Question

Find the values of sin2u, cos2u, and tan2u given the figure.

Answer 1

Givens

y = 2

x = 1

z(the hypotenuse) = √(2^2 + 1^2)  = √5

Cos(u) = x value / hypotenuse = 1/√5

Sin(u) = y value / hypotenuse = 2/√5

Solve for sin2u

Sin(2u) = 2*sin(u)*cos(u)

Sin(2u) = 2($\dfrac{1}{\dsqrt{5}} * \frac{2}{\dsqrt{5}} = \dfrac{2}{5}$) = 4/5

Solve for cos(2u)

cos(2u) = - sqrt(1 - sin^2(2u))

Cos(2u) = - sqrt(1 - (4/5)^2 )

Cos(2u) = -sqrt(1 - 16/25)

cos(2u) = -sqrt(9/25)

cos(2u) = -3/5

Solve for Tan(2u)

tan(2u) = sin(2u) / cos(2u) = 4/5// - 3/5 = - 0.8/0.6 = - 1.3333 = - 4/3

Notes

One: Notice that you would normally rationalize the denominator, but you don't have to in this case.  The formulas are such that they perform the rationalizations themselves.

Two: Notice the sign on the cos(2u). The sin is plus even though the angle (2u) is in the second quadrant. The cos is different. It is about 126 degrees which would make it a negative root (9/25)

Three: If you are uncomfortable with the  tan, you could do fractions.

$\text{tan(2u)} = \dfrac{\dfrac{4}{5}}{\dfrac{-3}{5} } =\dfrac{4}{5} *\dfrac{5}{-3} =\dfrac{4}{-3}$