Cosine Function Properties

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.