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Lecture Day 09
for 2/08/2011
Generalizing Trigonometry,
part Tres
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More Generalizations
If a point (x,y) in the plane is on the unit circle then that points
coordinates give the values of the trigonometric functions for
any angle whose terminal side goes through that point.
But what if the point is not on the unit circle? In that case, we can
normalize the point by dividing its ______________ by the
points norm, r, its _______________ from the origin.
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Trig. Functions defined by Normalized Pts.
Let (x, y) be a point in space. Let r =
Then,
Notice:
1) From the last slide we know ___________ is a point on the unit circle.
2) This definition of the 6 trig. functions agrees with the unit circle definition
when r = ___
3) And agrees with the right triangle definition if the point (x, y) is in
quadrant ___.
4) If x = 0, secant and tangent are _____________.
5) If y = 0, cosecant and cotangent are _____________.
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Evaluating the Trig functions for Any Point
The point
lies on the terminal ray of the angle
standard position.
in
Find the six trigonometric function using the normalized point
Definition. What is the Normalized Point?
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Examples
1) Let point (-5, -12) on the terminal side of the angle . Find the
sine, cosine, and tangent values of the angle .
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Examples
2) If
and lies in Quadrant IV, find the secant,
cosecant, and cotangent trigonometric function values.
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Reference Angles & Triangles
Let be an angle in standard position. The reference angle to
is the acute angle formed by the terminal side of and the
x-axis.
The reference triangle to is the triangle whose hypotenuse
___________ the terminal ray of and whose base
___________ the x-axis.
Some individuals prefer to label the base length and height as
_________________ when the leg of the triangle matches up
with a negative axis. The hypotenuse is always ____________.
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Reference Angle & Triangle Cases
The following four cases are all that are possible:
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Trig function Values
The values of the Trig. Functions evaluated at the angle , are ________
times the values of Trig. Function evaluated using the reference angle
or using right triangle trigonometry on the reference triangle. The
following table describes all cases.
To evaluate the trig function of an arbitrary angle, locate the angle in
the plane and identify the reference angle. Then either
1) use one of the formula in the table above or
2) 2) draw a reference triangle and account for positive and negative.
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Examples
Find the reference angle for
1)
2)
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More Examples
Use reference angles or reference triangles to find
1)
2)
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Even More Examples
Use reference angles or reference triangles to find
3)
4)
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The Examples Just Keep Coming
1) Given
2) If
and
and
, find
is in Quadrant III, find
.
.
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Word Problem Examples
Ex. A giant redwood tree casts a shadow 600 feet long. Find the
height of the tree if the angle of elevation of the sun is 30
degrees.
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Another Word Problem Example
Ex. From a point on the ground 500 feet from the base of a
building, an observer find the angle of elevation to the top of
the building is 30 degrees and that the angle of elevation to
the top of a flagpole atop the building is 32 degrees. Find the
height of the building and the length of the flagpole.
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So what the heck do I have to memorize
Memorize the cardinal
(the book calls them
quadrant) angles and
the special triangle
angles in QI, (in Maroon)
The special angles in QI
are the reference angles
to those in QII, QIII, and
QIV (in Dark Blue).
Symmetry can used to
find all remaining
terminal points.
(Maroon for Memorize,
Dark Blue for Derive)
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