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7.5-7.7 Notes
Geometry
Solving Right Triangles
Solve for x.
x
1)
2)
3)
x
12
4)
10
6
10
x
x
5
12
60
You can solve a right triangle (meaning – you can find the measurements of all 6 parts) if you know:
 2 side lengths
 1 side length and one acute angle measurement
Trigonometry
The origins of trigonometry trace back to the cultures of the ancient Egyptians, Babylonians and Indus
Valley civilizations, over 4000-5000 years ago. Trigonometry (from the Greek trigonon meaning 3
angles, and metro meaning measure) is a branch of mathematics dealing with angles, triangles, and trig
functions including sine, cosine and tangent. Why study triangles? Because they are the basic
building blocks from which any shape (with straight boundaries) can be constructed. There are an
enormous number of applications of trig:
triangulation, which is used in astronomy to measure the distance to nearby stars geography, to
measure distances between landmarks
satellite navigations systems
astronomy (and hence navigation, on the oceans, in aircraft, and in space)
music theory
acoustics
optics
analysis of financial markets
electronics
probability theory
statistics
biology
medical imaging (CAT scans and ultrasound)
pharmacy
chemistry
number theory
cryptology
seismology
meteorology
oceanography
physical sciences
land surveying and geodesy (the branch of applied mathematics that determines the exact position of
points and figures, and areas of large portions of the earth’s surface, the shape and size of the earth and
the variations of terrestrial gravity and magnetism)
architecture
phonetics
economics
electrical engineering
mechanical engineering
civil engineering
computer graphics
cartography
crystallography
In geometry, we only study the branch of trig referred to as “Right Triangle Trigonometry”
The ratio of any two sides of a triangle is called a trigonometric ratio.
Three of the most common ratios are:
Sine
The calculator converts these functions to:
_____________
Cosine
_____________
Tangent
_____________
Trig ratios are related to the acute angles of a right triangle, not the right angle.
sin A =
cos A =
Examples:
tan A =
B
a) sin A =
b) cos A =
Use the calculator to find the
following to decimal places:
12
a) sin 43 =
c) tan A =
d) sin B =
C
5
A
b) cos 84 =
c) tan 45 =
e) cos B =
f) tan B =
A
Find all 6 measurements of the triangle:
32
C
Find all 6 measurements of the triangle:
B
5
A
10
C
Examples:
a) sin A = .63
A=
13
b) cos B = 0.1034
B=
B
c) tan C = 1.23
C=
Examples: Applications
a) A 16 foot ladder is propped against a building. The angle it forms with the ground is 55 . How
far up the side of the building is the ladder?
b) A ramp is to be built from a location 4 meters from the base of a loading dock. The dock is 1.2
meters up from the ground. What is the measure of the angle the ramp makes with the ground?