Trig Functions of Real Numbers Characteristics of the six trig graphs (5.3)(2)

POD If sin θ = 4/5, and θ is in quadrant II, find cos θ sin (π-θ) What can we say about the sine of any obtuse angles? How about the sine of opposite angles?

Review from last time Using the unit circle and the graphs on the handout or calculator, compare cos (30°) cos (-30°) sin (π/4) sin (-π/4) tan (π/6) tan (-π/6) What might that tell us about the nature of these functions?

Consider a reciprocal function What do you think the graph of y = csc θ would look like? Let’s build it off of the sine graph.

Consider a reciprocal function Start with the sine graph. Plot reciprocal y-values for x-values. Where do we not get y-values?

Consider a reciprocal function See how the ranges of the reciprocal functions are related? If we remove the sine graph, we have this. Where are the vertical asymptotes? What are the domain and range? Is it even, odd, neither?

Consider a reciprocal function y = csc (x) an odd function asymptotes at x = ±πn where sin(θ) = 0

Consider another reciprocal function How would the graph of y = sec θ compare with this?

Consider another reciprocal function How would the graph of y = sec θ compare with this? Where are the vertical asymptotes? What are the domain and range? Even, odd, or neither?

Consider another reciprocal function y = sec θ an even function vertical asymptotes at x = π/2±πn, where cos(θ) = 0

Consider the third reciprocal function y = cot θ Where are the vertical asymptotes? Why? What are the domain and range?

Consider the third reciprocal function y = cot θ vertical asymptotes at x = ±πn, where sin(θ) = 0 or where tan (θ) = 0

Summary chart– do we need to do this? Fill in the chart below for the characteristics of the trig functions. Function domain range even/odd symmetric element

Summary chart—let’s do this. Fill in the chart below for the characteristics of three primary trig functions. Function period amplitude asymptotes The full chart for all six trig functions is on p. 401.

The Chart

Formulas for negative angles Since sine and tangent are odd functions, sin(-x) = -sin(x) tan(-x) = -tan(x) csc(-x) = -csc(x) cot(-x) = -cot(x) In other words, change the sign of the angle, change the sign of the trig value. You can see this especially clearly on the graph.

Formulas for negative angles Since cosine is an even function cos(x) = cos(-x) sec(x) = sec(-x) In other words, change the sign of the angle, the trig value stays the same. You can see this on the graph.

Practice an identity Use the negative angle formulas to verify the identity.

Practice an identity Use the negative angle formulas to verify the identity.

Finally… … an interesting graph. On calculators, graph f(x) = sin(x)/x on the interval . What does the graph do as and ?

Finally… Although we know there is a hole at x = 0, it appears that as x approaches 0 from either direction.

Finally… An interesting result from this interesting graph is that, if x is in radians and close to 0, then which means that for very small angles. Test if for x = .03, .02, .01.