Algebra 2 Chapter 13.4-13.8 Flashcards

Study Notes

Sine Function

Defined as ( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} )

Graph of a Sine Function

A periodic wave that oscillates above and below the x-axis, with peaks and troughs.

Period

The time required to complete one full cycle of a wave.

Amplitude

Half the height of the wave from the midline to the peak (or trough).

Sine Function Equation

General form: ( y = a \sin(b \theta) ), where ( a ) is amplitude and ( b ) affects the period.

Graph of a Cosine Function

Similar to the sine function, but it starts at its maximum value.

Amplitude Formula

Calculated as ( \text{Amplitude} = |a| ), where ( a ) is the coefficient in the function's equation.

Period Formula

Determined by ( \text{Period} = \frac{2\pi}{b} ), where ( b ) is the coefficient of ( \theta ).

Cosine Function Equation

General form: ( y = a \cos(b \theta) ), where ( a ) indicates amplitude.

Identify Period, Amplitude, Range: ( y = 2\cos(0.5t) )

Period: ( \frac{2\pi}{0.5} = 4\pi )
Amplitude: ( 2 )
Range: ([-2, 2])

Identify Period, Amplitude, Range: ( y = 16\cos\left(\frac{3\pi}{2}t\right) )

Period: ( \frac{2\pi}{\frac{3\pi}{2}} = \frac{4}{3} )
Amplitude: ( 16 )
Range: ([-16, 16])

Cosecant

Denoted as ( \csc \theta ), it is the reciprocal of the sine function.

Secant

Denoted as ( \sec \theta ), it is the reciprocal of the cosine function.

Cotangent

Denoted as ( \cot \theta ), defined as ( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} ).

Cosecant Calculation: ( \csc(60°) )

Evaluated as ( \csc(60°) = \frac{1}{\sin(60°)} = \frac{1}{\frac{\sqrt{3}}{2}} ).
Rationalized result: ( \frac{2\sqrt{3}}{3} ).

Cosecant Graph

A periodic function characterized by vertical asymptotes and repeating cycles.

Secant Graph

Also periodic with vertical asymptotes, the graph oscillates between its maximum and minimum values based on cosine oscillations.

Cotangent Graph

Exhibits periodic behavior with vertical asymptotes, alternating between positive and negative values across its domain.

Description

This quiz features flashcards focused on essential concepts from Algebra 2, specifically chapters 13.4, 13.5, and 13.8. Explore key terms such as sine function, amplitude, and period to enhance your understanding of trigonometric functions. Perfect for review and study ahead of exams.

Algebra 2 Chapter 13.4-13.8 Flashcards

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Questions and Answers

What is the sine function represented as?

What is the amount of time it takes to complete one cycle?

What is the amplitude of a wave?

What is the sine function equation?

What is the formula for amplitude?

What is the period formula?

What is the cosine function equation?

What is the period of y = 2cos(0.5t)?

What is the amplitude of y = 16cos(3π/2t)?

What is the range of y = 16cos(3π/2t)?

What is csc(60°)?

What is the definition of cosecant?

What is the definition of secant?

What is the definition of cotangent?

What is the relationship for cotangent in terms of cosine and sine?