cosec theta-cot theta=1/2. Find cos(theta). if 0

To solve the equation cosec(θ) - cot(θ) = 1/2 and find cos(θ), we can start by expressing cosec(θ) and cot(θ) in terms of sine and cosine. Recall that cosec(θ) is the reciprocal of sin(θ), and cot(θ) is the ratio of cos(θ) to sin(θ). This will help us rewrite the equation in a more manageable form.

Rewriting the Equation

The identities we need are:

• cosec(θ) = 1/sin(θ)
• cot(θ) = cos(θ)/sin(θ)

Substituting these into the equation gives us:

1/sin(θ) - cos(θ)/sin(θ) = 1/2

Combining the Terms

We can combine the left side since they share a common denominator:

(1 - cos(θ))/sin(θ) = 1/2

Cross-Multiplying

Next, we can cross-multiply to eliminate the fraction:

2(1 - cos(θ)) = sin(θ)

Using the Pythagorean Identity

We know from the Pythagorean identity that sin²(θ) + cos²(θ) = 1. Therefore, we can express sin(θ) in terms of cos(θ):

sin(θ) = √(1 - cos²(θ))

Substituting this back into our equation gives:

2(1 - cos(θ)) = √(1 - cos²(θ))

Squaring Both Sides

To eliminate the square root, we square both sides:

4(1 - cos(θ))² = 1 - cos²(θ)

Expanding and Rearranging

Expanding the left side:

4(1 - 2cos(θ) + cos²(θ)) = 1 - cos²(θ)

This simplifies to:

4 - 8cos(θ) + 4cos²(θ) = 1 - cos²(θ)

Now, bringing all terms to one side gives us:

5cos²(θ) - 8cos(θ) + 3 = 0

Solving the Quadratic Equation

This is a standard quadratic equation in the form of ax² + bx + c = 0, where:

• a = 5
• b = -8
• c = 3

We can apply the quadratic formula:

cos(θ) = [ -b ± √(b² - 4ac) ] / 2a

Plugging in the values:

cos(θ) = [ 8 ± √((-8)² - 4 * 5 * 3) ] / (2 * 5)

Calculating the discriminant:

64 - 60 = 4

Now substituting back:

cos(θ) = [ 8 ± 2 ] / 10

Finding the Values of cos(θ)

This gives us two potential solutions:

• cos(θ) = (8 + 2) / 10 = 10 / 10 = 1
• cos(θ) = (8 - 2) / 10 = 6 / 10 = 0.6

Considering the Range of θ

Now, it's essential to consider the context of the problem. The values of cos(θ) must be within the range of -1 to 1. The value cos(θ) = 1 corresponds to θ = 0, while cos(θ) = 0.6 is valid for angles in the first and fourth quadrants.

Hence, the solutions are:

• cos(θ) = 1 (θ = 0°)
• cos(θ) = 0.6 (θ ≈ 53.13° or θ ≈ 306.87°)

To sum up, the values of cos(θ) that satisfy the original equation are 1 and 0.6.