Trigonometric Functions The unit circle. Radians vs. Degrees Computing Trig Ratios Trig Identities Functions Definitions Effects Applications
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Trigonometric Functions
The unit circle.Radians vs. DegreesComputing Trig Ratios
Trig IdentitiesFunctions
DefinitionsEffectsApplications
Review
hypotenuse
oppositesin
hypotenuse
adjacentcos
cos
sintan
adjacent
opposite
opposite
adjacent
hypotenuse
Starting with a right triangle, like the one pictured on the right, three basic trig ratios are defined as follows:
Review
opposite
adjacent
hypotenuse
opposite
hypotenuse
sin
1csc
adjacent
hypotenuse
cos
1sec
opposite
adjacent
tan
1cot
Three additional trig ratios are defined from the basic ratios as follows:
Table of Contents
The Unit Circle
360
2
deg
rees
radians
Consider the unit circle: a circle with a radius equal to one unit, centered at the origin.
The unit circle has a circumference: 2C
30°
2
2
23
43
6
45°
Radians relate directly to degrees:The distance around the unit
circle, starting at the point (1, 0)
equals the angle formed between
the x-axis and the radius drawn
from the origin to a point along
the unit circle.
60°
Distance around the unit circle is measured in radians.
The Unit CircleRadians vs. Degrees
180deg
rees
radiansThe conversion from radians to degrees or
the other way around uses the equation:
180120
x
Convert 120° to radians by solving the equation:
120180 x
180
120
180
180x
3
2x
Cross multiply to solve for x:
The Unit CircleRadians vs. Degrees
180deg
rees
radiansThe conversion from radians to degrees or
the other way around uses the equation:
Convert radians to degrees by solving the equation:4
5
1804
5
x
Cross multiply to solve for x:
x41805
x41805
x41805
x
4
4
4
1805
x455
225x
The Unit CircleComputing Trig Ratios
1
x
y
hypotenuse = 1x = cos y = sin tan = y/x
The trigonometric ratios can be computed using the unit circle.
To form the trig ratios, we need a right triangle inscribed in the unit circle, with one vertex placed at the origin so that the perpendicular sides are parallel to the x-axis & y-axis.
This triangle has the following relationships:
Notice that tan is the same as the slope of the line radiating out of the origin!
1
1
0
1/2
1/2
23
22
23
22
The Unit CircleComputing Trig RatiosUsing the newly defined relationship, the trig
ratios are determined by reading the x & y values off the graph.
x = cos y = sin tan = y/x
Note the pattern:Values increase
from 0 to 1 according to integral square roots.
angle sine
0 2
0
2
1 2
1
2
2 2
2
2
3 2
3
1 2
4
sin x cos x tan x
0 0 1 0
/6 2
1 2
3 3
1
/4 2
2 2
2 1
/3 2
3 2
1 3
/2 1 0
The Unit CircleComputing Trig Ratios
These trig ratios are summarized in the following table:
Table of Contents
Trig identities
In the first and forth quadrants x is positive while y changes sign.
As is swept up and down away from the positive x-axis, only its sign changes.
These characteristics lead to the following relationships:
x
cos (-) = cos ()sin (-) = -sin () tan (-) = -tan ()
Trig identities
cos (-) = -cos ()sin (-) = sin () tan (-) = -tan ()
y From the first to the second quadrants x changes sign while y remains positive.
As is swept up away from the positive and negative x-axis, equal angle sweeps are related as: : -.
