Questions and Answers
Find the period, range, and amplitude of the cosine function $y = \frac{3}{2} \cos \frac{t}{2}$.
Sketch one cycle of the cosine function $y = -\cos 3\theta$. Which graph correctly represents this?
What is the graph of one cycle of a cosine curve with amplitude 2, period $2π$, and $a < 0$?
Model the motion of a simple spring with a cosine function. The bottom of the spring has a maximum height of 7 feet 4 inches and a minimum height of 6 feet 2 inches. It takes 2 seconds for the spring to expand from its minimum length to its maximum length. What is a cosine function that models the spring's length in inches above and below its average, resting position?
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What are all solutions to the equation $2 \cos \theta = 1$ for $0 ≤ \theta ≤ 2\pi$? Round to the nearest hundredth.
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Study Notes
Cosine Function Properties
- The cosine function has a period, range, and amplitude, which can be derived from the general form ( y = A \cos(Bt) ).
- For ( y = \frac{3}{2} \cos \frac{t}{2} ), key attributes include:
- Period: ( 4\pi ) (calculated as ( \frac{2\pi}{B} ) where ( B = \frac{1}{2} ))
- Range: (-\frac{3}{2} \leq y \leq \frac{3}{2})
- Amplitude: (\frac{3}{2})
Graphing the Cosine Function
- To sketch the function ( y = -\cos(3\theta) ):
- The cosine wave is inverted due to the negative sign, indicating it reaches a maximum at negative peak values.
- Various graphs can illustrate one complete cycle of the function, with specific images linked for visual reference.
Characteristics of Cosine Graphs
- When analyzing one cycle of a cosine curve with an amplitude of 2 and a period of ( 2\pi ) where ( a < 0 ):
- The cosine function is reflected over the x-axis.
- An example corresponds to a specific image link demonstrating this curve.
Modeling Real-World Applications
- The motion of a spring can be described with a cosine function, modeling its height variation over time.
- For a spring with a maximum height of 88 inches and a minimum height of 74 inches:
- Average resting height is ( 81 ) inches.
- The corresponding cosine function can be expressed as ( f(t) = 14 \cos(\pi/2 t) ) where the period is the time taken to go from max to min height.
Solving Cosine Equations
- Solutions to the equation ( 2 \cos(\theta) = 1 ) over the interval ( 0 \leq \theta \leq 2\pi ) can be found by isolating ( \theta ):
- The solutions round to approximately ( 1.05 ) and ( 5.24 ).
- Understanding how to manipulate and solve cosine equations transforms structured visualization into practical problem-solving skills in trigonometry.
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Description
This quiz explores the key attributes of the cosine function including its period, range, and amplitude. It covers essential graphing techniques and characteristics of cosine waves, illustrating various transformations. Students will apply their understanding through problem-solving and visual representation.
