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1.1 Angles, Degrees and Special Triangles (1 of 24)
1.1 Angles, Degrees and Special Triangles
Definitions
An angle is formed by two rays with the same
end point. The common endpoint is called the
vertex of the angle, and the rays are called the
sides of the angle.
In the figure, the vertex of angle θ (theta) is
labled O, and A and B are points on each side of
θ . The angle is denoted in two ways: angle θ
and angle AOB.
Think of θ as being formed by rotating ray
OA about its vertex to side OB. In this case,
OA is called the initial side of θ and OB is
called the terminal side of θ . When the
rotation from the initial to the terminal side is
counterclockwise, the the angle is positive,
otherwise the angle is negative.
An angle formed by rotating the rotating a ray
one complete revolution has a measure of
360 0 .
1.1 Angles, Degrees and Special Triangles (2 of 24)
Common Angles
A right angle has measure 90 0 , a straight angle is 180 0 , an acute
angle measures between 0 0 and 90 0 , and an obtuse angle is over
90 0 . Two angles that are complementary have a sum of 90 0 , and
two angles that are supplementary have a sum of 180 0 .
Example 1
Give the complement and supplement of each angle.
a. 40 0
b. 110 0
c. θ
1.1 Angles, Degrees and Special Triangles (3 of 24)
Triangles
A triangle is a three sided polygon. It is
customary to label the triangle so that side a
opposite angle A, side b opposite angle B and
side c opposite angle C.
In an equalteral triangle all three sides are of
equal length and all three angles are equal. An
isoseles triangle has two equal sides and two
equal angles. A scalene triangle no equal
sides and no equal angles. An acute triangle has three acute
angles. An obtuse triangle has exactly one obtuese angle. A right
triangle has one right angle.
1.1 Angles, Degrees and Special Triangles (4 of 24)
Pythagorean Theorem
In any right triangle, the square of the length of the hypoteneuse
(longest side) is equal to the sum of the squares of the legs (the
other two sides).
Example 2
Solve for x is the triangle shown.
1.1 Angles, Degrees and Special Triangles (5 of 24)
Example 3
One of the chair lifts at a Tahoe ski resort has a vertical rise of
1,170 feet. If the length of the chair lift is 5,750 feet, find the
horizontal distance covered by a person on the lift (round to the
nearest foot).
The 300 - 600 - 900 Triangle
In any right triangle that also has a 30 0 angle and a
60 0 angle, the hypotenuse is always twice the shortest
side, and the middle-length side is always the shortest
side times 3 .
Proof:
1.1 Angles, Degrees and Special Triangles (6 of 24)
Example 4
If the shortest side of a 30 0 − 60 0 − 90 0 triangle is 5, find the other
two sides. Express the answers in exact form and then round to the
nearest two decimal places.
Example 5
A ladder is leaning against a wall. The top of the ladder is 4 feet
above the ground and the bottom of the ladder make an angle of
60 0 with the ground. Express the answers in exact form and then
round to the nearest two decimal places.
a.
How long is the ladder?
b.
How far from the wall is the bottom of the ladder?
1.1 Angles, Degrees and Special Triangles (7 of 24)
The 450 - 450 - 900 Triangle
In any right triangle that also two 45 0 angles,
then the legs have equal length and the
hypotenuse is 2 times the length leg.
Example 6
A 10-foot rope connects the top of a
tent pole to the ground. If the rope
makes an angle of 45 0 with the
ground, find the length of the tent
pole.
1.2 The Rectangular Coordinate System (8 of 24)
1.2 The Rectangular Coordinate System
y Example 1
3
Graph y = x − 2 .
2
2 -2
x 2
-2
1.2 The Rectangular Coordinate System (9 of 24)
Vertex Form of a Parabola
Any parabola can be described by the equation
y = a(x − h)2 + k .
The vertex of the parabola is at (h, k). If a > 0, then the parabola
opens upward, and if a < 0, the parabola opens upward. The value
of a also determines how wide or narrow the parabola is.
Example 2
At a county fair a human
cannonball is show from a canon.
He reached a height of 70 feet
before landing in a net 160 feet
from the canon. Sketch the path of
the human cannonball and find its
equation.
1.2 The Rectangular Coordinate System (10 of 24)
The Distance Formula
The distance r between any two points
(x1 , y1 ) , (x2 , y2 ) in the rectangular
coordinate system is given by
r = (x2 − x1 ). 2 + (y2 − y1 )2
Example 3
Find the distance between the points
(−1, 5) and (2, 1) .
