2.
Review Starting with a right triangle, like the one pictured on the right, three basic trig ratios are defined as follows:
3.
Review Three additional trig ratios are defined from the basic ratios as follows: Table of Contents
4.
The Unit Circle Consider the unit circle: a circle with a radius equal to one unit, centered at the origin. The unit circle has a circumference: 30 ° 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 .
5.
The Unit Circle Radians vs. Degrees The conversion from radians to degrees or the other way around uses the equation: Convert 120 ° to radians by solving the equation: Cross multiply to solve for x :
6.
The Unit Circle Radians vs. Degrees The conversion from radians to degrees or the other way around uses the equation: Cross multiply to solve for x : Convert radians to degrees by solving the equation:
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!
8.
The Unit Circle Computing Trig Ratios Using 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.
9.
The Unit Circle Computing Trig Ratios These trig ratios are summarized in the following table: Table of Contents
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:
cos (- ) = cos ( ) sin (- ) = -sin ( ) tan (- ) = -tan ( )
11.
Trig identities cos ( - ) = -cos ( ) sin ( - ) = sin ( ) tan ( - ) = -tan ( ) 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:
12.
Trig identities - Examples : a.) second quadrant: b.) fourth quadrant: c.) third quadrant:
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 ,
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:
Frequency:
Time between high tides: 12 hrs. 24 min. = 12.4 hrs.
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