Find the exact values of sin 2theta, cos 2 theta and tan 2 theta when cot theta = 4/3
No idea
Tan(2 theta)=-1
To find the exact values of sin 2θ, cos 2θ, and tan 2θ, when cot θ = 4/3, we can start by using the given cot θ value to find the value of tan θ.
We know that cot θ is the reciprocal of tan θ (cot θ = 1/tan θ).
Given cot θ = 4/3, we can find tan θ by taking the reciprocal of cot θ:
tan θ = 1/(cot θ) = 1/(4/3) = 3/4.
Now that we have the value of tan θ, we can use it to find the values of sin 2θ and cos 2θ using the double angle formulas.
The double angle formulas for sin 2θ and cos 2θ are:
sin 2θ = 2sin θ * cos θ
cos 2θ = cos^2 θ - sin^2 θ
To find sin θ and cos θ, we can use the Pythagorean identity, sin^2 θ + cos^2 θ = 1.
From tan θ = 3/4, we can see that the opposite side of the right triangle is 3 and the adjacent side is 4. Therefore, the hypotenuse can be found using the Pythagorean theorem:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 3^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25
hypotenuse = √25 = 5
Therefore, sin θ = opposite/hypotenuse = 3/5, cos θ = adjacent/hypotenuse = 4/5.
Now we can substitute these values into the double angle formulas:
sin 2θ = 2sin θ * cos θ = 2(3/5)(4/5) = 24/25
cos 2θ = cos^2 θ - sin^2 θ = (4/5)^2 - (3/5)^2 = 16/25 - 9/25 = 7/25
Finally, we can find tan 2θ by using the identity: tan 2θ = (2tan θ) / (1 - tan^2 θ):
tan 2θ = (2 * (3/4)) / (1 - (3/4)^2) = (6/4) / (1 - 9/16) = (6/4) / (16/16 - 9/16)
tan 2θ = (6/4) / (7/16) = (6/4) * (16/7) = 96/28 = 24/7.
Therefore, the exact values of sin 2θ, cos 2θ, and tan 2θ when cot θ = 4/3 are:
sin 2θ = 24/25
cos 2θ = 7/25
tan 2θ = 24/7.