Unit 2 Angles, Triangles, and Trigonometryfaculty.ung.edu/.../2013-3-fall/1113notes_unit-2-40p.pdfHARTFIELD – PRECALCULUS UNIT 2 NOTES | PAGE 1 Unit 2 Angles, Triangles, and Trigonometry

HARTFIELD – PRECALCULUS UNIT 2 NOTES | PAGE 1

Unit 2 Angles, Triangles, 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) Solving Right Triangle

(17) Angles of Inclination, Depression, and Elevation

(20) Special Right Triangles

(21) Trigonometry of Angles

(23) Signed values of Trigonometric Ratios

(24) Reference Angles

(26) Reference Angles and Trigonometric Ratios

(28) Solving Oblique Triangles

Law of Sines

(31) Law of Cosines

(36) Ambiguous Case of Law of Sines

Know the meanings and uses of these terms:

Degree

Radian

Angle in standard position

Quadrantal angle

Coterminal angles

Sector of a circle

Reference angle

Oblique triangle

Review the meanings and uses of these terms:

Angle

Vertex of an angle

Ray

Intecepted arc

Central angle of a circle

Right triangle

Angle Sum Theorem


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

Definition: A right triangle is a triangle with a 90 angle (right angle).

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

tanadjacent leg

adjacent leg

cotopposite leg

hypotenuse

secadjacent 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 cot

sec csc

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

Ex.:

sin cos

tan cot

sec csc


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

Ex.: 2

cos5

sin

tan cot

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


Solving Right Triangles

Given a right triangle, if: 1. the length of two sides are known, or

2. the length of one side and the measure of one acute angle are known,

then it is possible to find the remaining side lengths and angle measures. A triangle is said to be solved when all sides and angles are known.

Each trigonometric ratio is formed by one angle and two sides. With two pieces of information, it is possible to create an equation with only one unknown. The inverse of a trigonometric ratio is necessary when you are solving for an angle.

When given two sides, find the length of the third side by the Pythagorean Theorem.

If an acute angle is known, the other acute angle can be found using the Angle Sum Theorem.

* Angle Sum Theorem: The sum of the angle

measurements in a triangle is 180.

Sides of a triangle should always be labeled with lower case letters. Vertices (and by extension the angles formed at the vertices) should always be labeled with upper case letters matching the side opposite the angle.


Solve the right triangle. Approximate as necessary to five digits.

Ex. 1:

7

52


Ex. 2 12

8



Ex. 3: The length of the hypotenuse is 15 units & the larger of the two acute angles

measures 64.

Sketch the triangle given in example 3.


Angles of Elevation, Depression and Inclination

Definition: An angle of elevation is an angle above horizontal that an observer must look to see an object that is higher than the observer.

Definition: An angle of depression is an angle below horizontal that an observer must look to see an object that is lower than the observer.

Definition: An angle of inclination is an angle formed by a line above horizontal.

We usually use an angle of inclination in the context of an object without an observer, such as the incline of a mountain. However this is not always the case.


Ex. 1: Bill is looking out the window of his third floor apartment, 20 feet about the ground, as illustrated at right. (Note, picture is not drawn to scale.)

A: Bill spots a $100 bill on the ground outside his

apartment. If the angle of depression is 18, how far from Bill’s apartment building is the money? Approximate to the nearest foot.

B: Bill sees an airplane flying in the distance. Small planes in this area are usually flying at 10,000 feet above sea level. Bill’s apartment building is at 1200 feet above sea level. If the

angle of elevation to see the plane is 24, by line of sight how far away is the airplane? Approximate to the nearest tenth of mile.


Ex. 2: George is standing near a tree that is casting a long shadow.

A: The tree stands 40 feet high and the shadow is 100 feet long. At what angle of inclination, with respect to the ground, is the sun based on this shadow? Approximate to the nearest tenth of a degree.

B: George’s eyes are 5 feet above the ground. He spots a cardinal in the tree and the angle of

elevation for him to see the bird is 32. If George is 30 feet from the spot beneath the cardinal, how high up is the bird? Approximate to the nearest tenth of a foot.


Special Right Triangles

45-45-90 Triangle

sin45

cos45

tan45

cot45

sec45

csc45

30-60-90 Triangle

sin30 sin60

cos30 cos60

tan30 tan60

cot30 cot60

sec30 sec60

csc30 csc60


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 ry 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: cos290

Ex. 2: tan 570

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

Ex. 3: sin2390

Ex. 4: 27

cos5


Solving Oblique Triangles

Defintion: An oblique triangle is any triangle that is not a right triangle; i.e., it does not have a right angle.

Given a side length and two additional pieces of information (side lengths or angle measurements), it is possible to solve any triangle for which a solution exists. When given an oblique triangle with two known angle measurements, it is possible to use the Law of Sines to find a unique triangle solution.

Law of Sines The Law of Sines is a statement of proportionality: in any triangle each ratio formed by the sine of an angle to the length of a side opposite the angle is equal.

sin sin sinA B C

a b c

B

a

c

C

A b

Observation: The largest angle of a triangle is always opposite of the longest side.


Solve the triangle. Approximate as necessary to five digits.

Ex. 1:

60

84

25



Ex. 2: b = 1000 A = 22 C = 95

Sketch the triangle from example 2.


Law of Cosines

When given an oblique triangle with all three sides known or an oblique triangle where the one known angle is between two known sides, it is possible to use the Law of Cosines to find a unique triangle solution.

B

a

c

C

A b

The Law of Cosines is a set of statements amending the Pythagorean Theorem such that it can be applied to any triangle: the square of a side of a triangle is equal to sum of the squares of the other sides minus twice the product of the other sides and the cosine of the first side.

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

c a b ab C

a b c bc A

b c a ca B

After using the Law of Cosines to find an unknown side or an unknown angle, it is possible to find the remaining sides or angles using the Law of Sines. The Law of Sines should never be used to find an obtuse angle however; either use the Law of Cosines or the Angle Sum Theorem.



55

10 18

Ex. 1:



Ex. 2: a = 12 c = 30 B = 28




Ex. 3: 10

20

12



Ex. 4: a = 20 b = 24 c = 32



Ambiguous Case of the Law of Sines

If you are given the lengths of two sides and the measure of angle opposite one of the sides, it is possible for three scenarios to exist:

1. A unique solution exists for one triangle; that is, exactly one third side length and two additional angle measures satisfy the given information.

2. Two parallel solutions exist, each with a third side length and two angle measures, that create triangles satisfying the given information.

3. No triangle can be formed using the given information.

The number of triangles satisfying the given information can be determined based on what happens when the Law of Sines is applied to find the measure opposite of one of the side lengths:

1. If an angle measure exists and is less than the given angle measure, exactly one triangle satisfies the given information.

2. If an angle measure exists and is of greater measure than the given angle, two triangles will satisfy the given information. One triangle will use the initially found measure while the second triangle will use the supplement of the found measure.

3. If no angle measure exists, then no triangle will satisfy the given information.


Solve all possible triangles that satisfy the information given below. Approximate as necessary to five digits.

Ex.1: a = 16 b = 12 A = 65



Ex. 2: b = 14 c = 12 C = 22



Ex. 3: a = 17 c = 12 C = 79



Ex. 4: a = 21 b = 25 A = 29

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Unit 2 Angles, Triangles, and Trigonometryfaculty.ung.edu/.../2013-3-fall/1113notes_unit-2-40p.pdfHARTFIELD – PRECALCULUS UNIT 2 NOTES | PAGE 1 Unit 2 Angles, Triangles, and Trigonometry