MATH 107

Sections 5.1

Angles and their Measure

The initial side is always located on the positive-x-axis;

the vertex is always the origin.

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ANGLES

An angle in a rectangular coordinate system is in standard position if its vertex is at the origin and its initial side is the positive x-axis.

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ANGLES

An angle in standard position is said to lie in a quadrant if its terminal side lies in that quadrant.

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MEASURING ANGLES BY USING DEGREESAn acute angle has measure between 0° and 90°.

A right angle has measure 90°, or one-fourth of a revolution.

An obtuse angle has measure between 90° and 180°.

A straight angle has measure 180°, or half a revolution.

Angle Measure

Acute 0° < θ < 90°

Right 90°

Obtuse 90° < θ < 180°

Straight 180°

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EXAMPLE 1 Drawing an Angle in Standard Position

Draw each angle in standard position.

a. 60° b. 135° c. 240° d. 405°

Solution

a. Because 60 = (90),

a 60° angle is of a

90° angle.3

23

2

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EXAMPLE 1 Drawing an Angle in Standard Position

Solution continued

b. Because 135 = 90 + 45, a 135º angle is a counterclockwise rotation of 90º, followed by half a 90º counterclockwise rotation.

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EXAMPLE 1 Drawing an Angle in Standard Position

Solution continued

c. Because 240 = 180 60, a 240º angle is a clockwise rotation of 180º, followed by a clockwise rotation of 60º.

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EXAMPLE 1 Drawing an Angle in Standard Position

Solution continued

d. Because 405 = 360 + 45, a 405º angle is one complete counterclockwise rotation, followed by half a 90º counterclockwise rotation.

Answers on next slide.

C

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CONVERTING BETWEENDEGREES AND RADIANS

radians 180

degree 1

degrees 180

radian 1

radians 180

degrees 180

radians

Degrees to radians:

Radians to degrees:

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EXAMPLE 3 Converting from Degrees to Radians

Convert each angle in degrees to radians.

a. 30° b. 90° c. 225° d. 55°

Solution

a.

b.

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EXAMPLE 3 Converting from Degrees to Radians

Solution continued

d.

c.

(a) 30° (b) 120° (c) - 60° (d) 270° (e) 104 °

Answers on next slide.

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EXAMPLE 4 Converting from Radians to Degrees

a. radians3

Convert each angle in radians to degrees.

Solution180º 180

a. radians 60º3 3 3

º

180º3 3 3b. radians 180º 135º

4 4 4

c. 1 radian 1180

7 ºº

5 .3

radians 4

3 b.

5(a) radian (b) radian (c) radians (d) 5 radians

3 2 6

Answers on next slide.

s s

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ARC LENGTH FORMULA

Where: r is the radius of the circle

θ is in radians.

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EXAMPLE 6 Finding Arc Length of a Circle

A circle has a radius of 18 inches. Find the length of the arc intercepted by a central angle with measure 210º.

Solution

MATH 107

Sections 5.2

Right Triangle Trigonometry

SOH CAH TOA

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EXAMPLE 1Finding the Values of Trigonometric Functions

Find the exact values for the six trigonometric functions of the angle in the figure.

Solution

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EXAMPLE 1

Solution continued

Finding the Values of Trigonometric Functions

Now, with c = 4, a = 3, and b = , we have7

Examples

Find exact values of 6 trig functions for right triangle with opposite side of length 4 and hypotenuse of 5

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EXAMPLE 3

Solution

Finding the Trigonometric Function Values for 45°.

Use the figure to find sin 45°, cos 45°, and tan 45°.

2

2

2

1

hypotenuse

opposite45sin

2

2

2

1

hypotenuse

adjacent45cos

11

1

adjacent

opposite45tan

Example

Find the other 5 trig fcts of θ given θ is acute angle of right triangle and cos θ = 1/3

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SOH CAH TOA

Common trigonometric values

***Have these memorized, but be able to re-derive them if necessary.***

Also remember the two special right triangles and their ratios of sides:

45-45-90 degree triangle: 1, 1, 2 ratio of sides

30-60-90 degree triangle: 1, 3, 2 ratio of sides

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PYTHAGOREAN IDENTITIES

1cossin 22 θ

22 sectan1 θ

22 csccot1 θ

The cofunction, reciprocal, quotient, and Pythagorean identities are called the Fundamental identities.

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APPLICATIONSAngles that are measured between a line of sight and a horizontal line occur in many applications and are called angles of elevation or angles of depression.

If the line of sight is above the horizontal line, the angle between these two lines is called the angle of elevation.

If the line of sight is below the horizontal line, the angle between the two lines is called the angle of depression.

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EXAMPLE 8

A surveyor wants to measure the height of Mount Kilimanjaro by using the known height of a nearby mountain. The nearby location is at an altitude of 8720 feet, the distance between that location and Mount Kilimanjaro’s peak is 4.9941 miles, and the angle of elevation from the lower location is 23.75º. See the figure on the next slide. Use this information to find the approximate height of Mount Kilimanjaro (in feet).

Measuring the Height of Mount Kilimanjaro

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EXAMPLE 8 Measuring the Height of Mount Kilimanjaro

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SolutionThe sum of the side length h and the location height of 8720 feet gives the approximate height of Mount Kilimanjaro. Let h be measured in miles. Use the definition of sin , for = 23.75º.

EXAMPLE 8 Measuring the Height of Mount Kilimanjaro

h = (4.9941) sin θ = (4.9941) sin 23.75°h ≈ 2.0114

9941.4hypotenuse

oppositesin

h