Sec 1.5(b) Trigonometric functions and their inverse

Important! They are periodic, or repeating, and therefore model many naturally occurring periodic processes. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

(1). Radian Measure Why do we need to introduce the Radian Measure? What is radian measure? Definition: The radian measure of the angle ACB at the center of the unit circle (Figure 1.63) equals the length of the arc that ACB cuts from the unit circle. If is measured in radian, then for any other circle with radius r, the length of the arc Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

(2). Six basic trigonometric functions Definition: If the angle is in standard position in a circle of radius r, then we can define the six basic trigonometric functions as follows Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Periodicity of the Trigonometric Functions When an angle of measure θ and an angle θ +2π of measure are in standard position, their terminal rays coincide. Therefore they have the same trigonometric function values: Similarly, cos(θ-2 π) = cos θ , sin(θ-2 π) = sin θ, and so on. repeating behavior ---periodicity Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Definition of the periodic function Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Period of trigonometric functions The tangent function and cotangent function have the period p = π, and the other four functions have the period p = 2π. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graphs of trigonometric functions Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Even and odd property of the trigonometric function The above graphs reveals that the cosine and secant functions are even and the other four functions are odd: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Other important formulas Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Inverse Trig Functions

Consider the sine function. You can see right away that the sine function does not pass the horizontal line test. But we can come up with a valid inverse function if we restrict the domain as we did with the previous function. How would YOU restrict the domain?

Take a look at the piece of the graph in the red frame. We are going to build the inverse function from this section of the sine curve because: This section picks up all the outputs of the sine from –1 to 1. This section includes the origin. Quadrant I angles generate the positive ratios and negative angles in Quadrant IV generate the negative ratios. Lets zoom in and look at some key points in this section.

I have plotted the special angles on the curve and the table.

The new table generates the graph of the inverse. The domain of the chosen section of the sine is So the range of the arcsin is To get a good look at the graph of the inverse function, we will “turn the tables” on the sine function. The range of the chosen section of the sine is [-1 ,1] so the domain of the arcsin is [-1, 1].

Note how each point on the original graph gets “reflected” onto the graph of the inverse. etc. You will see the inverse listed as both:

In the tradition of inverse functions then we have: Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians). The thing to remember is that for the trig function the input is the angle and the output is the ratio, but for the inverse trig function the input is the ratio and the output is the angle.

The other inverse trig functions are generated by using similar restrictions on the domain of the trig function. Consider the cosine function: What do you think would be a good domain restriction for the cosine? Congratulations if you realized that the restriction we used on the sine is not going to work on the cosine.

The chosen section for the cosine is in the red frame The chosen section for the cosine is in the red frame. This section includes all outputs from –1 to 1 and all inputs in the first and second quadrants. Since the domain and range for the section are the domain and range for the inverse cosine are

The other trig functions require similar restrictions on their domains in order to generate an inverse. Like the sine function, the domain of the section of the tangent that generates the arctan is y=arctan(x) y=tan(x)

arcsin(x) arccos(x) arctan(x) Domain Range The table below will summarize the parameters we have so far. Remember, the angle is the input for a trig function and the ratio is the output. For the inverse trig functions the ratio is the input and the angle is the output. arcsin(x) arccos(x) arctan(x) Domain Range When x<0, y=arcsin(x) will be in which quadrant? y<0 in IV When x<0, y=arccos(x) will be in which quadrant? y>0 in II y<0 in IV When x<0, y=arctan(x) will be in which quadrant?

Inverse Sine Function Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. f(x) = sin x does not pass the Horizontal Line Test and must be restricted to find its inverse. y x y = sin x Sin x has an inverse function on this interval. Inverse Sine Function

The range of y = arcsin x is [–/2 , /2]. The inverse sine function is defined by y = arcsin x if and only if sin y = x. Angle whose sine is x The domain of y = arcsin x is [–1, 1]. The range of y = arcsin x is [–/2 , /2]. Example: This is another way to write arcsin x. Inverse Sine Function

Inverse Cosine Function f(x) = cos x must be restricted to find its inverse. y x y = cos x Cos x has an inverse function on this interval. Inverse Cosine Function

Inverse Cosine Function The inverse cosine function is defined by y = arccos x if and only if cos y = x. Angle whose cosine is x The domain of y = arccos x is [–1, 1]. The range of y = arccos x is [0 , ]. Example: This is another way to write arccos x. Inverse Cosine Function

Inverse Tangent Function f(x) = tan x must be restricted to find its inverse. y x y = tan x Tan x has an inverse function on this interval. Inverse Tangent Function

Inverse Tangent Function The inverse tangent function is defined by y = arctan x if and only if tan y = x. Angle whose tangent is x The domain of y = arctan x is . The range of y = arctan x is (–/2 , /2). Example: This is another way to write arctan x. Inverse Tangent Function

 

Find the exact value of each expression without using a calculator Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians. Work out the answers yourself before you click.

Answers for problems 1 – 9. Negative ratios for arccos generate angles in Quadrant II. y x 1 2 The reference angle is so the answer is

y x -1 2 14. x 1 2 y 15.