Questions and Answers
What is the sine function represented as?
What is the amount of time it takes to complete one cycle?
Period
What is the amplitude of a wave?
1/2 height of wave
What is the sine function equation?
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What is the formula for amplitude?
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What is the period formula?
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What is the cosine function equation?
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What is the period of y = 2cos(0.5t)?
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What is the amplitude of y = 16cos(3π/2t)?
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What is the range of y = 16cos(3π/2t)?
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What is csc(60°)?
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What is the definition of cosecant?
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What is the definition of secant?
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What is the definition of cotangent?
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What is the relationship for cotangent in terms of cosine and sine?
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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.
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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.
