https://r2.erweima.ai/mathbot/image/1726263308491kms5f63l.jpg

Let's work through the second equation, number **15**, from your uploaded image:

### Problem 15: 
\[
2\cos^2 \theta + \cos\theta - 1 = 0, \quad 0^\circ \leq \theta \leq 360^\circ
\]

This is a quadratic equation in terms of \(\cos \theta\), and can be solved using factoring or the quadratic formula.

1. **Step 1: Solve the quadratic equation:**

Let \( u = \cos \theta \). The equation becomes:
\[
2u^2 + u - 1 = 0
\]

To factor this quadratic equation, look for two numbers that multiply to give \(2 \times (-1) = -2\) and add up to \(1\). The factors are \(2u^2 + 2u - u - 1 = 0\).

Group and factor:
\[
(2u - 1)(u + 1) = 0
\]

So, the solutions for \(u\) are:
\[
u = \frac{1}{2} \quad \text{or} \quad u = -1
\]

2. **Step 2: Solve for \(\theta\):**

Now, substitute back \( u = \cos \theta \):

- For \(\cos \theta = \frac{1}{2}\), the general solutions are:
\[
\theta = 60^\circ \quad \text{or} \quad \theta = 300^\circ
\]

- For \(\cos \theta = -1\), the solution is:
\[
\theta = 180^\circ
\]

3. **Step 3: Final solution:**

The solutions for \(\theta\) in the interval \(0^\circ \leq \theta \leq 360^\circ\) are:
\[
\theta = 60^\circ, 180^\circ, 300^\circ
\]

Would you like a detailed explanation on any specific part of this? Also, here are five related questions for further practice:

1. How do you solve a quadratic trigonometric equation involving sine instead of cosine?
2. What is the general solution for \(\cos \theta = -\frac{1}{2}\) in radians?
3. How can you derive the trigonometric identity for \(\cos^2 \theta + \sin^2 \theta = 1\)?
4. Can quadratic equations in trigonometry always be factored?
5. What is the general solution for \(\sin \theta = \frac{1}{2}\)?

**Tip:** Always check the unit (degrees or radians) for the domain of the trigonometric function before solving.

2\cos^2 \theta + \cos\theta - 1 = 0, 0^\circ \leq \theta \leq 360^\circ

Learn how to solve the quadratic trigonometric equation 2cos²θ + cosθ - 1 = 0 in the interval 0° ≤ θ ≤ 360°. This step-by-step guide explains factoring, finding the cosine values, and identifying angles. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation