Transcript Sin - Turner School District USD #202

Chapter 1
Trigonometric Functions
Section 1.1 Basic Concepts
Section 1.2 Angles
Section 1.3 Angle Relationships
Section 1.4 Definitions of Trig Functions
Section 1.5 Using the Definitions
Section 1.1 Basic Concepts
In this section we will cover:
• Labeling Quadrants
• Pythagorean Theorem
• Distance Formula
• Midpoint Formula
• Interval Notation
• Relations
• Functions
The Coordinate Plane
Horizontal
Quadrant II
Quadrant I
(-,+)
(+,+)
Quadrant III
Quadrant IV
(-,-)
(+,-)
x
abscissa
Vertical
y
ordinate
Pythagorean Theorem
B
a
leg
a2 + b2 = c2
C
leg
b
A
Distance Formula
a = (x2 – x1)
b = (y2 – y1)
c = √ a2 + b2
or
distance = √ (x2 – x1)2 + (y2 – y1)2
Midpoint Formula
The Midpoint Formula: The midpoint of a
segment with endpoints (x1 , y1) and (x2 , y2)
has coordinates
Interval Notation
•
Set-builder notation
{x|x<5} the set of all x such that x is less than 5
•
Interval notation
(-∞, 5) the set of all x such that x is less than 5
(-∞, 5] the set … x is less than or equal to 5
the first is an open interval
the second is a half-opened interval
[0, 5] is an example of a closed interval
Relations and Functions
A relation is a set of points.
A dependent variable varies based on an
independent variable.
For example y = 2x y is the dependent variable
x is the independent variable
A relation is a function if each value of the
independent variable leads to exactly one
value of the dependent variable.
The values of the dependent variable
represent the range.
The values of the independent variable
represent the domain.
A relation is a function if a vertical line
intersects its graph in no more than one
point. (Vertical Line Test)
Section 1.2 Angles
In this section we will cover:
• Basic terminology
• Degree measure
• Standard position
• Co terminal Angles
Basic Terminology
• line - an infinitely-extending onedimensional figure that has no curvature
• segment - the portion of a line between
two points
• ray - the portion of a line starting with a
single point and continuing without end
• angle - figure formed through rotating a
ray around its endpoint
Basic Terminology (cont)
•
•
•
•
initial side - ray position before rotation
terminal side - ray position after rotation
vertex - point of rotation
positive rotation - counterclockwise
rotation
• negative rotation - clockwise rotation
• degree - 1/360th of a complete rotation
Basic Terminology (cont)
• acute angle - angle with a measure
between 0° and 90°
• right angle - angle with a measure of 90°
• obtuse angle - angle with a measure
between 90° and 180°
• straight angle - angle with a measure of
180°
• complementary - sum of 90°
• supplementary - sum of 180°
Basic Terminology (cont)
• minute - ‘ , 1/60th of a degree
• second – “ , 1/60th of a minute, 1/3600th of a
degree
• standard position - an angle with a vertex at the
origin and initial side on the positive abscissa
• quadrantal angles - angles in standard position
whose terminal side lies on an axis
• co terminal angles - angles having the same
initial and terminal sides but different angle
measures
Section 1.3 Angle Relationships
In this section we will cover:
• Geometric Properties
– Vertical angles
– Parallel lines cut by a transversal
• Corresponding angles
• Same side interior and exterior angles
• Applying triangle properties
– Angle sum
– Similar triangles
Geometric Properties
• Vertical angles are formed when two lines
intersect. They are congruent which
means they have equal measures.
• When parallel lines are cut by a third line,
called a transversal, the result is to sets of
congruent angles.
1
2
3 4
6
5
7
8
Geometric Properties (cont
1
2
3 4
6
5
7
8
So here angles 1, 4, 5, and 8 are congruent
and angles 2, 3, 6, and 7 are congruent.
Corresponding pairs are / 1 & / 5, / 2 & / 6,
/ 3 & / 7, and / 4 & / 8.
Triangle Properties
The sum of the interior angles of a triangle
equal 180°.
