cos Θ - 1  -    sin Θ__    = 2  cot Θ

          sin Θ        cos Θ - 1

We are given the equation:

\[
\frac{\cos \theta - 1 - \sin \theta}{\sin \theta} = 2 \cot \theta
\]

We aim to simplify and solve this equation.

### Step 1: Express \(\cot \theta\) in terms of \(\sin \theta\) and \(\cos \theta\)
Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). So the right-hand side becomes:

\[
2 \cot \theta = 2 \cdot \frac{\cos \theta}{\sin \theta} = \frac{2 \cos \theta}{\sin \theta}
\]

Now our equation becomes:

\[
\frac{\cos \theta - 1 - \sin \theta}{\sin \theta} = \frac{2 \cos \theta}{\sin \theta}
\]

### Step 2: Multiply both sides by \(\sin \theta\)
This eliminates the denominators:

\[
\cos \theta - 1 - \sin \theta = 2 \cos \theta
\]

### Step 3: Simplify the equation
Rearrange the terms to combine like terms:

\[
\cos \theta - 1 - \sin \theta - 2 \cos \theta = 0
\]

\[
- \cos \theta - \sin \theta - 1 = 0
\]

### Step 4: Solve for \(\theta\)
Now, we have the equation:

\[
\cos \theta + \sin \theta = -1
\]

We can solve this equation for specific values of \(\theta\). One known solution is \(\theta = \frac{3\pi}{4}\), since \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\), and their sum is:

\[
-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = -1
\]

Thus, one solution is \(\theta = \frac{3\pi}{4}\).

Would you like more details on this process or other possible solutions?

Here are 5 related questions:
1. How do we graphically represent trigonometric identities?
2. What are other possible values of \(\theta\) that satisfy this equation?
3. Can this equation be solved using a graphical or numerical method?
4. How do we verify the trigonometric identities used here?
5. How does the unit circle help in solving trigonometric equations?

**Tip**: When solving trigonometric equations, it's useful to remember fundamental identities such as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) for simplifying expressions.

cos Θ - 1  - sin Θ / sin Θ = 2 cot Θ / cos Θ - 1

This article walks through solving the trigonometric equation (cos Θ - 1 - sin Θ) / sin Θ = 2 cot Θ. It covers important concepts such as the cotangent identity and basic trigonometric relationships, and it provides a detailed, step-by-step solution. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation