Question

Suppose that cos(t) = 8/17 and t is in the 4th quadrant. Evaluate sin(t+t) to 2 decimal places.

Answer 1

well without much more information provided this looks like a double angle question

so sin(t + t) = sin(2t)

there is a double angle identity for this

sin(2t) = 2sin(t)cos(t)

so all you need to do is find sin(t)

since you are in the 4th quadrant sin will be negative...

you have a right triangle with adjacent side of 8 and hypotenuse of 17

find the length of the opposite side using Pythagoras' theorem..


`o^2 = 17^2 - 8^2`

so the opposite side is 15

then
`sin(t) = -(15)/(17)`

so sin(2t) is


`sin(2t) = 2* (- (15)/(17))* (8)/(17)`

hope it makes sense... just evaluate it for the answer

Answer 2

sin(t+t) = 0.83

Explanation:

cos (t) = 8/17 we know that cos x = adjacent side / hypotenuse

(You can take a look at the image below to give you an idea of how it looks)

The problem asks us to evaluate sin (t + t) = sin (2t), however we have a formula to help us find this in terms of t:

sin (2t) = 2(sint)(cost)

So now we need to find sin(t)

• If we need to find sin (t) we will need to find the opposite side, but in the figure we can see that to find it we can use the Pythagoras Theorem for the opposite side (OS).
• Writing this as an equation and solving for the opposite side we get:

OS² = 17² - 8² = 289 - 64 = 225

OS = √225

OS = 15

But since the angle is in the 4th quadrant, then the OS = -15.

Therefore, sin(t) = opposite side / hypotenuse = -15/17

Now we have sin (t) = -15/17 and cos(t) = 8/17 and we can substitute this in the formula for sin (2t)

`sin(2t)= 2(sint)(cost)\\sin(2t)=2((-15)/(17))((8)/(17))\\  sin(2t)= (-240)/(289) \\sin(2t)=-0.8304`

Therefore, sin(2t) = 0.83 (rounded to two decimal places)