Trigonometric Functions

Amplitude and Vertical Shift

• Amplitude: The amplitude is 2, which is placed in front of the trigonometric function.
• Vertical Shift: A vertical shift of -5 is added at the end of the equation, indicating the midline's position.

Finding the B Value from the Period

• The general form to find the (b) value is 2\pi / b.
• When to use the formula 2\pi / b:
  • Use this formula when you need to find the (b) value using the given period.
  • If you are given a graph and asked to find the period, you simply observe where the wave completes one full cycle on the graph.

Solving for B

• Given the period (\pi), set up the equation: 2\pi / b = \pi
• Swap (b) and (\pi) to isolate (b): b = 2\pi / \pi
• Simplify: b = 2

Graphing Trigonometric Equations

• Given an equation, the first step in graphing is to plot the midline.
• The midline for the equation is at (y = -5).
• Determine the amplitude to find how high and low the wave will go from the midline. In this case, the amplitude is 2, so the wave oscillates between (y = -3) and (y = -7).
• Determine the number of points needed to graph (typically five) to represent one full period of the wave.

Finding the Period from the Equation

• If the b-value is given use the formula 2\pi / b to find the period.

Determining Intervals on the Graph

• If the period ends at (\pi) and starts at zero:
  • The middle point is (\pi / 2).
  • The point between 0 and (\pi / 2) is (\pi / 4).
• Counting intervals:
  • If the first interval is (\pi / 4), then you count in increments of (\pi / 4).
  • (1\pi / 4, 2\pi / 4, 3\pi / 4, 4\pi / 4) (which simplifies to (\pi)).

Graphing Cosine

• For a positive cosine function, start at the maximum.
• Plot points at each interval: max, midline, min, midline, max, completing one full period.
• If the function is negative cosine, start at the minimum and go down first.

Understanding the Impact of Interval Specifications

• Even if intervals are specified (e.g., between (-2\pi) and (2\pi)), the basic shape of the graph remains the same within one period.
• Repeat the graph to fill the specified intervals.

Graphing Sine

• For a sine function, begin at the midline.
• For a positive sine function, after the midline, go upwards towards the maximum.
• For a negative sine function, after the midline, go downwards towards the minimum.

B Value

• If there is no visible number for (b), the value of (b) is 1.
• If (b = 1), the period is 2\pi / 1 = 2\pi.

Tangent

• Tangent ratio: Opposite / Adjacent

• In terms of a coordinate plane with coordinates ((x, y)) and radius (r), tangent is (y / x).

• In trigonometric terms, using the unit circle, tangent is sine / cosine.

• If given a coordinate (e.g., ((0.6, 0.8))), tangent (\theta = y / x). Take the (y) value and put it over the (x) value to find the tangent ratio.

• Key Trigonometric Ratios:

  • Sine: Opposite / Hypotenuse
  • Cosine: Adjacent / Hypotenuse
  • Tangent: Opposite / Adjacent

Graphing Tangent

• Unit Circle Coordinates:
  • Right: (1, 0)
  • Left: (-1, 0)
  • Up: (0, 1)
  • Down: (0, -1)

Radians

• Angles in Radians:
  • Right: 0
  • Up: (\pi / 2)
  • Left: (\pi)
  • Down: (3\pi / 2)
  • Full Circle: (2\pi)
  • Counting: (\pi / 4, 2\pi / 4, 3\pi / 4, 4\pi / 4) etc.

Negative Angles

• To find negative angles, move clockwise instead of counterclockwise.
  • Positive (\pi / 2) is at the top; negative (\pi / 2) is at the bottom.
  • Positive (\pi) and negative (\pi) are at the same spot (left).

Tangent Values

• Tangent is (y / x).

• Any value divided by zero is undefined, creating an asymptote.

  • At zero: Tangent ratio is (0 / 1 = 0).
  • At negative (\pi / 2): Tangent ratio is (-1 / 0 = \text{undefined}).
  • At negative (\pi): Tangent ratio is (0 / -1 = 0).

• Isosceles triangles sides are equal.

• If (y / x) when both sides are the same (= 1).

Characteristics of the Tangent Graph

• Range of a tangent function: negative infinity to positive infinity.
• Tangent has an endless infinite range.

Homework

• Go to Desmos.
• Complete slides 6 to 12.