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Math 4S. Parker
Spring 2013
Trig Foundations
The Trig You Should Already Know
Three Functions: Sine Cosine Tangent
The Trig You Should Already Know
Definitions: Sine = opp/hyp
Cosine = adj/hyp
Tangent = opp/adj
The Trig You Should Already Know
All the trig you have studied so far has been based upon the sides of a ________ triangle.
right
The Trig You Should Already Know
So far you have used trig to find: missing sides using sin /cos /tan
missing angles using sin -1 / cos -1 / tan -1
The Trig You Will Learn You will find that trig functions can be defined: by the sides of a right triangle (prior knowledge)
based upon other trig functions
based upon the unit circle
The Trig You Will Learn There are six (6) trig functions:SineCosineTangentCosecantSecantCotangent
The three you
already know
The Three Reciprocal Definitions Cosecant =
Secant =
Cotangent =
Given One Trig Function, Find Others
Write definitions of given and needed functions.
Use Pythagorean Theorem to find missing side.
adjacent
hypotenu
se
op
posi
te
x˚
Angles in Standard Position
Vertex is always at the origin.
Initial side is always on the positive x axis.
Terminal side is the ending side.
Angles in Standard Position
Positive angle = counterclockwise
0˚
90˚
180˚
270˚
Angles in Standard Position
Negative angle = clockwise
0˚
−270˚
−180˚
−90˚
Angles in Standard Position
Quadrantal angle = angle not in a quadrant: 0˚, 90˚, 180˚, 270˚, 360˚, etc.
Quadrantal angles will not use reference angles.
Coterminal Angles Coterminal angles always differ by a multiple of 360.
Every angle has an infinite number of coterminal angles.
The interval given determines how many and which coterminal angles may be used.
Reference AnglesAll reference angles are acute.
An acute angle does not need a reference angle (or is considered its own reference).
Quadrantal angles NEVER use reference angles.
Reference AnglesFinding Reference Angles:
1st Quadrant:No ref. angle
2nd Quadrant:180 − angle3rd Quadrant:angle − 180
4th Quadrant:360 − angle
Reference Angles for Angles > 360˚
If the given angle is greater than 360˚, first find a coterminal that falls in the interval 0˚≤ x < 360˚.
Now find the reference angle based upon the coterminal angle.
Reference Angles for Angles < 0˚
If the given angle is negative, first find a coterminal that falls in the interval 0˚≤ x < 360˚.
Now find the reference angle based upon the coterminal angle.
Remember: What is true about ALL reference angles?
Radians and Degrees Common agreements: C = 2r Circle has 360˚ Unit circle has radius = 1
So unit circle has C = 2 and 360˚
If 2 = 360, = 180.
Radians and Degrees
Radians are typically given in terms of but do not have to be.
Radians to Degrees =
#Rad •
Radians and Degrees
Degrees to Radians =
#Deg •
Common Degrees and Radians
30˚ = 45˚= 60˚ = 90˚ =
As the semester goes along, we will use degrees and radians interchangeably.
Trig and the Unit Circle
sin = cos = tan = (cos , sin )
Tangent
By sides tan =
By unit circle: tan =
By other trig functions
tan =
Cotangent (cot)
By sides cot =
By unit circle: cot =
By other trig functions
cot = cot =
Cosecant (csc)
By sides csc =
By unit circle: csc =
By other trig functions
csc =
Secant (sec)
By sides sec =
By unit circle: sec =
By other trig functions
sec =
Trig With Reference Angles
If angle given is not acute, first find the reference angle.
Consider whether the trig function is positive or negative in this quadrant.
Find answer based upon showing these two pieces of information.
Trig With Reference Angles
Find sin 150˚. Ref angle = 180 – 50 = 30
150 is in 3rd quadrant where sine is negative
sin 150˚ = − sin 30˚ =
Point on the Terminal Side Draw a representation of the angle in the proper quadrant.
Do NOT use negatives in labeling side lengths.
must be the angle at the origin
Point on the Terminal Side The hypotenuse will always be the missing side.
Pay attention to quadrant to decide whether answer is positive or negative.
Use trig definitions to find answer.
