having trouble with these type of problems

Question

Consider the following.

t= -4π/3

(a) Find the reference number t for the value of t.

t=

(b) Find the terminal point determined by t.

(x,y)=

Answer 1

In Mathematics, we never use the term "reference number". That term is use mostly in business transaction and databases. So I'm assuming that you are looking for reference angle because the t is a measure of angle in radians.

Reference angle is the positive acute angle (less than 90˚ or less than π /2) with the x-axis that can represent an angle of any measure.

(1)If the angle measure is greater the 2π or 360˚ , then you keep on subracting 2π or 360˚ until you reach the smallest positive angle measure.

(2) if the angle measure is less than 0, then you keep on adding 2π or 360˚ until you reach the smallest positive angle measure.

so for t= -4π / 3,

 θ = -4π / 3 + 2π = 2π / 3. This is greater than π / 2 and located on the 2nd quadrant.

If 0 < θ < π / 2 (1st quadrant), the reference angle is θ,

if π / 2 <  θ < π (2nd quadrant), the reference angle is π - θ,

if π <  θ < 3π / 2 (3rd quadrant), the reference angle is  θ- π,

if 3π / 2 <  θ< 2π (4th quadrant), the reference angle is 2π -  θ.

Since 2π / 3 is in π / 2 < θ < π (2nd quadrant), then the reference angle is:

(a) π - 2π / 3 = π / 3

For (b), we can have only one value of (x,y) if the radius is given. Since it is not given, therefore we can have several values of (x,y) but they are all proportional. I'm going to assume you are referring an angle in a unit circle (meaning radius = 1)

(x,y) = (-1/2, √3 / 2)