find cos(θ)ˏsin(θ)ˏtan(θ), if cot (2θ)=5/12 with 0≤2θ≤π

Answer this Que

if cot 2θ = 5/12, the 2θ is in QI or QIII. But, since 2θ<π, we must be in QI.

so, sin2θ = 12/13, cos2θ = 12/13

Now use the half-angle formulas to find sinθ, cosθ, tanθ.

To find the values of cos(θ), sin(θ), and tan(θ), we need to use the given information that cot(2θ) = 5/12 and 0 ≤ 2θ ≤ π.

First, let's rewrite cot(2θ) in terms of sine and cosine using the identity cot(θ) = cos(θ)/sin(θ):

cot(2θ) = 5/12
cos(2θ)/sin(2θ) = 5/12

Now, we can use the double angle formulas to express cos(2θ) and sin(2θ) in terms of cos(θ) and sin(θ). The double angle formulas are:

cos(2θ) = cos²(θ) - sin²(θ)
sin(2θ) = 2sin(θ)cos(θ)

Let's substitute the expressions for cos(2θ) and sin(2θ) into the equation:

(cos²(θ) - sin²(θ))/(2sin(θ)cos(θ)) = 5/12

Now, let's simplify this equation:

(cos²(θ) - sin²(θ))/(2sin(θ)cos(θ)) = 5/12
(cos²(θ) - sin²(θ))/(2sin(θ)cos(θ)) - 5/12 = 0

At this point, we have a quadratic equation in terms of sin(θ) and cos(θ). To solve this equation, we can substitute a variable, such as x, for cos(θ), and use trigonometric identities to relate sin(θ) to x.

Let's substitute x = cos(θ) and rewrite sin(θ) in terms of x:

sin(θ) = √(1 - cos²(θ))
= √(1 - x²)

Now, let's replace sin(θ) and cos(θ) with their equivalent expressions in the quadratic equation:

(x² - (1 - x²))/(2√(1 - x²)x) - 5/12 = 0

Simplifying this equation, we get:

(2x² - 1)/(2√(1 - x²)x) - 5/12 = 0

Now, let's cross multiply and simplify:

12(2x² - 1) - 5(2√(1 - x²)x) = 0
24x² - 12 - 10√(1 - x²)x = 0

At this point, we have a quadratic equation in terms of x. By solving the quadratic equation, we can find the value(s) of x (corresponding to cos(θ)) that satisfy the equation.

Once we find the value(s) of x, we can substitute that into the expression for sin(θ) = √(1 - x²) to find the corresponding value(s) of sin(θ).

Finally, we can use the values of cos(θ) and sin(θ) to calculate the value of tan(θ) using the equation tan(θ) = sin(θ)/cos(θ).