Can you show us what you have tried and any attempts (even partial)?
- Determine 2 positive and 2 negative co-terminal angles for a related acute angle of 20° in quadrant III.
- Using the CAST rule, state the sign of each of the following:
- sin 220°
- cos 320°
- tan 112°
- Determine θ if cos θ = 1/2 and θ is an angle in quadrant IV.
- The CAST Rule describes for a quadrant which ratios have positive values. Describe why the tangent ratio is positive only in quadrants I and III.
- Given a point on the x-y plane, explain how you would determine a principal angle that would correspond to a terminal arm containing that point.
To start you off, consider that for the quadrants (in counter clock-wise order starting from the top right quadrant) you have the following signs for the sines and cosines that follow the relationship:
sgn(sin(x)) = +, +, -, -
sgn(cos(x)) = +, -, -, +
where sgn is the sign (positive or negative) and sin(x), cos(x) are trig functions.
To understand this you need to consider when the x or y co-ordinate is negative or positive in each quadrant and that the quadrants in terms of angles are divided up into 0,90,180,270 in terms of their relation to quadrants 1,2,3,4.
Just remember that when the arm of an angle rotates we get an angle. The rotating arm is always considered positive the sign of the other. Two arms of the triangle is taken depending on the part of axis along which it is measured. For example if the arm of triangle is going upwards form x axis it is taken as positive otherwise negative. Similarly if the other side of the triangle is measured towards right of y axis then it is positive else negative. Going by this logic we see that the sign of both the sides of right angle triangle is positive in first quadrant and negative in fourth quadrant. That is the reason we have tangent positive in first. And fourth quadrant.