If 5secθ- 12cosecθ = 0, find the values of secθ, cosθ and sinθ.

Solution

We know that sec(θ) = 1/cos(θ) and cosec(θ) = 1/sin(θ), so we can rewrite the given equation as:

5/cos(θ) – 12/sin(θ) = 0

Multiplying both sides by cos(θ) * sin(θ), we get:

5sin(θ) – 12cos(θ) = 0

Dividing both sides by 5cos(θ), we get:

sin(θ)/cos(θ) = 12/5

This is the same as tan(θ) = 12/5.

Now we can use the Pythagorean theorem to find the value of the hypotenuse:

sin²(θ) + cos²(θ) = 1

Substituting sin(θ)/cos(θ) = 12/5, we get:

(12/5)² + cos²(θ) = 1

Solving for cos(θ), we get:

cos(θ) = ± √[1 – (12/5)²] = ± √(119)/5

Since 5sec(θ) – 12cosec(θ) = 0, we know that either sec(θ) = 12/5 or cosec(θ) = 5/12.

If sec(θ) = 12/5, then cos(θ) = 5/12, which contradicts the value we just found for cos(θ).

Therefore, cosec(θ) = 5/12 and sin(θ) = 12/5.

Finally, we can use the definition of sec(θ) = 1/cos(θ) to find the value of sec(θ):

sec(θ) = 1/cos(θ) = 1/[(± √(119))/5] = ± 5/√(119)

So, the values of sec(θ), cos(θ), and sin(θ) are:

sec(θ) = ± 5/√(119), cos(θ) = ± √(119)/5, sin(θ) = 12/5.