Trig Functions of Real Numbers

Moving past triangles and angles, to consider circular functions (5.3)(1)

POD

Simplify.

)tan1)(sin1( 22

POD

Simplify. Two approaches:

1))(sec(cos

)tan1)(sin1(22

22

1sincos

cos

sin1)(cos

)tan1)(sin1(

22

2

22

22

Before, seen trig values in…

… right triangles, with ratios for acute angles. Remember SOH-CAH-TOA.

… then in the unit circle, including a rotation of all angles.

… finally as a graph of variables: independent (angles) and dependent (trig values of the angle).

Today, starting to consider…

… trig functions of real numbers, so that we can consider sin 2

as the sine of an angle of 2 radians,

or the sine of the number 2.

Review

Remember how we look at cosine and sine as the x- and y-coordinates of a point traveling around the unit circle.

Again, what are the trig ratios with this view?

cos θ = sec θ = sin θ = csc θ =tan θ = cot θ =

.(x, y)

r

θ

Review

Remember how we look at cosine and sine as the x- and y-coordinates of a point traveling around the unit circle.

Again, what are the trig values in a unit circle?

cos θ = x sec θ = 1/xsin θ = y csc θ = 1/ytan θ = y/x cot θ = x/y

.(x, y)

r

θ

Use it

Find the trig values for θ. How would you know this is this a unit circle?

cos θ = sec θ =

sin θ = csc θ =

tan θ = cot θ =

.(4/5, 3/5)

θ

Use it

Find the trig values for θ. How would you know this is this a unit circle?

cos θ = 4/5 sec θ = 5/4

sin θ = 3/5 csc θ = 5/3

tan θ = 3/4 cot θ = 4/3

(3/5)2 + (4/5)2 = 1

.(4/5, 3/5)

θ

Use it

Find the trig values for -θ. (Hint: what are the coordinates of the point rotated that amount?)

cos -θ = sec -θ = sin -θ = csc -θ =tan -θ = cot - θ =

What has changed from the original trig values?

.(4/5, 3/5)

θ

-θ

Use it

Find the trig values for -θ. (Hint: what are the coordinates of the point rotated that amount?)

cos -θ = 4/5 sec -θ = 5/4sin -θ = -3/5 csc -θ = -5/3tan -θ = -3/4 cot - θ = -4/3

What has changed from the original trig values?

.(4/5, 3/5)

θ

-θ

Use it again

What are the coordinates of the point when rotated (θ+π)? Find the trig values for this new angle.

cos (θ+π)= sec (θ+π)= sin (θ+π)= csc (θ+π)=

tan (θ+π)= cot (θ+π)=

Again, what has changed? How do these values compare to those for the angle (θ-π)? Why?

.(4/5, 3/5)

θ

θ+π

Use it again

What are the coordinates of the point when rotated (θ+π)? Find the trig values for this new angle.

cos (θ+π)= -4/5 sec (θ+π)= -5/4 sin (θ+π)= -3/5 csc (θ+π)= -5/3

tan (θ+π)= 3/4 cot (θ+π)= 4/3The signs for cosine and sine have changed from the original values, since the rotation moves from quadrant 1 to 3.

Trig values for (θ+π) and (θ-π) are the same, since the angles are coterminal.

.(4/5, 3/5)

θ

θ+π

Beyond the unit circle

We’ve built that unit circle, and seen the animated sine wave. The graph of y = sin x is also on the handout. What do those waves represent on the graph? What is another main characteristic of the sine graph?

One characteristic relates to domain, and one to range. Which is which?

What are the domain and range of the graph, and how does that compare to our discussion the other day?

How does this apply to the previous three slides?Is the graph odd, even, or neither?

Beyond the unit circle

How often to the waves repeat in y = sin x? Why?

Finish this series of equalities:

y = sin x = sin (x ± _____) = sin (x ± ______)

Let’s build a graph of y = cos x, and compare it to this one.

If there’s time…

… let’s discuss the graph of y = tan x. You have it on the back side of the handout, and we’ll talk about one period of the graph.

What is the period for this graph?

What are the domain and range of

y = tan x?