Acute – 3 acute angles
Right – 2 acute and one right angle
Obtuse – 1 obtuse and two acute angles
Equilateral – all sides (and angles) equal
Isosceles – two equal sides (and angles)
Scalene – no equal sides (or angles)
Triangle Properties (cont)
Corresponding parts of congruent triangles
are congruent.
Corresponding angles of similar triangles
are congruent.
Corresponding sides of similar triangles are
in proportion.
Section 1.4 Definitions of
Trigonometric Functions
In this section we will cover:
• Trigonometric functions
– Sine
– Cosine
– Tangent
• Quadrantal angles
–Cosecant
–Secant
–Cotangent
Trigonometric Functions
•
•
•
•
•
•
Sine = opposite /hypotenuse = y/r
Cosine = adjacent/hypotenuse = x/r
Tangent = opposite/adjacent = y/x
Cosecant = hypotenuse/opposite = r/y
Secant = hypotenuse/adjacent = r/x
Cotangent = adjacent/opposite = x/r
Special Triangles
Special Trig Values
sin
cos
tan
csc
sec
cot
30à
1/2
ñ3/2
ñ3/3
2
45à
ñ2/2
ñ2/2
1
ñ2
90à
1
0
Und
1
ñ2
60à
ñ3/2
1/2
ñ3
2ñ3
3
2
2ñ3
3
ñ3
1
ñ3/3
0
Und
Trigonometric Functions Values
for Quadrant Angles
sin
cos
tan
csc
sec
cot
0à
0
1
0
90à
1
0
Undefined
180à
0
-1
0
270à
-1
0
Undefined
Undefined
1
Undefined
-1
1
Undefined
-1
Undefined
Undefined
0
Undefined
0
Section 1.5 Using the Definitions of
Trigonometric Functions
In this section we will cover:
• The reciprocal identities
• Signs and ranges of function values
• The Pythagorean identities
• The quotient identities
The Reciprocal Identities
sin £ =
1
csc
£
=
csc £
1
sin £
cos £ =
1
sec
£=
sec £
1
cos £
tan £ =
1
cot
£ =£
cot
1
tan £
Signs and Ranges of
function values
£ in
Quadrant
sin £ cos £ tan £ cot £ sec £ csc £
I
+
+
+
+
+
+
II
+
-
-
-
-
+
III
-
-
+
+
-
-
IV
-
+
-
-
+
-
All Students Take Calculus
x<0
y>0
r>0
x<0
y<0
r>0
Quadrant II
Quadrant I
(-,+)
(+,+)
Sin & Csc
are positive
All functions
are positive
Tan & Cot
are positive
Cos & Sec
are positive
Quadrant III
Quadrant IV
(-,-)
(+,-)
x>0
y>0
r>0
x>0
y<0
r>0
Ranges for Trig Functions
For any angle £ for which the indicated functions exist:
1.
2.
3.
-1 < sin £ < 1 and -1 < cos £ < 1;
tan £ and cot £ may be equal to any real number;
sec £ < -1 or sec £ > 1 and
csc £ < -1 or csc £ > 1
(Notice that sec £ and csc £ are never between -1 and 1.)
The Pythagorean Identities
Remember in a right triangle
a2 + b2 = c2
or using x, y, and r
r
x2 + y2 = r2
y
Dividing by r2
x
x2 + y2 = r2
r2 r 2 r 2
or
r
y
cos2θ + sin2θ = 1
or
θ
x
sin2θ + cos2θ = 1
This is our first trigonometric identity
Basic trigonometric identities
cos2θ + sin2θ
1
cos2θ cos2θ= cos2θ
or
1 + tan2θ = sec2θ
r
or
tan2θ
+1 =
y
sec2θ
θ
x
Basic trigonometric identities
cos2θ + sin2θ
1
= sin2θ
2
2
sin θ sin θ
or
cot2θ + 1 = csc2θ
r
or
1 +
cot2θ
=
y
csc2θ
θ
x
The quotient Identities
tan £ =
sin £
=
cos £
y
x
cot £ =
cos £
sin £
x
y
=