Slide 1-1By Y. Ath

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Section 1Angles

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Basic Terminology

Line AB.

Line segment AB

Ray AB

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Math symbols (Greek alphabets)

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An angle’s measure is generated by a rotation about the vertex.

The ray in its initial position is called the initial side of the angle.

The ray in its location after the rotation is the terminal side of the angle.

Basic Terminology

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Positive angle: The rotation of the terminal side of an angle is counterclockwise.

Negative angle: The rotation of the terminal side is clockwise.

Basic Terminology

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Measuring Angle

Two units for measuring angle: (1)degrees, minutes, seconds (DMS) or decimal degrees (2)radians

0

0

Formula:

(1) 1 60 minutes, 1 minute = 60 seconds

(2) 180 (3.14159...) radians

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The most common unit for measuring angles is the degree.

Degree Measure

A complete rotation of a ray gives an angle whose measure is 360°.

of complete rotation gives an angle whose measure is 1°.

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Example

Convert to degrees, minutes, seconds

Convert to radians

'''

'' .

' ..

365145:Ans

36or seconds366060

51or minutes65160860

0

radian 0.8 :Ans

radian 0.800408 180

860.45

18086.45

00

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Example

Convert to decimal degrees

'''3214270

0227.242

degrees 242222.273600

32

60

1427

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Angles are classified by their measures.

Degree Measure

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Complementary, Supplementary Angles

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For an angle measuring 40°, find the measure of (a) its complement and (b) its supplement.

Example 1 FINDING THE COMPLEMENT AND THE SUPPLEMENT OF AN ANGLE

(a) To find the measure of its complement, subtract the measure of the angle from 90°.

Complement of 40°

(b) To find the measure of its supplement, subtract the measure of the angle from 180°.

Supplement of 40°

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Find the measure of each marked angle.

Example 2 FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Since the two angles form a right angle, they are complementary.

Combine like terms.

Divide by 9.

Determine the measure of each angle by substituting 10 for x:

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Find the measure of each marked angle.

Example 2 FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES (continued)

Since the two angles form a straight angle, they are supplementary.

The angle measures are and .

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Quadrantal Angles

angles. quadrantal are on, so and,27018090,0 measures with angles assuch

axis,-yor axis- xon the lies sides terminalhoseposition w standardin Angles0000 ,,

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Coterminal Angles

Angles are coterminal if their initial and terminal sides are the same.

Example000 340,380 ,20

,...2,1,0

A angle toangles coterminal

(given) degreesin angle

360

:Formula

k

A

A

kAA

c

c

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Coterminal Angles

To find an expression that will generate all angles coterminal with a given angle, add integer multiples of 360° to the given angle.

For example, the expression for all angles coterminal with 60° is

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Coterminal Angles

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Example ANALYZING THE REVOLUTIONS OF A CD PLAYER

CD players always spin at the same speed. Suppose a player makes 480 revolutions per min. Through how many degrees will a point on the edge of a CD move in 2 sec?

The player revolves 480 times in 1 min or times = 8 times per sec.

In 2 sec, the player will revolve times. Each revolution is 360°, so a point on the edge of the CD will revolve in 2 sec.

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Section 1.2 Angles Relationships and Similar Angles

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Vertical Angles

Parallel Lines

Alternate interior Angles

Alternate exterior Angles

Corresponding Angles

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Find the measures of angles 1, 2, 3, and 4, given that lines m and n are parallel.

Example FINDING ANGLE MEASURES

Angles 1 and 4 are alternate exterior angles, so they are equal.

Subtract 3x.Add 40.Divide by 2.

Angle 1 has measureSubstitute 21 for x.

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Example FINDING ANGLE MEASURES (continued)

Angle 4 has measureSubstitute 21 for x.

Angle 2 is the supplement of a 65° angle, so it has measure .

Angle 3 is a vertical angle to angle 1, so its measure is 65°.

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Angle Sum of a Triangle

The sum of the measures of the angles of any triangle is 180°.

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Example APPLYING THE ANGLE SUM OF A TRIANGLE PROPERTY

The measures of two of the angles of a triangle are 48 and 61. Find the measure of the third angle, x.

The third angle of the triangle measures 71°.

The sum of the angles is 180°.

Add.

Subtract 109°.

