Finding exact value without calculator?

Question

If tan theta = 7/24 when pi <= theta <= 3pi/2 find the exact values of sin(2theta)

Answer 1

Hi Mario

 

tanθ = 7/24    for  θ  between π and 3π//2 in the third quadrant. Note that sinθ and cosθ are negative in this quadrant.

 

sin(2θ) = 2sinθcosθ 

 

tan θ = 7/24 =opposite/ adjacent,

sinθ = opposite/ hypotenuse,

cosθ = adjacent/hypotenuse

 

hypotenuse =√[(7)² + (24)²] =√625 = 25

 

sin(2θ) = 2(-7/25)(-24/25)

 

sin(2θ)=336/625

Answer 2

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

 

Given -

Tan(theta)=7/24

So.sin(theta)=-7/25.       - sign because the angle lies in third quadrant.

 

cos(theta)=-24/25.         - sign because the angle lies in third quadrant

 

So,

 

Sin(2theta)=2*(-7/25)*(-24/25)

                = 336/625