Math Problem Statement

Express 3sin(x) - 2cos(x) in the form ksin(x + α)°.

Solution

You are asked to express 3sinx2cosx3\sin x - 2\cos x in the form ksin(x+α)k\sin(x + \alpha)^\circ. 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:

Rsin(x+α)=Rsinxcosα+RcosxsinαR\sin(x + \alpha) = R\sin x \cos \alpha + R\cos x \sin \alpha

We want to match this with the given expression 3sinx2cosx3\sin x - 2\cos x.

Step 2: Match Coefficients

Comparing Rsinxcosα+RcosxsinαR\sin x \cos \alpha + R\cos x \sin \alpha with 3sinx2cosx3\sin x - 2\cos x, we get:

Rcosα=3(coefficient of sinx)R\cos \alpha = 3 \quad \text{(coefficient of } \sin x \text{)} Rsinα=2(coefficient of cosx)R\sin \alpha = -2 \quad \text{(coefficient of } \cos x \text{)}

Step 3: Solve for RR and α\alpha

To find RR, we use the Pythagorean identity:

R=(Rcosα)2+(Rsinα)2=32+(2)2=9+4=13R = \sqrt{(R\cos \alpha)^2 + (R\sin \alpha)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}

So, R=13R = \sqrt{13}.

To find α\alpha, we take the tangent of α\alpha:

tanα=RsinαRcosα=23\tan \alpha = \frac{R\sin \alpha}{R\cos \alpha} = \frac{-2}{3}

Thus,

α=tan1(23)33.69\alpha = \tan^{-1} \left( \frac{-2}{3} \right) \approx -33.69^\circ

Final Expression

The expression 3sinx2cosx3\sin x - 2\cos x can be written as:

13sin(x33.69)\sqrt{13}\sin(x - 33.69^\circ)

Would you like more details or have any questions on this?

Here are 5 related questions:

  1. How do you apply similar transformations for other trigonometric expressions?
  2. Can this approach work with functions involving cos(x)\cos(x) instead of sin(x)\sin(x)?
  3. How do you find α\alpha when the ratio is more complex than 23\frac{-2}{3}?
  4. Can this method be used for other combinations of sine and cosine terms?
  5. What would the transformation look like if the coefficients were both negative?

Tip: The identity Rsin(x+α)R\sin(x + \alpha) helps simplify trigonometric expressions into a single sinusoidal form, often used in solving equations and analyzing oscillations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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