Math Problem Statement

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

Solution

We are tasked with finding 2sinxcosx2 \sin x \cos x given that cosx=23\cos x = -\frac{2}{3} and that the angle xx lies in the second quadrant. To do this, let's follow these steps:

Step 1: Recall the identity for 2sinxcosx2 \sin x \cos x

The expression 2sinxcosx2 \sin x \cos x is a known trigonometric identity: 2sinxcosx=sin(2x)2 \sin x \cos x = \sin(2x) So we need to first find sinx\sin x, and then we can calculate sin(2x)\sin(2x).

Step 2: Find sinx\sin x

We know that cosx=23\cos x = -\frac{2}{3} and that the angle xx is in the second quadrant. In the second quadrant, sinx\sin x is positive.

Using the Pythagorean identity: sin2x+cos2x=1\sin^2 x + \cos^2 x = 1 Substitute cosx=23\cos x = -\frac{2}{3} into this equation: sin2x+(23)2=1\sin^2 x + \left( -\frac{2}{3} \right)^2 = 1 sin2x+49=1\sin^2 x + \frac{4}{9} = 1 sin2x=149=9949=59\sin^2 x = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9} sinx=59=53\sin x = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} Since xx is in the second quadrant, sinx\sin x is positive, so: sinx=53\sin x = \frac{\sqrt{5}}{3}

Step 3: Find sin(2x)\sin(2x)

Now that we have sinx\sin x and cosx\cos x, we can use the double-angle identity for sine: sin(2x)=2sinxcosx\sin(2x) = 2 \sin x \cos x Substitute the known values: sin(2x)=2(53)(23)\sin(2x) = 2 \left( \frac{\sqrt{5}}{3} \right) \left( -\frac{2}{3} \right) sin(2x)=2×259=459\sin(2x) = 2 \times \frac{-2\sqrt{5}}{9} = \frac{-4\sqrt{5}}{9}

Final Answer:

2sinxcosx=sin(2x)=4592 \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 sinx\sin x and cosx\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 xx given sinx\sin x or cosx\cos x?

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

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Pythagorean Identity
Double-Angle Identity

Formulas

2sinxcosx = sin(2x)
sin^2x + cos^2x = 1
sin(2x) = 2sinxcosx

Theorems

Pythagorean Identity
Double-Angle Identity

Suitable Grade Level

Grades 10-12