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Chapter 2: Problem 11

Schreibe in der Form \(y=A \sin (\omega t+\alpha):\) (zeichnerische L枚sung reicht aus) a) \(y=-2 \sin (5 t)+3 \cos (5 t)\) b) \(y=\sin (\pi t)-\cos (\pi t+\pi / 4)\)

Short Answer

Expert verified
a) \(y = \sqrt{13}\sin(5t - 0.588)\), b) \(y = \sqrt{2}\sin(蟺t - 蟺/4)\)

Step by step solution

01

Rewrite the equation y = -2sin(5t) + 3cos(5t)

Firstly, recognize that the equation can be rewritten in phasor form if it matches the pattern \(A\sin(wt+伪)\) by using the formula \(\sqrt{(-2)^2 + 3^2} = \sqrt{13}\) to solve for \(A\) and \(\arctan(-2/3) = -33.69^\circ\) will yield the phase offset \(伪\). Convert the phase offset from degrees to radians => \(\alpha = -33.69^\circ \times \pi/180 = -0.588\) rad.
02

Check if the obtained solution are correct for equation y = -2sin(5t) + 3cos(5t)

Check by plugging in the values of \(A\), \(\omega\), and \(\alpha\) into \(A\sin(\omega t + \alpha)\). It should match the original equation. If it does, we've found our solution.
03

Rewrite the equation y = sin(蟺t) - cos(蟺t + 蟺/4)

Similarly, for the equation use the formula \(\sqrt{1^2 + 1^2} = \sqrt{2}\) to solve for \(A\). Calculate phase offset using \(\arctan(-1/1) = -45^\circ\), and convert the phase offset from degrees to radians: \(\alpha = -45^\circ \times 蟺/180 = -蟺/4\) rad.
04

Check if the obtained solution are correct for equation y = sin(蟺t) - cos(蟺t + 蟺/4)

Check by plugging in the values of \(A\), \(\omega\), and \(\alpha\) into \(A\sin(\omega t + \alpha)\). It should match the original equation. If it does, then you've found your solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sinusoidal Function Transformation
Sinusoidal function transformation is all about converting a trigonometric expression into a different form that is often easier to work with. In this exercise, we aim to express trigonometric equations in the form \(y = A \sin(\omega t + \alpha)\), which makes them more manageable to analyze or graph.
  • Start by identifying the coefficients in front of the sine and cosine terms.
  • The aim is to transform the original equation into a single sine equation with a specific amplitude, frequency, and phase shift.
  • The combined effect of the sine and cosine components can be expressed as a single sine wave with a phase shift.
To do this, we use trigonometric identities and properties, such as the sum formula for sines and cosines. By converting expressions into phasor form, where angles and magnitudes characterize the sine wave, you can simplify and solve these equations more efficiently.
Phasor Representation
Phasor representation is a useful tool in analyzing trigonometric functions. Phasors can simplify the process of adding or subtracting sine and cosine waves.
  • A phasor is a complex number that represents the magnitude and phase of a sinusoidal function.
  • In the case of combining \(-2 \sin(5t) + 3 \cos(5t)\), the phasor provides a visual means to represent and combine these sinusoidal components.
  • To find a phasor, calculate its magnitude \(A = \sqrt{(-2)^2 + 3^2} = \sqrt{13}\).
  • The angle or phase is found using the arctangent function, \(\alpha = \arctan\left(-\frac{2}{3}\right)\).
Representing complex trigonometric expressions in a single phasor helps us consolidate them into a streamlined form of \(y = A \sin(\omega t + \alpha)\), which encapsulates all necessary information about the sinusoidal wave for further analysis.
Phase Angle Calculation
Calculating the phase angle is a critical part of transforming trigonometric functions into a single sinusoidal expression. The phase angle determines how much a wave is shifted horizontally compared to a standard sine or cosine wave.
  • The phase angle \(\alpha\) is calculated using the arctangent function.
  • This often involves determining the ratio of the sine and cosine coefficients, such as \(-\frac{2}{3}\) in the task.
  • Use \(\arctan(-\frac{2}{3})\) to find the angle in degrees and then convert it to radians as needed: \(-33.69^\circ\) becomes \(-0.588\) radians.
  • Ensure to adjust the angle to match the required quadrant for accurate representation.
Understanding the phase angle helps in precisely positioning the resulting sinusoidal wave within the coordinate system, making it indispensable in fields like signal processing and alternating current circuit analysis.

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Most popular questions from this chapter

Beweisen Sie: \(f: \mathbb{R} \rightarrow \mathbb{R}\) mit \(f(x)=a x^{3}\) ist f眉r \(a>0\) streng monoton wachsend und f眉r \(a<0\) streng monoton fallend.

Man zerlege in ganzrationale Anteile und Partialbr眉che: a) \(f(x)=\frac{3 x^{4}-3 x^{3}-10 x^{2}+16 x+5}{x^{2}-x-2}\) b) \(f(x)=\frac{18 x^{4}-7 x^{3}-35 x^{2}-8 x+24}{(x-2)\left(x^{2}+2 x+2\right)(3 x-4)}\) c) \(f(x)=\frac{x^{5}}{(x-1)^{3}(x+2)}\)

Beweisen Sie: Stimmen zwei ganzrationale Funktionen \(n\)-ten Grades \(f\) und \(g\) an \(n+1\) Stellen 眉berein, dann gilt \(f=g .\)

Gegeben ist die Zuordnungsvorschrift \(x \mapsto y=f(x)=-x^{2}+4 x-3\). Wie lautet die Umkehrung der Zuordnung? W盲hlen Sie den Definitionsbereich (maximal) und die Zielmenge so, da脽 die Umkehrfunktion \(f^{-1}\) existiert. Geben Sie \(f^{-1}\) an!

Geben Sie die mittelbaren Funktionen \(g \circ f\) und \(f \circ g\) an, falls diese existieren! a) \(f:[0,1] \rightarrow[-1,4]\) mit \(f(x)=5 x-1, \quad g:[-1,1] \rightarrow[0,1] \operatorname{mit} g(x)=\sqrt{1-x^{2}}\) b) \(f: \mathbb{R} \backslash\\{3\\} \rightarrow \mathbb{R} \backslash\\{0\\}\) mit \(f(x)=\frac{2}{x-3}, \quad g: \mathbb{R} \backslash\\{0\\} \rightarrow \mathbb{R} \backslash\\{-1\\} \operatorname{mit} g(x)=\frac{7-x}{x}\) c) \(f: \mathbb{R} \rightarrow \mathbb{R}\) mit \(x \mapsto x^{3} \quad g: \mathbb{R} \rightarrow \mathbb{R}_{0}^{+}\)mit \(x \mapsto|x|\) d) \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\)mit \(x \mapsto \frac{1}{x} \quad g: \mathbb{R} \rightarrow \mathbb{R}_{0}^{+}\)mit \(x \mapsto|x|\)
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