These characteristics lead to the following relationships:
Trig identities - Examples:
6cos
6
7cos
6sin
6
5sin
4tan
4
7tan
a.) second quadrant:
b.) fourth quadrant:
c.) third quadrant:
6sin
2
3
4tan
1
6cos
6cos
6cos
2
3
Trig identities
sin2 + cos2 = 1
(1,0)x
y (cos ,sin )
1
Combining the Pythagorean Theorem with the properties of the right triangle inscribe in the unit circle we get the following trig identity, relating sine to cosine:
Note that when x = sin, 21cos x
(1,0)
(1 ,tan )
sec
1
Trig identities
sec2 = 1 + tan2Cosecant and Cotangent
are similarly related:
csc2 = 1 + cot2
A similar triangle combined with the Pythagorean Theorem produces the trig identity relating tangents to secants:
Trig identities
These other trig identities can also be derived from the unit circle:
cos(-) = coscos + sinsincos(+) = coscos - sinsincos(2) = cos2 - sin2sin(+) = sincos + cossinsin(-) = sincos - cossin
These trig identities are useful to solve problems such as:
43cos
12cos
4sin
3sin
4cos
3cos
Proof
Table of Contents
Functions
Consider the ratio expressed as a function:
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
xxfy sin:sin We can graph the function on the Cartesian
coordinates:
Functions - Definition
The function:
x
y
-1
-0.5
0
0.5
1
xxf sinhas the domain: ,
and range: 1,1
Functions - Definition
The function: xxf coshas the domain: ,
and range: 1,1
x
y
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Functions - Definition
The function: xxf tanhas the domain: ,...
2
3,
2,0
x
and range: ,
x
y
-4
-3
-2
-1
0
1
2
3
4
y = Asin (Bx-C)+DAmplitude (A):
Distance between minimum and maximum values.
Frequency (B): Number of intervals required for one complete cycle
Period (2/B): Length of interval containing one complete cycle
Phase Shift (C): Shift along horizontal axis.
Vertical Shift (D): Shift along vertical axis.
Functions - Effects
y = A(sin (Bx-C)
Examples:
Functions - Amplitude (A)
x
y
-3
-2
-1
0
1
2
3xy sin3
xy sin3
1
x
y
-3
-2
-1
0
1
2
3
x
y
-3
-2
-1
0
1
2
3
x
y
-3
-2
-1
0
1
2
3
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
y = A(sin (Bx-C)
Examples:
Functions – Frequency/Period (B)
xy 3sin
xy
3
1sin
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
Period = 2/3
Period = 6
x
y
-1
-0.5
0
0.5
1
y = A(sin (Bx-C)
Examples:
Functions – Phase (C)
3sin
xy
2sin
xy x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
x
y
-1
-0.5
0
0.5
1
xcos
What does the sine curve represent?Periodic Behavior:
SoundWaves, TidesSpringsCyclic growth and decay
Consider the waves in the ocean,The amplitude effect their heightChoppy water is caused a high frequencyFlat seas indicate that there is a low frequency
and amplitude
Functions - Applications
Low tide occurs in some port at 10:00 am on Monday and again at 10:24 pm that same night. At low tide the water level is 1 foot and at high tide it measures 7 feet. What is the sine function that represents the water level?
Functions - Applications
Amplitude:The difference between low and high tide is 7-1=6 feet.
The amplitude is half that difference: 6/2=3 feet
Vertical Shift:The average water level: .4
2
71ft
Frequency:Time between high tides: 12 hrs. 24 min. = 12.4 hrs.
Period : 507.04.12
2
ttf 507.0sin34
Practice:1. Express 135 in radians:
360
2
135
135180
180
135
4
3
2. Convert 4/3 radians to degrees:
1803
4
60
14
240
1803
4
Express the following trig ratios as multiples of a simple radical expression:
Practice:
3cos
2
3sin
4
3tan
12
sin2
sin
14
tan4
tan
2
1
3cos
Express the following trig ratios as multiples of a simple radical expression:
Practice:
3
2sin
3
2tan
4
3cos
2
3
3sin
3sin
2
2
4cos
4cos
33
tan3
tan3
2tan
4sin1
xy
42sin21
xy
xy 2sin32
Match the curve to the equation:
Practice:
x
y
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
-1
0
1
2
3
4
A.
B.
C.
B
A
C
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