Need help solving this math problem with trigonometric functions :)

Question

Suppose the terminal point determine by t lies in the second quadrant and is (-24/25, y).

 

(a) Find y.

 

(b) Find the value of the 6 trigonometric functions at 837π + t

 

I believe the answer to a. is found by doing this:

x^2 + y^2 = 1

(-24/25)^2 + y^2 = 1 

y^2 = 1 - 0.9216

y = .28

so the answer would be (-24/25, .28) 

 

But I am not sure how to solve question b. I would really appreciate some help with question b.

Answer 1

Hi Chloe,

 

Your answer to a looks correct.

 

For part b, all you need to figure out is what is the final rotation around the unit circle for 837π + t. Since the period is 2pi, we are only interested in the remainder after division by 2pi to figure out where we end in the unit circle. 

837pi when divided by 2pi leaves a remainder of 1pi. So, we have pi+t. All you need to do is move pi radians from the terminal point of t, which should bring us to the fourth quadrant. 

Now, you can find all the trigonometric functions paying attention to the signs in this quadrant. Remember x axis is positive and y axis is negative in the fourth quadrant.

I hope this all makes sense.

Answer 2

Your answer to (a) is correct.  I would convert 0.28 to a fraction, though, to be consistent with the x value of -24/25:

 

y = 0.28 = 28/100 = (7*4)/(25*4) = 7/25

 

So the answer is (-24/25, 7/25).

 

For b, determine the value of angle t using the inverse tangent:

 

t = tan^-1( (7/25)/(-24/25) ) = tan^-1(-7/24)

 

Use your calculator to get the answer.  837pi is a radian measure.  There are 2pi radians in one full circle, so 837pi is 418 fill circles around plus an extra pi.  You can ignore the 418 full circles around because it just brings you back to your starting point at 0 radians.  So 837pi + t = pi + t.  So just add pi radians (or 180 degrees if using degrees) to t.  The new endpoint for pi+t would be (24/25,-7/25).  To find the other trig functions of t+pi, you must use the reference angle which is the smallest angle between the new terminator line (for pi+t) and the x-axis.  The sine, for example would be:

 

sin(t+pi) = (-7/25)/1 = -7/25

cos(t+pi) =