4.4 Graphing sin and cos Functions

5–Minute Check 1 Let (–5, 12) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ. Let, where cos θ < 0. Find the exact values of the five remaining trigonometric functions of θ.

Let (–5, 12) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ.

Let, where cos θ < 0. Find the exact values of the five remaining trigonometric functions of θ.

sinusoid amplitude frequency phase shift vertical shift midline

Key Concept 1

Sine Function y Rad

Equations of Sine and Cosine

Graph Vertical Dilations of Sinusoidal Functions Describe how the graphs of f (x) = sin x and g (x) = 2 sin x are related. Then find the amplitude of g (x), and sketch two periods of both functions on the same coordinate axes. Maximum : Minimum: X-intercepts: Period:

Maximum : Minimum: X-intercepts: Period:

Cosine function Rad x

Equations of Sine and Cosine

Describe how the graphs of f (x) = cos x and g (x) = 5 cos x are related. Maximum : Minimum: X-intercepts: Period:

Describe how the graphs of f (x) = cos x and g (x) = –6 cos x are related. Maximum : Minimum: X-intercepts: Period:

Key Concept 3

Graph Horizontal Dilations of Sinusoidal Functions Describe how the graphs of f (x) = cos x and g (x) = cos are related. Then find the period of g (x), and sketch at least one period of both functions on the same coordinate axes. Extrema: Intercepts: Increments: Period:

Graph Horizontal Dilations of Sinusoidal Functions Sketch the curve through the indicated points for each function, continuing the patterns to complete one full cycle of each.

Describe how the graphs of f (x) = sin x and g (x) = sin 4x are related. Extrema: Intercepts: Increments: Period:

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Homework answers

Key Concept 5

Steps for graphing Sin and Cos

Graph Horizontal Translations of Sinusoidal Functions Amplitude: |a| = |2| or 2 State the amplitude, period, and phase shift of. Then graph two periods of the function. In this function, a = 2, b = 5, and c =. Period: Phase shift: X

X

State the amplitude, period, frequency, and phase shift of y = 4 cos Amp: b: c: Per: Increments: PS: x I want to get a common denominator for c, increments and ps

Amp: b: c: Per: Increments: PS:

Amp: b: c: Per: Increments: PS:

Amp: b: c: Per: Increments: PS:

Amp: b: c: Per: Increments: PS:

Amp: b: c: Per: Increments: PS:

Key Concept 4

State the amplitude, period, frequency, phase shift, and vertical shift of.

Use Frequency to Write a Sinusoidal Function MUSIC A bass tuba can hit a note with a frequency of 50 cycles per second (50 hertz) and an amplitude of Write an equation for a cosine function that can be used to model the initial behavior of the sound wave associated with the note. The general form of the equation will be y = a cos bt, where t is the time in seconds. Because the amplitude is 0.75, |a| = This means that a = ±0.75. The period is the reciprocal of the frequency or. Use this value to find b.

Answer:Sample answer: y = 0.75 cos 100 π t Use Frequency to Write a Sinusoidal Function By arbitrarily choosing the positive values of a and b, one cosine function that models the initial behavior is y = 0.75 cos 100πt. Solve for |b|. |b| = 2π(50) or 100π Solve for b. period = Period formula

MUSIC In the equal tempered scale, F sharp has a frequency of 740 hertz. Write an equation for a sine function that can be used to model the initial behavior of the sound wave associated with F sharp having an amplitude of 0.2.

Key Concept 7