week 1

Analytic Trigonometry (Week 1)

Irianto

Universiti Teknikal Malaysia Melaka

[email protected]

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 theline and extends indefinitely in one direction. The starting point V of a rayis 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 theterminal side. The angle formed is identified by showing the direction andamount 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; ifthe 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 ofa rectangular coordinate system and its initial side coincides with thepositive x-axis.

Figure : Standard Position



Quadrant

When an angle is in standard position, the terminal side will lie either in aquadrant, in which case we say that lies in that quadrant, or will lie onthe 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 arcof the circle equal in length to the radius of the circle.

1 radian ≈ 57.2950.



NOTE

1800 = π radians;

10 = π1800

radian

1 radian = 1800

π

Degrees 0 30 45 60 90 180 360

Radians 0 π6

π4

π3

π2 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 OABsin θ = Opposite

Hypotenuse = bc

cos θ = AdjacentHypotenuse = a

c

tan θ = OppositeAdjacent = b

a

csc θ = 1sin θ , sec θ = 1

cos θ , cot θ = 1tan θ

Figure : Right Triangle Trigonometry



Theorem

Cofunctions of complementary angles are equal.sinβ = b

c = cosα; cosβ = ac = sinα

cscβ = cb = secα; secβ = c

a = cscα

tanβ = ba = cotα; cotβ = a

b = tanα

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



PPPPPPPPPθT-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 AcuteAngle

Commonly Used Ratios 300, 450, and 600 Angles

sin 600 =√32 ; cos 600 = 1

2 ; tan 600 =√

3

sin 300 = 12 ; cos 300 =

√32 ; tan 300 = 1√

3

sin 450 = 1√2

=√22 ; 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 quadrant900 → 1800 2nd quadrant1800 → 2700 3rd quadrant2700 → 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 thesame 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 angleformed by the terminal side of θ and either the positive x-axis or thenegative 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, thensin θ = ± sinα; cos θ = ± cosα; tan θ = ± tanαcsc θ = ± cscα; sec θ = ± secα; cot θ = ± cotαwhere 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


week 1

Education

measure angle

quadrantal angle

content angles

positive angles

measure standard

measure positive

measure quadrant

negative angles