Transcript Chapter 1 Right Triangle Ratios

Chapter 1
Right Triangle Ratios
1.1 Angles, Degrees, and Arcs
1.2 Similar Triangles
1.3 Trigonometric Ratios and
Right Triangles
1.4 Right Triangle Applications
1.1 Angles, Degrees and Arcs
Angles
Degree measure of angles
Angles and arcs
Approximation of Earth’s circumference
Approximation of the diameters of the sun
and moon
Angles
 Degree measure: An angle formed by one
complete rotation of the terminal side in a
counterclockwise direction has measure 360º.
And angle with measure 1º is formed by 1/360 of
the complete rotation.
Right Triangle Ratios
Special Angles
Angle Measurement
Degrees, minutes and seconds are used
to measure angles.
1 degree = 60 minutes
1 degree = 3600 seconds
1 minute = 60 seconds
Convert 12º6’23” to degrees:
6’ = (6/60)º and 23” = (23/3600)º
12º6’23” = (12 + 6/60 + 23/3600)º
= 12.106º
Angles and Arcs
 The circumference of a circle C, the length of an
arc s, and the central angle subtended by the
arc q are related by q / 360º = s / C
Eratosthenes’ approximation: 7.5º/360º = 500 mi./C
C = 24,000 miles Actual value today: 24, 875 miles
1.2 Similar Triangles
Euclid’s Theorem and similar triangles
Applications
Euclid’s Theorem
 If two triangles are similar, their corresponding
sides are proportional.
Calculating Tree Height
A tree casts a shadow of
32 ft. and a yardstick
(3.0 ft.) casts a shadow
of 2.2 ft. Find the
height of the tree.
Solution:
x / 3 = 32 / 2.2
x = 3(32 / 2.2)
x ≈ 44 ft. to the nearest
integer
1.3 Trigonometric Ratios and Right Angles
Pythagorean Theorem
Trigonometric ratios
Calculator evaluation
Solving right triangles
Pythagorean Theorem
 In a right triangle the side opposite the right
angle is called the hypotenuse. If the other two
legs are a and b and the hypotenuse is c, this
relation is true:
a2 + b 2 = c2
Trigonometric Ratios
Complementary Relationships
Solving Right Triangles
 Given the measures of two sides or of one side and an
acute angle of a right triangle, the measures of the other
sides and angles can be found.
Example:
90º - 35.7º = 54.3º
b = (124m) sin 35.7º ≈ 72.4m
to three significant digits
a = (124m) cos 35.7º ≈ 101m
to three significant digits
1.4 Right Triangle Applications
Mine shaft
Length of air-to-air fueling hose
Astronomy
Mine Shaft Application
tan q =
opposite / adjacent
tan 20º = x / 310 ft.
x = (310 ft.) (tan 20º )
≈ 110 ft. to three
significant digits
Air-to-Air Fueling Hose
q = 32º and b = 120 ft.
sec 32º = c/b
c = 120ft. / cos 32º ≈ 140 ft. to three significant digits
Astronomy
sin 46º = x / 93,000,000 miles
x = 93,000,000 sin 46º ≈ 67,000,000 miles to two
significant digits