Math Problem Statement
Solution
You are asked to express in the form . Here's how to proceed:
Step 1: General Formula
We can express a linear combination of sine and cosine functions as a single sine function using the following identity:
We want to match this with the given expression .
Step 2: Match Coefficients
Comparing with , we get:
Step 3: Solve for and
To find , we use the Pythagorean identity:
So, .
To find , we take the tangent of :
Thus,
Final Expression
The expression can be written as:
Would you like more details or have any questions on this?
Here are 5 related questions:
- How do you apply similar transformations for other trigonometric expressions?
- Can this approach work with functions involving instead of ?
- How do you find when the ratio is more complex than ?
- Can this method be used for other combinations of sine and cosine terms?
- What would the transformation look like if the coefficients were both negative?
Tip: The identity helps simplify trigonometric expressions into a single sinusoidal form, often used in solving equations and analyzing oscillations.
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Math Problem Analysis
Mathematical Concepts
Trigonometric Identity
Sine and Cosine
Formulas
Rsin(x + α) = Rsin(x)cos(α) + Rcos(x)sin(α)
tan(α) = Rsin(α)/Rcos(α)
Pythagorean Identity: R = √(Rcos(α)^2 + Rsin(α)^2)
Theorems
Trigonometric Identity for Sine and Cosine Combination
Suitable Grade Level
Grades 10-12