Determine the following using the identities

Question

Using the Identities determine the following

a. 𝑠𝑖𝑛(2𝜃), given that 𝑡𝑎𝑛(𝜃) = 5/3 and 𝑠𝑖𝑛(𝜃) < 0

b. 𝑡𝑎𝑛 ( 𝜃/2 ), given that 𝑡𝑎𝑛(𝜃) = − √5/2 and 𝜋/2 < 𝜃 < pi

Answer 1

Note that sin(2θ) will have the same sign as tan(θ), so we didn't need the sign of sin(θ).

If tan(θ) = 5/3 and sin(θ) is negative, so is cos(θ). Then, sin(θ) = -5/√34 = -5√34/34 and cos(θ) = -3/√34 = -3√34/34 (think of a 5-3-√34 right triangle)

Then, sin(2θ) = 2sin(θ)cos(θ) = 2 * -5√34/34 * -3√34/34 = 15/17

tan(θ/2) = √((1-cos(θ))/(1+cos(θ)))

If tan(θ) = -√5/2 and π/2 < θ < π, cos(θ) will be negative, and we will have a 2-√5-3 right triangle, so cos(θ) = -2/3; also π/4 < θ/2 < π/2, so tan(θ) > 1.

Then, √((1-cos(θ))/(1+cos(θ))) = √((1--2/3)/(1+-2/3)) = √((5/3)/(1/3)) = √5

Answer 2

sine is negative and tangent is positive in quadrant 3

So pi < T < 3*pi/2

tan T = 5/3 ----> y=opposite = 5 and x=adjacent = 3

hypothesis = sqrt( 9 + 25) = sqrt(34)

sin T = 5/sqrt(34) = 5*sqrt(34)/ 34

cos T = 3/sqrt(34) = 3 * sqrt(34)/ 34

tan T = 5/3

sin (2T) = 2 sin T cos T = 2 ( 5 * sqrt(34))/34 * (3 * sqrt(34))/34

= 30 sqrt(34)/34

= (15/17) sqrt(34)

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In quadrant 2, sine is positive, cosine and tangent are negative

y=opposite = sqrt(5)

x=adjacent = -2

hypothesis = sqrt( 4+ 5) = 3

sin T = sqrt(5)/3

cos T = -2/3

tan T/2 = sin T / (1 + cos T)

= sqrt(5)/3 divided by ( 1 + -2/3)

= sqrt(5)/3 divided by 1/3

= sqrt(5)