1. Graphs of Tangent, Cotangent, Secant, and Cosecant
4.5
Graphs of Tangent, Cotangent, Secant, and Cosecant

2. Quiz: 4-4 1. Vertical stretch/shrink 2. Horizontal stretch/shrink
3. Horizontal translation (phase shift)
4. Vertical translation

3. What you’ll learn about
The Tangent Function
The Cotangent Function
The Secant Function
The Cosecant Function
… and why
This will give us functions for the remaining
trigonometric ratios.

4. UNIT CIRCLE
30º A C B 1
60º 3 2 C
r = 1
60º 2 y
30º A x B

5. UNIT CIRCLE Vertical asymptote 1 -90º -60º -30º 0º 30º 60º 90º -1

6. UNIT CIRCLE
Pattern repeats 1
-90º 0º
90º -1

7. Your turn: Your turn: Domain = ? Range = ? Continuous? Symmetry?
bounded?
Vertical/horizontal asymptotes?
End behavior?

8. f(x) = sin x Remember this?
radians
radians
The y-coordinate of the graph is the distance above/below
the x-axis on the unit circle.
The x-coordinate of the graph is the
distance around the unit circle.
Same thing for cos x.

9. How do we construct the Tangent Function’s graph?
Sin x
y = Sin x: the vertical
distance above/below
the x-axis (on the unit circle).
This distance is graphed as
the y-coordinate on this graph.
x = cos x: the horizontal
distance to the right/left of
the y-axis (on the unit circle).
Cos x
This distance is also graphed as
the y-coordinate on this graph.

10. Construct the Tangent Function using the graph of ‘y’ and ‘x’
y = Sin x: (on unit circle)
y = Sin x
x = cos x: (on unit circle)
Using the graphs of the left:
x = Cos x

11. The Tangent and Cotangent Functions
Tan x
Cot x

12. Asymptotes and Zeros of the Tangent Function
X-intercepts: sin x = 0
Asymptotes: cos x = 0
Rules for Tangent: y = tan (bx – c) + d
a = none
period =
A full period of tan occurs between 2 consecutive asymptotes.
To find 2 consecutive asymptotes: bx – c = and bx – c =
The graph starts low, ends high, ½ way thru period is where it crosses the x-axis.

13. The Tangent Function X-intercepts: sin x = 0 Asymptotes: cos x = 0
Amplitude: doesn’t have one (look at the graph)
A full period of tan(x) occurs between 2
asymptotes which occur at odd multiples of
Period:
Shrink factor causes asymptotes to be odd multiplies of:

14. Describe how tan(x) is transformed to graph: -tan(2x)
1. Amplitude: n/a
2. Period:
3. Vertical Asymptotes: odd multiples of
4. -1 coefficient: causes reflection across x-axis.
Tan(x)
-tan(2x)

15. The Sine and Cosecant Functions
Sine (x)
Cosecant (x)

16. The Cosecant Function: y = 2 csc 2x
To graph graphing shifts, rewrite the equation as it’s reciprocal, then find zero’s, max/mins (amp), and period.
1. Vertical stretch: factor of 2
2. Vertical asymptotes: even multiples of
3. Horizontal shrink factor of ½, changes period to
sin(x) csc(x)
2csc(2x)

17. The Cosine and Secant Functions
Cos x
Sec x

18. Solve Trig function Algebraically
Find: ‘x’ such that and sec(x) = -2
Converting to unit circle:
r = x = -½
That looks familiar! 3
30º 1
60º 2 -½
60º 1
240º
x = 240º

19. Solving a Trig equation graphically
Find the smallest number ‘x’ such that:
1. Graph and
2. Find the point of intersection

20. Basic Trigonometry Functions

21. HOMEWORK
Section 4-5