a sinx b cosx
How to simplify or express the expression a \sin x + b \cos x?
Answer:
The expression a \sin x + b \cos x is a common trigonometric form that can be rewritten into a single sine or cosine function with a phase shift. This simplification makes it easier to analyze and solve equations involving these expressions.
Key Concept:
Any expression of the form:
a \sin x + b \cos x
can be rewritten as:
R \sin (x + \alpha) \quad \text{or} \quad R \cos (x - \beta)
where:
- R = \sqrt{a^2 + b^2} is the amplitude,
- \alpha and \beta are phase angles defined by the coefficients a and b.
How to find R and \alpha?
We use the following relationships:
- Amplitude:
[
R = \sqrt{a^2 + b^2}
]
- Phase angle \alpha:
[
\alpha = \arctan \left( \frac{b}{a} \right)
]
or equivalently,
[
\cos \alpha = \frac{a}{R}, \quad \sin \alpha = \frac{b}{R}
]
Complete derivation:
We want to write:
[
a \sin x + b \cos x = R \sin (x + \alpha)
]
Using the sine addition formula:
[
R \sin (x + \alpha) = R (\sin x \cos \alpha + \cos x \sin \alpha)
]
Matching coefficients:
[
a = R \cos \alpha, \qquad b = R \sin \alpha
]
From this, we get:
- R = \sqrt{a^2 + b^2}
- \cos \alpha = \dfrac{a}{R}, \sin \alpha = \dfrac{b}{R}
Summary table:
| Expression | Equivalent Form | Amplitude R | Phase Angle \alpha |
|---|---|---|---|
| a \sin x + b \cos x | R \sin(x + \alpha) | R = \sqrt{a^2 + b^2} | \alpha = \arctan \left(\frac{b}{a}\right) |
Additional notes:
- If required, the expression can also be written in the cosine form:
[
a \sin x + b \cos x = R \cos(x - \beta)
]
where
[
\beta = \arctan\left(\frac{a}{b}\right), \quad \text{with} \quad R = \sqrt{a^2 + b^2}
]
- This method is used extensively to solve trigonometric equations, perform integration, or analyze oscillations in physics and engineering.
Example:
Express the function
[
3 \sin x + 4 \cos x
]
in the form R \sin(x + \alpha).
- Compute R:
[
R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
]
- Compute \alpha:
[
\alpha = \arctan\left(\frac{4}{3}\right) \approx 53.13^\circ \quad \text{or} \quad 0.927 \text{ radians}
]
- Therefore,
[
3 \sin x + 4 \cos x = 5 \sin(x + 0.927)
]
Summary:
The expression a \sin x + b \cos x can be rewritten as R \sin(x + \alpha), where R = \sqrt{a^2 + b^2} and \alpha = \arctan(\frac{b}{a}). This conversion simplifies computations and provides deeper insight into the behavior of trigonometric expressions.