Question

Solve the following equation for θ where 0≤θ<π. sec(−2θ+2π5)=csc(−2θ+33π10)

Answer 1

Thus, the solution for θ in the given range
`$0 \leq \theta < \pi$ is $\theta=(4 \pi)/(5)$`

To solve the trigonometric equation
`\(\sec(-2\theta + (2\pi)/(5)) = \csc(-2\theta + (33\pi)/(10))\), where \(0 \leq \theta < \pi\),` we'll follow these steps:

1. Express in terms of sine and cosine: Convert secant and cosecant to their equivalent in terms of sine and cosine.

`\[ \sec(x) = (1)/(\cos(x)) \quad \text{and} \quad \csc(x) = (1)/(\sin(x)) \]`

2. Rewrite the equation: Substitute the expressions for secant and cosecant.

3. Manipulate the equation: Try to simplify and solve for
`\(\theta\).`

4. Consider the domain: Ensure the solutions are within
`\(0 \leq \theta < \pi\).`

Let's start solving it step by step.

Step 1: Express in Terms of Sine and Cosine

`\[(1)/(\cos(-2\theta + (2\pi)/(5))) = (1)/(\sin(-2\theta + (33\pi)/(10)))\]`

Step 2: Rewrite the Equation

`\[\sin(-2\theta + (33\pi)/(10)) = \cos(-2\theta + (2\pi)/(5))\]`

Step 3: Manipulate the Equation

Use the identity
`\(\sin(x) = \cos((\pi)/(2) - x)\)` to rewrite the sine term as a cosine:

`\[\cos((\pi)/(2) - (-2\theta + (33\pi)/(10))) = \cos(-2\theta + (2\pi)/(5))\]`

Simplify both sides:

`\[\cos(2\theta - (23\pi)/(10)) = \cos(-2\theta + (2\pi)/(5))\]`

Step 4: Solve for θ

For
`\(\cos(A) = \cos(B)\), the general solutions are \(A = B + 2n\pi\) or \(A = -B + 2n\pi\) for any integer \(n\)`. Apply this to our equation:

`\[2\theta - (23\pi)/(10) = -2\theta + (2\pi)/(5) + 2n\pi \quad \text{or} \quad 2\theta - (23\pi)/(10) = 2\theta - (2\pi)/(5) + 2n\pi\]`

Solving these equations will give us the values of
`\(\theta\)` in the given domain. Let's calculate these solutions.

After solving the equations, we find the following:

1. For the first equation, the solution is
`\(\theta = (\pi(20n + 27))/(40)\).`

2. The second equation does not yield any additional solutions.

Since we are interested in solutions where
`\(0 \leq \theta < \pi\), we need to find the values of \(n\) for which \(\theta\)` falls within this range.

Let's calculate these specific values for n and the corresponding values of
`\(\theta\).`

The valid values of
`\( n \) that provide solutions for \( \theta \) within the range \( 0 \leq \theta < \pi \) are \( n = -1 \) and \( n = 0 \). Correspondingly, the solutions for \( \theta \)` are:

1. For
`\( n = -1 \), \( \theta = (7\pi)/(40) \).`

2. For
`\( n = 0 \), \( \theta = (27\pi)/(40) \).`

These are the solutions for the given trigonometric equation within the specified domain.