Download Chapter 4 - Clayton School District

Chapter 4
Trigonometric Functions
Chapter 4:
●
●
●
●
●
●
●
4.1-Right Triangle Trigonometry
4.2-Degrees and Radians
4.3-Trigonometric Functions on the Unit Circle
4.4-Graphing Sine and Cosine Functions
4.5-Graphing Other Trigonometric Functions
4.6-Inverse Trigonometric Functions
4.7-The Law of Sines and the Law of Cosines
4.1
Right Triangle Trigonometry
4.1 Concepts
●
●
●
●
●
Learning Trig Ratios
Learning the Special Right Triangles
Finding sides of Right Triangles with the Trig ratios
Finding angles of Right Triangles with the Trig ratios
Solving word problems with Trig
4.1 Vocab!
●
●
●
●
●
Trigonometric Ratios= see below
Sine, Cosine, Tangent, Cosecant, Secant, Cotangent
Special Right Triangles= 30-60-90, 45-45-90
Inverse Trig Functions= see second dot
Angles of Depression and Elevation=
4.1 Formulas
● opp/hyp, adj/hyp, opp/adj, hyp/opp, hyp/adj, adj,opp
● 30-60-90
● 45-45-90
4.2
Degrees and Radians
4.2 Concepts
● Converting radians to degrees
● Converting degrees to radians
● Use the measures of angles to solve word problems
4.2 vocabulary
● vertex- a common endpoint for two noncollinear rays
● initial side- the starting position of a ray is the initial side
of an angle
● terminal side-the rays position after rotation
● standard position- when there is an angle at the origin
and its initial side along the positive x-axis
4.2 Formulas
●
●
●
●
●
●
●
●
radians/180 =radians
180/radians=degrees
degrees+360n=degrees possible
radians+2n=radians possible
s=r(theta)
A=½r^2(theta)
v=s/t
w=(theta)/t
4.3
Trigonometric Functions on the Unit Circle
4.3 Concepts
●
●
●
●
●
●
Evaluate trigonometric functions given a point
Evaluate trigonometric functions of quadrantal angles
Find reference angles
Use reference angles to find trigonometric functions
Use one trigonometric function to find others
Find trigonometric values using the unit circle (the one you know by heart)
4.3 Vocabulary
quadrantal angle-an angle in standard position that has a terminal side that lies on one of the
coordinate axes
reference angle- the acute angle formed by the terminal side of an angle in standard position
and the x-axis
unit circle- circle of radius 1 centered at the origin of a coordinate system
circular function- a trigonometric function defined as the function of the real number system
using the unit circle
periodic function- a function with values that repeat at regular intervals
period- for a function y=f(t), the smallest positive number c for which f(t+c)= f(t)
4.3 Formulas
●
●
●
●
●
●
●
r=√(x2 +y2) ≠0
tan(theta)= y/x
cot(theta)= x/y
sin(theta)=y/r
csc(theta)=r/y
cos(theta)= x/r
sec(theta) r/x
4.4
Graphing Sine and Cosine Functions
a hilarious joke….
"sine and cosine had to choose their way of travel. Which
way did they go?"
...
"They went off on a tangent!"
4.4 Concepts
●
●
Graphing Sine Functions (with transformation)
○ vertical dilation (change in amplitude)
○ reflections over x - axis (add negative the equation ex. -3cos(x))
○ horizontal transformation (shifting left or right)
○ vertical transformations (shifting up or down)
Graphing Cosine Functions (with transformation)
○ vertical dilation (change in amplitude)
○ reflections over x - axis (add negative the equation ex. -3cos(x))
○ horizontal transformation (shifting left or right)
○ vertical transformations (shifting up or down)
Parent Graphs
SINE PARENT GRAPH y=sin(x)
COSINE PARENT GRAPH y=cos(x)
4.4 Vocabulary
sinusoid: any transformation of a sine graph
amplitude: half the distance between the max and min
midline: the line that becomes the reference line or
equilibrium point when the graph is shifted up or down
frequency: the number of cycles in one interval
phase shift: function shifts to the left or right
period: time it take of the function to make one cycle
vertical shift: graph goes through vertical transformation
(up or down)
4.4 Formulas
f(x) = (a)sin(b(x-c)) +d
a ----- amplitude (vertical stretch/shrink)
b ----- change in period
c ----- horizontal shift
d ----- vertical shift
How to Find the Period:
2(π)/ b
f(x) = (a)cos(b(x-c)) +d
a ----- amplitude (vertical stretch/shrink)
b ----- change in period
c ----- horizontal shift
d ----- vertical shift
How to Find the Period:
2(π)/ b
Example Problems
f(x) = -3 sin (π/2(x+2π) - 1
f(x) = 4 cos (2(x- π/4) + 1
Solution:
Solution:
4-5
Graphing Other Trigonometric Functions (Tangent and
Reciprocals)
Topics
Topics:
● Graphing tangent functions
● Graphing cotangent functions
● Graphing secant functions
● Graphing cosecant functions
Formulas
Formulas:
●
y= a tan (bx+c)
●
y= a cot (bx+c)
●
P= pi/|b| (for tangent and cotangent)
●
P= 2pi/|b| (for secant and cosecant)
●
y= a sec (bx+c)+d
●
y= a csc (bx+c)+d
Terms
period of a tangent function: the distance between any two
consecutive vertical asymptotes
4.6
Inverse Trigonometry Functions
Concepts
●
●
y=sin-1 X is the same as y=arcsin X
This means the angle between the restricted domains and ranges
Formulas
●
●
●
Inverse Sine of X: The angle between -Pi/2 and Pi/2
Y=sin-1 X
Domain: [-1,1] Range: [-Pi/2, Pi/2]
●
●
●
Inverse Cosine of X: The angle between 0 and Pi
Y=cos-1 X
Domain: [-1, 1] Range: [0, Pi]
●
●
●
Inverse of Tangent of X: The angle between -Pi/2 and Pi/2
Y=tan-1 X
Domain: (-infinity, infinity) Range: (-Pi/2, Pi/2)
4.7 Law of Sines and
Cosines!!!!
Concepts
●
●
●
●
Using Law of Sines to Solve a Triangle
Using Law of Cosines to Solve a Triangle
The Ambiguous Case
Finding the Area of a Triangle
Vocab
●
●
●
●
●
Law of Sines= a formula
Law of Cosines= another formula
The Ambiguous Case= SSA
Heron's Formula= the area
Oblique Triangles= not right
Formulas
●
●
●
●
a2 = b2 + c2 - 2bc Cos(A)
s sqrt(s-a)(s-b)(s-c)
s = 1/2(a+b+c)
sinA/a= sinB/b= sinC/c
The AMBIGUOUS Case
A good way to set it up (thanks Olivia)
THE RULES
Why you should study...
The End