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Types of Triangles

Acute triangle

Right triangle

Obtuse triangle

Equilateral triangle

Three equal anglesThree equal sides

Isosceles triangle

Two equal sidesTwo equal angles

Scalene triangle

No equal sidesNo equal angles

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Proportion, Similar Triangles

Proportion: One ratio or one fraction equals another

Similar Triangles: Two triangles are similar if and only if corresponding sides are in proportion and the corresponding angles are congruent.

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6

4

3 )( a

3 ,248 8

6

4 )( xx

xb

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Example FINDING ANGLE MEASURES IN SIMILAR TRIANGLES

In the figure, triangles ABC and NMP are similar.Find the measures of angles B and C.

Since the triangles are similar, corresponding angles have the same measure.

B corresponds to M, so angle B measures 31°.

C corresponds to P, so angle C measures 104°.

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Example FINDING SIDE LENGTHS IN SIMILAR TRIANGLES

Given that triangle ABC and triangle DFE are similar, find the lengths of the unknown sides of triangle DFE.

Similar triangles have corresponding sides in proportion.

DF corresponds to AB, and DE corresponds to AC, so

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Example FINDING SIDE LENGTHS IN SIMILAR TRIANGLES (continued)

Side DF has length 12.

EF corresponds to CB, so

Side EF has length 16.

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Sections 1.3 & 1.4 Trigonometric Functions

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Six Trigonometric Functions Given a Point

r

y)sin( )1(

r

x)cos()2(

x

y)tan()3(

y

r

)sin(

1)csc()4(

x

r

)cos(

1)sec()5(

y

x

)tan(

1)cot()6(

Try to remember

SOH-CAH-TOA

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The terminal side of angle in standard position passes through the point (8, 15). Find the values of the six trigonometric functions of angle .

Example FINDING FUNCTION VALUES OF AN ANGLE

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Example FINDING FUNCTION VALUES OF AN ANGLE (continued)

We can now find the values of the six trigonometric functions of angle .

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Example FINDING FUNCTION VALUES OF AN ANGLE (continued)

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Using a Calculator

Function values of quadrantal angles can be found with a calculator that has trigonometric function keys. Make sure the calculator is set in degree mode.

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Caution

One of the most common errors involving calculators in trigonometry occurs when the calculator is set for radian measure, rather than degree measure. Be sure you know how to set your calculator in degree mode.

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Signs of Function Values

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Pythagorean Theorem

222 ryx

1sincos 1 )1( 2222

r

y

r

x

Pythagorean Identities

2222

sectan1 1 )2(

x

r

x

y

22

22

csc1cot 1 )3(

y

r

r

x

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Quotient Identities

tan/

/

cos

sin )7(

x

y

rx

ry

cot/

/

sin

cos )8(

y

x

ry

rx

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Suppose that angle is in quadrant II and Find the values of the other five trigonometric functions.

Example FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT

Choose any point on the terminal side of angle .

Let r = 3. Then y = 2.

Since is in quadrant II,

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Example FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)

Remember to rationalize

the denominator.

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Example FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)

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Example USING IDENTITIES TO FIND FUNCTION VALUES

Choose the positive square root since sin θ >0.

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Example

To find tan θ, use the quotient identity

USING IDENTITIES TO FIND FUNCTION VALUES (continued)

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Example USING IDENTITIES TO FIND FUNCTION VALUES

Find sin θ and cos θ, given that and θ is in quadrant III.

Since θ is in quadrant III, sin θ and cos θ will both be negative. It is tempting to say that since

and

then sin θ = –4 and cos θ = –3. This is incorrect, however, since both sin θ and cos θ must be in the interval [–1,1].

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Example

Use the identity to find sec θ. Then use the reciprocal identity to find cos θ.

Choose the negative square root since sec θ <0 when θ is in quadrant III.Secant and cosine are reciprocals.

USING IDENTITIES TO FIND FUNCTION VALUES (continued)

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Example

Choose the negative square root since sin θ <0 for θ in quadrant III.

USING IDENTITIES TO FIND FUNCTION VALUES (continued)

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Example

This example can also be worked by sketching θ in standard position in quadrant III, finding r to be 5, and then using the definitions of sin θ and cos θ in terms of x, y, and r.

USING IDENTITIES TO FIND FUNCTION VALUES (continued)