Question

I will give someone a quadrillion awesome points...
(comments)

Answer 1

Watch......

Answer 2

cos (2t) = 2 cos^2t - 1 = 2(-9/11)^2 - 1 = 162/121 - 1 = 41/121

Answer 3

sin (2t) = 2 sin t cos t = 2(-9/11)(- sqrt(40)/11)

Answer 4

Let me make some comments for the above:

for cos(2t) ..I used one of the double-angle formulas which is 2 cos^2 (t) -1 and I substituted.

Answer 5

Easier....draw a picture. The angle is in the 3rd quadrant. Both sine and cosine are negative there. Use the pythagorean theorem to find the length of the missing side. Then just use trig ratios.

Answer 6

The second one, the double-angle formula for sin(2t) = 2 sin t cos t..now we know that cos t = -9/11, the question is what is the sin t? Answer is that t is in the third quadrant, as stated in the stem of the question, so sin in 3rd quadrant is negative. and to find the value, set up a right triangle and find the missing side to be sqrt(40).

Do the first 2 parts make sense? Any questions before I move on?

Answer 7

No diagram needed for cos(2t). Just use the formula and substitute. Period.

Answer 8

No formulas needed if you look at the diagram.

Answer 9

For sin(2t) you need a diagram as I stated above. Because you are missing sin t and you arent given that in the stem of the question.

Answer 10

The moment you mentioned the formula, it became a lot clearer. So no questions pertaining to the top 2.

Answer 11

The diagram is also a good way to go.

Answer 12

OK..lets move along to part (3).

cos (1/2 t)..thats the half-angle formula

Answer 13

sin (1/2 t) = +/- sqrt ((1-cos t)/2)
= - sqrt (1- -9/11)/2)
= - sqrt (20/11 divided by 2)
= - sqrt (20/22)

Answer 14

You can compute that on the calculator.

Now, one note on this part.

Answer 15

The formula for sin (1/2t) and cos(1/2 t) both have a + and - in front of the sqrt expression. You have to select which one, + sqrt or - sqrt. To know how to pick which one, well, we know the angle t is in the 3rd quadrant, so 1/2 of a third quadrant angle will always lie in the second quadrant. so cos (1/2 t) is negative in this problem as cos is negative in quadrant 2. Thats why I have a - sign in front of the sqrt.

Answer 16

Now, the last part, sin (1/2t)...same idea..but there will be a + in front of the square root sign because sin in quadrant 2 is positive.

so it will be + sqrt ((1+ cos t)/2)) = + sqrt (1-9/11)/2))
= + sqrt (1/11))

Answer 17

I gave exact answers, not decimals, as required in the stem of the question.

Answer 18

You can go through my arithmetic to see if I made any error, as I did them all in my head. Feel free to write to me if you have any questions whatsoever or if you dot understand something.

Answer 19

For now, I understand I'll take time to revise. Your time, patience and work is appreciated

Answer 20

Always use a formula where the formula gives you the direct solution. Thats what I did.

I use a diagram when the diagram is needed, like in part (2) above.

Answer 21

Thanks a lot.

Answer 22

Welcome. Make sure I didnt make an arithmetic mistake.

Answer 23

I dont think so, but I am human.