Math Problem Statement
how to find 2sinxcosx, given cos x = -2/3, the domain is the second quadrant
Solution
We are tasked with finding given that and that the angle lies in the second quadrant. To do this, let's follow these steps:
Step 1: Recall the identity for
The expression is a known trigonometric identity: So we need to first find , and then we can calculate .
Step 2: Find
We know that and that the angle is in the second quadrant. In the second quadrant, is positive.
Using the Pythagorean identity: Substitute into this equation: Since is in the second quadrant, is positive, so:
Step 3: Find
Now that we have and , we can use the double-angle identity for sine: Substitute the known values:
Final Answer:
Let me know if you want further clarification or details!
Here are 5 related questions to explore:
- How can the Pythagorean identity be used to find other trigonometric values?
- What are the signs of and in different quadrants?
- How does the double-angle identity apply to different trigonometric functions?
- What is the significance of knowing the quadrant for trigonometric functions?
- How can you find the angle given or ?
Tip: Always check the quadrant carefully as it determines the sign of sine, cosine, and tangent values.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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