lecture 13 sections 5.1-5.2 angles & right triangles
TRANSCRIPT
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