Transcript Unit Circle - Denise Kapler

Unit Circle
And Trigonometric Functions
(x, y) = (cos Ɵ, sin Ɵ)
Trigonometry relies on
triangle proportionality.
Given right triangles with
congruent acute angles,
the trig function is built
from the proportionality
constant.
Proportionality constants
are written
within the image: sin θ,
cos θ, tan θ, where θ is
the common measure
of the acute angle.
Angle Measure
a. Degrees
b. Radians
Radian is a unit of angular
measure defined such that an
angle of one radian subtended
from the center of a unit circle
produces an arc with arc
length 1.
Subtended Angle:
The angle made by a
line, arc or object.
Example: The Subtended Angle of
the tree (from the person's point of
view) is 22°
Measuring in Radians.
Trig Ratios
Trig Ratios
Trig Ratios
Trig Ratios
y = sin(x)
using a unit circle
Compare Graphs of sin and cos
Naming convention
Angles: Capital Letters
Side lengths: Small Letter of Opposite Angle
Using Trig
What is the
height of the
tree on the left?
Using Trig
At 57" from the
base of a building
you need to look up
at 55° to see the
top of a building.
What is the height
of the building?
Using Trig
Find Reference Angles
Quadrant II
Quadrant IV
Quadrant III
Find Reference Angle or Reference Triangle
Sin 135°= Sin 45° =
2
2
Sin 300°= Sin -60° =
− 3
2
Reference Angle = 30°
Reference Angle = 55°
Angles greater than 360°
Positive and Negative Angles
Inverse Trig Functions
When you want to find the angle measure Ɵ:
arcsin(x) = sin-1(x) Read as: “the angle whose sine is x”
arccos(x) = cos-1(x)
arctan(x) = tan-1(x)
The range of the Inverse Functions is limited as follows.
Inverse Trig Function
The base of a ladder is placed 3 feet away from a 10foot-high wall, so that the top of the ladder meets the
top of the wall. What is the measure of the angle formed
by the ladder and the ground?
Here we a have a right triangle where we know the lengths of the two legs, that
is, the sides opposite and adjacent to the angle. So, we use the inverse tangent
function. If you enter this into a calculator set to "degree" mode, you get
If you have the calculator set to radian mode, you get