will be negative.

You can develop a table of the functions in the various quadrants to illistruate that. However, you asked for a proof, I don't know that I can give you one. It's based on definition.

I believe the second question will be answered by the first.

Answer 3

Draw and x/y coordinate system. Draw a circle of radius 1 unit centered on the origin. The radius is always positive, but the trig ratios will always involve either the projection of the radius on the x or y axis. The sign of the trig ratio is determined by whether that projection lies on the positive or negative x or y axis. The rule for division of signed numbers applies: +/+ = +; -/- = +; +/- = -; -/+ = - . From this you can find the sign of each ratio in each quadrant. Acute angles all lie in the 1st quadrant. Obtuse angles lie in quadrants 2,3,and 4. You measure the angle from the x - axis. An angle > 90º but < 180º lies in the 2nd quadrant.