5.6 Graphs of Other Trig Functions p all, all Review Table 5.6 on pg 601

Analysis of the Tangent Function by Domain: All reals except odd multiples of Range: Continuous on its domain Increasing on each interval in its domain Symmetry: Origin (odd function)Unbounded No Local Extrema H.A.: None V.A.: for all odd integers k End Behavior: and do not exist (DNE)

Analysis of the Tangent Function by How do we know that these are the vertical asymptotes? They are where cos(x) = 0!!! How do we know that these are the zeros? They are where sin(x) = 0!!! What is the period of the tangent function??? Period:

Analysis of the Tangent Function The constants a, b, h, and k influence the behavior of in much the same way that they do for the sinusoids… The constant a yields a vertical stretch or shrink. The constant b affects the period. The constant h causes a horizontal translation The constant k causes a vertical translation Note: Unlike with sinusoids, here we do not use the terms amplitude and phase shift…

Analysis of the Cotangent Function by The graph of this function will have asymptotes at the zeros of the sine function and zeros at the zeros of the cosine function. Vertical Asymptotes: Zeros:

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph four periods of the function. Start with the basic tangent function, horizontally shrink by a factor of 1/2, and reflect across the x-axis. Since the basic tangent function has vertical asymptotes at all odd multiples of, the shrink factor causes these to move to all odd multiples of. Normally, the period is, but our new period is. Thus, we only need a window of horizontal length to see four periods of the graph…

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph four periods of the function. by

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. Start with the basic cotangent function, horizontally stretch by a factor of 2, vertically stretch by a factor of 3, and vertically translate up 1 unit. The horizontal stretch makes the period of the function. The vertical asymptotes are at even multiples of.

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. by How would you graph this with your calculator? OR

The graph of the secant function The graph has asymptotes at the zeros of the cosine function. Wherever cos(x) = 1, its reciprocal sec(x) is also 1. The period of the secant function is, the same as the cosine function. A local maximum of y = cos(x) corresponds to a local minimum of y = sec(x), and vice versa.

The graph of the secant function

The graph of the cosecant function The graph has asymptotes at the zeros of the sine function. Wherever sin(x) = 1, its reciprocal csc(x) is also 1. The period of the cosecant function is, the same as the sine function. A local maximum of y = sin(x) corresponds to a local minimum of y = csc(x), and vice versa.

The graph of the cosecant function

Summary: Basic Trigonometric Functions FunctionPeriodDomainRange

Summary: Basic Trigonometric Functions FunctionAsymptotesZerosEven/Odd NoneOdd NoneEven Odd None Even Odd

Guided Practice Solve for x in the given interval  No calculator!!!  Third Quadrant Let’s construct a reference triangle: –1 2 Convert to radians:

Guided Practice Use a calculator to solve for x in the given interval.  Third Quadrant The reference triangle: Does this answer make sense with our graph?

Guided Practice Use a calculator to solve for x in the given interval. Possible reference triangles: 0.3 or

Whiteboard Problem Solve for x in the given interval  No calculator!!!

Whiteboard Problem Solve for x in the given interval  No calculator!!!