Week 1

Analytic Trigonometry (Week 1)
Irianto
Universiti Teknikal Malaysia Melaka
irianto@utem.edu.my
September 3, 2013
Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 1 / 26

Content
Angles and Their Measure

Content
Right Triangle Trigonometry

Content
Computing the Values of Trigonometric Functions of Acute Angle

Content
Computing the Values of Trigonometric Functions of Acute Angle
Trigonometric Functions of General Angles

Angle and Their Measure
Deﬁnition
A ray, or half-line, is that portion of a line that starts at a point V on the
line and extends indeﬁnitely in one direction. The starting point V of a ray
is called its vertex.
Figure : Ray

Angle
If two rays are drawn with a common vertex, they form an angle.
Initial and Terminal sides
We call one of the rays of an angle the initial side and the other the
terminal side. The angle formed is identiﬁed by showing the direction and
amount of rotation from the initial side to the terminal side.
Figure : Angle, Initial Side, and Terminal Side

Positive and Negative Angles
If the rotation is in the counterclockwise direction, the angle is positive; if
the rotation is clockwise, the angle is negative.
Figure : Angle

Standard Position
An angle is said to be in standard position if its vertex is at the origin of
a rectangular coordinate system and its initial side coincides with the
positive x-axis.
Figure : Standard Position

Quadrant
When an angle is in standard position, the terminal side will lie either in a
quadrant, in which case we say that lies in that quadrant, or will lie on
the x-axis or the y-axis, in which case we say that is a quadrantal angle.
Figure : Quadrant

The two commonly used measures for angles are degrees and radians.
Degree
One complete revolution = 3600
One quarter of a complete revolution = 900 = one right angle
One degree equals 60 minutes, i.e. 10 = 60 .
One minute equals 60 seconds, i.e. 1 = 60 .
Radian
One complete revolution 2π radians = 2πc
One radian is the angle subtended at the center of a circle by an arc
of the circle equal in length to the radius of the circle.
1 radian ≈ 57.2950.

NOTE
1800 = π radians;
10 = π
1800 radian
1 radian = 1800
π
Degrees 0 30 45 60 90 180 360
Radians 0 π
6
π
4
π
3
π
2 π 2π

Note
Let α and β be positive angles.
If α + β = 900, they are complementary angles.
If α + β = 1800, they are supplementary angles.
Figure : Complementary and Supplementary Angles

For any acute angle θ of a right angled triangle OAB
sin θ = Opposite
Hypotenuse = b
c
cos θ = Adjacent
Hypotenuse = a
c
tan θ = Opposite
Adjacent = b
a
csc θ = 1
sin θ , sec θ = 1
cos θ , cot θ = 1
tan θ
Figure : Right Triangle Trigonometry

Theorem
Cofunctions of complementary angles are equal.
sin β = b
c = cos α; cos β = a
c = sin α
csc β = c
b = sec α; sec β = c
a = csc α
tan β = b
a = cot α; cot β = a
b = tan α
Figure : Cofunctions

Trigonometric Ratios of Allied Angles
For θ <= 900 (acute)
sin (900 − θ) = cos θ
cos (900 − θ) = sin θ
tan (900 − θ) = cot θ
cot (900 − θ) = tan θ
csc (900 − θ) = sec θ
sec (900 − θ) = csc θ

PPPPPPPPPθ
T-ratios
sin cos tan
−θ − sin θ cos θ − tan θ
π/2 ± θ cos θ sin θ cot θ
π ± θ sin θ − cos θ ± tan θ
3π/2 ± θ − cos θ ± sin θ cot θ
2π ± θ ± sin θ cos θ ± tan θ

Computing the Values of Trigonometric Functions of Acute
Angle
Commonly Used Ratios 300
, 450
, and 600
Angles
sin 600 =
√
3
2 ; cos 600 = 1
2; tan 600 =
√
3
sin 300 = 1
2; cos 300 =
√
3
2 ; tan 300 = 1√
3
sin 450 = 1√
2
=
√
2
2 ; cos 450 = 1√
2
; tan 450 = 1
Figure : Commonly Used Ratios 300
, 450
, and 600

The Signs of the Trigonometric Functions
The Cartesian axes divide a plane into 4 quadrants:
00 → 900 1st quadrant
900 → 1800 2nd quadrant
1800 → 2700 3rd quadrant
2700 → 3600 4th quadrant
Figure : Quadrants in Cartesian Coordinates

Figure : Diagram of Trigonometric Sign in Every Quadrant
NOTE
Quadrantal Angles: 00, 900, 1800, 2700, 3600

Coterminal Angles
Two angles in standard position are said to be coterminal if they have the
same terminal side.
Figure : Coterminal Angle

Figure : Coterminal Angle
NOTE
θ is coterminal with θ ± 2πk , k is any integer.
The trigonometric functions of coterminal angles are equal.
Example: sin θ = sin θ ± 2πk

Reference Angles
Let θ denote a nonacute angle that lies in a quadrant. The acute angle
formed by the terminal side of θ and either the positive x-axis or the
negative x-axis is called the reference angle for θ.
Figure : Reference Angle

Theorem
If θ is an angle that lies in a quadrant and if α is its reference angle, then
sin θ = ± sin α; cos θ = ± cos α; tan θ = ± tan α
csc θ = ± csc α; sec θ = ± sec α; cot θ = ± cot α
where the + or - sign depends on the quadrant in which θ lies.

References
For next meeting please read
C. Young (2010)
Algebra and Trigonometry (second edition)
Wiley pp. 586–657.

The End

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