how to find 2sinxcosx, given cos x = -2/3, the domain is the second quadrant

We are tasked with finding \( 2 \sin x \cos x \) given that \( \cos x = -\frac{2}{3} \) and that the angle \( x \) lies in the second quadrant. To do this, let's follow these steps:

### Step 1: Recall the identity for \( 2 \sin x \cos x \)
The expression \( 2 \sin x \cos x \) is a known trigonometric identity:
\[
2 \sin x \cos x = \sin(2x)
\]
So we need to first find \( \sin x \), and then we can calculate \( \sin(2x) \).

### Step 2: Find \( \sin x \)
We know that \( \cos x = -\frac{2}{3} \) and that the angle \( x \) is in the second quadrant. In the second quadrant, \( \sin x \) is positive. 

Using the Pythagorean identity:
\[
\sin^2 x + \cos^2 x = 1
\]
Substitute \( \cos x = -\frac{2}{3} \) into this equation:
\[
\sin^2 x + \left( -\frac{2}{3} \right)^2 = 1
\]
\[
\sin^2 x + \frac{4}{9} = 1
\]
\[
\sin^2 x = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9}
\]
\[
\sin x = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
\]
Since \( x \) is in the second quadrant, \( \sin x \) is positive, so:
\[
\sin x = \frac{\sqrt{5}}{3}
\]

### Step 3: Find \( \sin(2x) \)
Now that we have \( \sin x \) and \( \cos x \), we can use the double-angle identity for sine:
\[
\sin(2x) = 2 \sin x \cos x
\]
Substitute the known values:
\[
\sin(2x) = 2 \left( \frac{\sqrt{5}}{3} \right) \left( -\frac{2}{3} \right)
\]
\[
\sin(2x) = 2 \times \frac{-2\sqrt{5}}{9} = \frac{-4\sqrt{5}}{9}
\]

### Final Answer:
\[
2 \sin x \cos x = \sin(2x) = \frac{-4\sqrt{5}}{9}
\]

Let me know if you want further clarification or details!

Here are 5 related questions to explore:
1. How can the Pythagorean identity be used to find other trigonometric values?
2. What are the signs of \( \sin x \) and \( \cos x \) in different quadrants?
3. How does the double-angle identity apply to different trigonometric functions?
4. What is the significance of knowing the quadrant for trigonometric functions?
5. How can you find the angle \( x \) given \( \sin x \) or \( \cos x \)?

**Tip**: Always check the quadrant carefully as it determines the sign of sine, cosine, and tangent values.

Learn how to calculate 2sinxcosx given that cos x = -2/3 and the angle is in the second quadrant. This trigonometry problem covers key concepts like the double-angle identity and Pythagorean identity. Ideal for students in 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