Unit 2 Intro to Angles and Trigonometry ASIC C ONLY...

HARTFIELD – PRECALCULUS UNIT 2 NOTES | PAGE 1

Unit 2 Intro to Angles and Trigonometry

(2) Definition of an Angle

(3) Angle Measurements & Notation

(4) Conversions of Units

(6) Angles in Standard Position

Quadrantal Angles; Coterminal Angles

(8) Arcs and Sectors of Circles

(10) Trigonometry of Right Triangles

(14) 45-45-90 Triangles

30-60-90 Triangles

(15) Trigonometry of Angles

(17) Signed values of Trigonometric Ratios

(18) Reference Angles

(20) Reference Angles and Trigonometric Ratios

This is a BASIC CALCULATORS ONLY unit.

Know the meanings and uses of these terms:

Degree

Radian

Angle in standard position

Quadrantal angle

Coterminal angles

Sector of a circle

Reference angle

Review the meanings and uses of these terms:

Angle

Vertex of an angle

Ray

Intecepted arc

Central angle of a circle


Definition of an Angle

(Geometric Definition)

Definition 1: The composition of two rays with a common endpoint.

Definition 2: The result of coincident rays where one ray has been rotated about its endpoint.

Definition: The vertex of an angle is the endpoint shared by the rays of the angle.

AOB at right

R1 and R2 are the rays

O is the vertex

If R2 is the ray that has been rotated out of coincidence with R1, we say that R1 is the initial side and R2 is the terminal side.

The measurement of an angle is quantified by the amount of rotation from the initial side to the terminal side. A counterclockwise rotation results in a positive measurement while a clockwise rotation results in a negative measurement.


Angle Measurements & Notation

There are two primary units used for measuring angles: degrees and radians. (There is also a historically interesting but functionally irrelevant third unit called grads.)

One degree is defined to be 1/360th of a complete rotation about a vertex. Thus an angle measuring

360 would involve the terminal side rotating completely back into coincidence with the initial side.

The most common symbol used to mark an angle

and identify its measurement is the Greek letter (theta).

One radian is defined to be a rotation in which the intercepted arc of the unit circle is length 1. Thus

an angle measuring 2 would involve the terminal side rotating completely back into coincidence with the initial side; by extension, this means one

radian is exactly 1/2 of a complete rotation about a vertex.


Basic Conversions of Degrees and Radians

180 = radians

90 = radians 270 = radians

45 = radians

30 = radians 60 = radians

1 = 180

radians 0.017453 radians

1 radian = 180

57.296

When working with degrees, always either use the

symbol or write the word degree.

When working with radians you may use the word radian, the abbreviation rad, or nothing at all.


More Converting of Measurements

To convert from degrees to radians, multiply by .180

To convert from radians to degrees, multiply by 180

.

Convert from degrees to radians:

Ex. 1: 40

Ex. 2: 225

Convert from radians to degrees:

Ex. 1: 5

6

radians

Ex. 2: 13

8

radians

Ex. 3: 5 radians


Angles in Standard Position

Definition: An angle is said to be in standard position if its vertex is at the origin of the coordinate plane and its initial side is on the positive x-axis.

Quadrantal Angles and Coterminal Angles

Definition: An angle is described as quadrantal if its terminal side is on an axis.

…, -90, 0, 90, 180, 270, 360, …

…, ,2

0, ,

2

,

3,

2

2, …

Definition: Angles are said to be coterminal if they have a common terminal side.

Example: 70, 430, -290 are coterminal measurements


For an angle measurement of c degrees, c + 360n, where n is any integer, will be a coterminal angle measurement.

Find a coterminal angle measurement in [0, 360).

Ex. 1: = 1000

Ex. 2: = 1975

For an angle measurement c radians, c + 2n, where n is any integer, will be a coterminal angle measurement.

Find a coterminal angle measurement in [0, 2).

Ex. 1: = 23

3

radians

Ex. 2: = 37

4

radians


Arcs and Sectors of Circles

Definition: An arc is a portion of a circle between two endpoints.

Definition: An intercepted arc is a portion of a circle whose endpoints are points on the rays of an angle.

Definition: A central angle of a circle is an angle whose vertex is at the center of the circle.

The length of an arc of a circle, represented by s, can be calculated using the radius r of the

circle and the measurement in radians of the center angle which subtends the arc:

s = r

Definition: A sector of a circle is a region in the interior of a circle bounded by a central angle and the arc it subtends.

The area of a sector of a circle, represented by As, can be calculated using the radius r of the

circle and the measurement in radians of the center angle which subtends the arc:

As = 12 r2


Calculate the length of the arc labeled s below and

the area of the sector bounded by and s.

Ex. 1: s

8 m

Calculate the radius of the circle labeled r below

and the area of the sector bounded by and s.

Ex. 2: 16 ft

r


Trigonometry of Right Triangles

Trigonometry of right triangles is based on relationships between an acute angle and the ratio formed by two sides of the triangle.

