Post on 30-Dec-2015
Section 2.1 Angles and Their Measure
Vertex Initial Side
Terminal si
de
Counterclockwise rotationPositive Angle
Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.
r
r
1 radian
Theorem Arc Length
For a circle of radius r, a central angle of radians subtends an arc whose length s is
s r
Relationship between Degrees and Radians
-> 1 revolution = 2 π radians
-> 180o = π radians
1801
1801
radian
radian
Announcements• Test Friday (Jan 30) in lab, ARM
213/219, material through section 2.2• Sample test posted...link from course
Web site• Bring picture ID…you will need to
scan your ID upon entering the lab• You may use a calculator up to TI 86.• You can’t your the book or notes
Section 2.2Right Angle Trigonometry
A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle.
cb
a
90
Initial side
Terminal side
a
bc
The six ratios of a right triangle are called trigonometric functions of acute angles and are defined as follows:
sine of
cosine of
tangent of
cosecant of
secant of
cotangent of
Function namesin
cos
tan
csc
sec
cot
Abbreviation Valueb c
a c
b a
c b
c a
a b
/
/
/
/
/
/
Pathagorean Theorem
a
bc a2 + b2 = c2
Find the value of each of the six trigonometric functions of the angle .
Adjacent
12 13
c = Hypotenuse = 13
b = Opposite = 12a b c2 2 2
a2 2 212 13
a2 169 144 25 a 5
a
b
c
Adjacent = 5
Opposite =
Hypotenuse =
12
13
sin OppositeHypotenuse
1213
cos AdjacentHypotenuse
513
tan OppositeAdjacent
125
csc HypotenuseOpposite
1312
sec HypotenuseAdacent
135
cot AdjacentOpposite
512
b a c2 2 2
cb
a
90
b
c
a
c
c
c
2
2
2
2
2
2
bc
ac
2 2
1
sin cos2 2 1
Pythagorean Identities
The equation
sin2θ + cos2 θ
along with
tan2 θ + 1 = sec2 θ
and
1 + cot2 θ = csc2 θ
are called the Pythagorean identities.
More Identities
tan
1cot
cos
1sec
sin
1csc
sin
coscot
cos
sintan
Reciprocal Identities
Quotient Identities
Complementary Angles Theorem
Cofunctions of complementary angles are equal.
Two acute angles are complementary if the sum of their measures is a right angle…90 degrees.
α
β
The angles α and β are complementary in a right triangle, α + β = 90 degrees.
Complementary Angles in Right Triangles
Cofunctions
)90sec(csc)90csc(sec
)90tan(cot)90cot(tan
)90sin(cos)90cos(sin
Degrees
Radians
2seccsc
2cscsec
2tancot
2cottan
2sincos
2cossin
Using the Complementary Angle Theorem
Find the exact value (no calculator) of the following expressions.
70cos20cos1.
50sin
40cos.
22 b
a
50sin
40cos.a
50sin)4090sin(40cos
150sin
50sin
50sin
40cos
70cos20cos1. 22 b
70sin20cos 22
70cos70sin1 22
)70cos70(sin1 22
0)1(1
so