Question

Find the exact values of sin(2u), cos(2u), and tan(2u) using the double-angle formulas.

3T
cot(u) = -4, 2
sin(2u)
cos(2u)
tan(2u)
-
=

Answer 1

sin(2u) = - 8/17

cos(2u) = 15/17

tan(2u) = -8/15

Explanation:

`\sf cot(u) = -4, (3\pi)/(3) < u < 2\pi`

To find the exact values of sin(2u), cos(2u), and tan(2u) using the double-angle formulas, we will need to know the value of sin(u) and cos(u).

We can use the Pythagorean identity to find sin(u) and cos(u):

`\sf sin^2(u) + cos^2(u) = 1`

Since cot(u) = -4, we know that cos(u) / sin(u) = -4. We also know that cos(u) > 0 because u is in the third quadrant. Therefore, we can say that:

`\sf cos(u) = -4 sin(u)`

Substituting this into the Pythagorean identity, we get:

`\sf (-4 sin(u))^2 + sin^2(u) = 1`

`\sf 16 sin^2(u) + sin^2(u) = 1`

`\sf 17 sin^2(u) = 1`

`\sf sin(u) = \pm (1)/(√(17))`

Since u is in the third quadrant, we know that sin(u) is negative.

Therefore,

`\sf sin(u) = - (1)/(√(17))`

`\sf cos(u) = -4 \left(- (1)/(√(17))\right) = (4)/(√(17))`

Now that we know sin(u) and cos(u), we can use the double-angle formulas to find sin(2u), cos(2u), and tan(2u):

`\sf sin(2u) = 2 sin(u) cos(u)`

`\sf cos(2u) = cos^2(u) - sin^2(u)`

`\sf tan(2u) = (2 sin(u) cos(u))/(cos^2(u) - sin^2(u))`

Substituting in
`\sf sin(u) = - (1)/(√(17))` and
`\sf cos(u) = (4)/(√(17))` , we get:

`\begin{aligned} \sf sin(2u) & = 2sin(u)cos(u)\\\\ & = 2 \cdot - (1)/(√(17)) \cdot (4)/(√(17))\\\\ & = (2\cdot -1\cdot 4 )/(√(17)\cdot √(17)) \\\\ & =- (8)/(17) \end{aligned}`

Now

`\begin{aligned} \sf cos(2u) & =\sf cos^2(u) - sin^2(u) \\\\ & = \left( (4)/(√(17)) \right)^2 - \left(- (1)/(√(17))\right)^2 \\\\ &= (16)/(17) - (1)/(17) \\\\ & = ( 16-1)/(17) \\\\ & = (15)/(17)\end{aligned}`

And Finally:

`\begin{aligned} \sf tan(2u)& = \sf (2 sin(u) cos(u))/(cos^2(u) - sin^2(u)) \\\\ & =( - (8)/(17) )/((15)/(17) ) \\\\ & = - \frac{ 8}{\cancel{17}}\cdot \frac{\cancel{17}}{15} \\\\ & =- ( 8)/(15)\end{aligned}`

Therefore, the exact values of sin(2u), cos(2u), and tan(2u) are:

sin(2u) = - 8/17

cos(2u) = 15/17

tan(2u) = -8/15