With respect to the

angle chosen in the triangle at left, the trigonometric ratios

of are defined:

The sine of is the ratio of the opposite leg to the hypotenuse.

The cosine of is the ratio of the adjacent leg to the hypotenuse.

The tangent of is the ratio of the opposite leg to the adjacent leg.

The cotangent of is the ratio of the adjacent leg to the opposite leg.

The secant of is the ratio of the hypotenuse to the adjacent leg.

The cosecant of is the ratio of the hypotenuse to the opposite leg.

opposite leg

sinhypotenuse

adjacent leg

coshypotenuse

opposite leg

tanad jacent leg

adjacent leg

cotopposite leg

hypotenuse

secad jacent leg

hypotenuse

cscopposite leg

It is important to remember that the ratios are based on the relative position of the legs of the right triangle with respect to the angle chosen.

If changes from one of the acute angles to the other acute angle, the roles of opposite and adjacent are switched.


Define the trigonometric ratios of below.

Ex:

sin cos

tan co t

sec csc

Find the third side of length using the Pythagorean Theorem, then define the trigonometric ratios.

Ex.:

sin cos

tan co t

sec csc


Sketch a triangle which satisfies the given ratio & then define the remaining trigonometric ratios.

Ex.: 2

cos5

sin

tan co t

sec csc

Observe that a physical model of a triangle demonstrates the consistency between the ratios and the angles:

1 unit


Express x and y as ratios in terms of .

Ex. 1:

15 x

y

Express x and y as ratios in terms of .

Ex. 2: 12

x y


45-45-90 Triangle

sin 45

cos 45

tan 45

cot 45

sec 45

csc 45

30-60-90 Triangle

sin 30 sin 60

cos 30 cos 60

tan 30 tan 60

cot 30 cot 60

sec 30 sec 60

csc 30 csc 60


Trigonometry of Angles

By placing one of the acute angles of a triangle at the origin in standard position, it is possible to relate the trigonometry of right triangles to angles in general.

We can eventually extend the definition of the trigonometric ratios by noting that it is possible to form a consistent definition even when the angle

is outside the interval (0, 90).

Definition: Let (x, y) be a point on the terminal

side of an angle in standard position. Let r be the distance from the origin to (x, y). Then:

sin cos

tan cot

sec csc

y x

r r

y x

x y

r r

x y

(x, y)


Find the trigonometric ratios of .

Ex. 1:

Find the trigonometric ratios of .

Ex. 2:

(-4, -10)

sin cos

tan cot

sec csc

sin cos

tan cot

sec csc


Signed values of trigonometric ratios

From the definition of each trigonometric ratio, it is possible to know in advance whether the ratio is going to be positive or negative.

Quadrant I sin, cos, tan, cot, sec, csc positive

x > 0, y > 0

(0, 90) or 20,

Quadrant II sin, csc positive

x < 0, y > 0 cos, tan, cot, sec negative

(90, 180) or 2,

Quadrant III tan, cot positive

x < 0, y < 0 sin, cos, sec, csc negative

(180, 270) or 32

,

Quadrant IV cos, sec positive

x > 0, y < 0 sin, tan, cot, csc negative

(270, 360) or 32

,2

Without determining the exact value, determine whether the trigonometric ratio is positive or negative.

Ex. 1 sin 200

Ex. 2 sec 300

Ex. 3 tan –80

Ex. 4 cos 1180


Reference Angles

Definition: Let be an angle in standard

position. Then the reference angle associated with is the acute angle formed by the terminal side of and the x-axis.

To find a reference angle for some angle :

If is in (0, 360) or (0, 2), and

the terminal side is in QI, then .

the terminal side is in QII, then 180 .

.

the terminal side is in QIII, then 180 .

.

the terminal side is in QIV, then 360 .

2 .

If is not in (0, 360) or (0, 2), find a coterminal angle c that is and then apply the rules above

substituting c for .

__

__

__

__


Find the reference angle for each given .

Ex. 1: 290

Ex. 2: 570

Find the reference angle for each given .

Ex. 3: 2390

Ex. 4: 27

5


Reference angles and trigonometric ratios

Any trigonometric ratio involving will have the same absolute value as the same trigonometric

ratio involving .

Since is acute, any trigonometric ratio involving

will have a positive value.

Thus any trigonometric ratio of can be defined

in terms of a trigonometric ratio of with an appropriate accommodation of its sign value based on the quadrant where the terminal side of is found.

Find the sine, cosine, and tangent of .

Ex.:

sin =

cos =

tan =


Rewrite each trigonometric ratio using a reference angle and then evaluate as possible.

Ex. 1: cos 290

Ex. 2: tan 570

Rewrite each trigonometric ratio using a reference angle and then evaluate as possible.

Ex. 3: sin 2390

Ex. 4: 27

cos5

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Unit 2 Intro to Angles and Trigonometry ASIC C ONLY...

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