1.2 The Rectangular Coordinate System (11 of 24)
Example 4
Find the distance between the point
(x, y) and the origin.
1.2 The Rectangular Coordinate System (12 of 24)
The Equation of a Circle
A circle is the set of all points in the plane
that are a fixed distance from a given fixed
point. The distance is the radius of the
circle r, and the fixed point is the center of
the circle (h, k) .
(x − h)2 + (y. − k)2 = r 2
Example 5
Verify the points (1 / 2, 1 / 2 ) and
(− 3 / 2, − 1 / 2) both lie on the circle of
radius 1 centered at the origin. This
circle is called the unit circle.
1.2 The Rectangular Coordinate System (13 of 24)
Angle in Standard Position
An angle is in standard position if its initial side is along the
positive x-axis and its vertex is at the origin.
Example 6
Draw angle θ = 45 0 in standard position
and find three points on the terminal
side θ .
Notation: When the terminal side of θ = 45 0 is in quadrant I, we
denote it θ ∈QI , read “angle theta is an element of
quadrant I.”
Quadrantal and Coterminal Angles
If the terminal side of an angle lies on one of the axes ( 90 0 , 180 0 ,
270 0 , 360 0 , etc . . .) the angle is called a quadrantal angle. Two
angles in standard position with the same terminal side are
coterminal angles.
1.2 The Rectangular Coordinate System (14 of 24)
Example 7
Draw −90 0 in standard position and find
two positive angles and two negative
angles that are coterminal to −90 0 .
Example 8
Find all angles coterminal to 120 0 .
Common Angles in
Trigonometry
(Unit Circle)
1.3 Definition of the Six Trigonometric Functions (15 of 24)
1.3 Definition of the Trigonometric Functions
1.3 Definition of the Six Trigonometric Functions (16 of 24)
Example 1
Find the six trigonometric functions of θ if θ
is in standard position and the point (-2, 3) is
on the terminal side of θ .
Example 2
Find the sine and cosine of θ = 45 0 .
Example 3
Find the six trigonometric functions of θ = 270 0 .
1.3 Definition of the Six Trigonometric Functions (17 of 24)
Example 4
What is greater, tan 30 0 or tan 40 0 .
How large can tan θ be?
Signs of Trigonometric Functions
Example 5
If sin θ = −5 /13 and θ terminates in quadrant III, find cos θ and
tan θ .
1.4 Introduction to Identities (18 of 24)
1.4 Introduction to Identities
Memorize The Six
Trigonometric Functions
Memorize The Reciprocal Identities
1.4 Introduction to Identities (19 of 24)
Memorize The Reciprocal Identities
Examples 1-6
3
1.
If sin θ = , find csc θ .
5
3
, find sec θ .
2
2.
If cos θ = −
3.
If tan θ = 2 , find cot θ .
4.
If csc θ = a , find sin θ .
5.
If sec θ = 1 , find cos θ .
6.
If cot θ = −1 , find tan θ .
1.4 Introduction to Identities (20 of 24)
Memorize The Ratio Identities
Examples 7
3
4
If sin θ = − and cos θ = , find tan θ and cot θ .
5
5
Notational Point:
sin 2 θ = (sin θ )
Examples 8-9
8.
3
If sin θ = − , find sin 2 θ .
5
9.
If cos θ =
1
, find cos 2 θ
2
2
1.4 Introduction to Identities (21 of 24)
Memorize the Pythagorean Identities
1.4 Introduction to Identities (22 of 24)
Example 10
If sin θ = 3 / 5 and θ terminates in quadrant II, find cos θ and
tan θ .
Example 11
If cos θ = 1 / 2 and θ terminates in quadrant IV, find the remaining
trigonometric functions evaluated at θ .
1.5 More on Identities (23 of 24)
1.5 More on Identities
Example 1
Write tan θ in terms of sin θ [and no other
trigonometric functions].
Example 2
Write sec θ tan θ in terms of sin θ and cos θ .
Example 3
Add
1
1
.
+
sin θ cos θ
1.5 More on Identities (24 of 24)
Example 4
Multiply (sin θ + 2)(sin θ − 5) .
Example 5
Substitute x = 3 tan θ in the expression
and simplify.
Example 6
Prove the following identity. That is, show the
following statement is true by transforming the
left side to the right side. cos θ tan θ = sin θ
Example 7
Prove the identity (sin θ + cos θ )2 = 1+ 2 sin θ cos θ
x2 + 9