trigonometric Functions, algebra homework help

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Pages 779 and 796 attached 13-1Right Triangle Trigonometry
Main Ideas
• Find values of
trigonometric functions
for acute angles.
Solve problems
involving right triangles.
GET READY for the Lesson
The Americans with Disabilities
Act (ADA) provides regulations
designed to make public buildings
accessible to all. Under this act,
the slope of an entrance ramp
angle A
designed for those with mobility
disabilities must not exceed a ratio
of 1 to 12. This means that for
every 12 units of horizontal run,
the ramp can rise or fall no more than 1 unit.
When viewed from the side, a ramp forms a right triangle. The slope
of the ramp can be described by the tangent of the angle the ramp
makes with the ground. In this example, the tangent of angle A is 12
1 ft
New Vocabulary
trigonometry
trigonometric functions
sine
cosine
tangent
cosecant
secant
cotangent
solve a right triangle
angle of elevation
angle of depression
12 ft
B
Trigonometric Values The tangent of an angle
is one of the ratios used in trigonometry.
Trigonometry is the study of the relationships
among the angles and sides of a right triangle.
hypotenuse
Reading Math
opposite
leg
Trigonometry
The word trigonometry is
derived from two Greek
words-trigon meaning
triangle and metra meaning
measurement
Consider right triangle ABC in which the
measure of acute angle A is identified by the
Greek letter theta, 8. The sides of the triangle
are the hypotenuse, the leg opposite 0, and the
leg adjacent to 8.
A
adjacent leg
с
Using these sides, you can define six trigonometric functions: sine,
cosine, tangent, cosecant, secant, and cotangent. These functions are
abbreviated sin, cos, tan, csc, sec, and cot, respectively.
KEY CONCEPT
Trigonometric Functions
If e is the measure of an acute angle of a right triangle, opp is the measure
of the leg opposite 6, adj is the measure of the leg adjacent to e, and hyp is
the measure of the hypotenuse, then the following are true.
sin 6 =
орр
adi
opp
COS 6 =
tan =
hyp
hyp
adj
hyp
hyp
adj
CSC O =
opp
cot =
adj
opp
Notice that the sine, cosine, and tangent functions are reciprocals of the
cosecant, secant, and cotangent functions, respectively. Thus, the following
are also true.
sec =
CSC A =
1
sin e
sec 8 =
cot 8 =
1
tan
cos
13-3
Trigonometric Functions
of General Angles
GET READY for the Lesson
Main Ideas
• Find values of
trigonometric functions
for general angles.
• Use reference angles
to find values of
trigonometric
functions.
New Vocabulary
A skycoaster consists of a large arch
from which two steel cables hang
and are attached to riders suited
together in a harness. A third cable,
coming from a larger tower behind
the arch, is attached with a ripcord.
Riders are hoisted to the top of the
larger tower, pull the ripcord, and
then plunge toward Earth. They
swing through the arch, reaching
speeds of more than 60 miles per
hour. After the first several swings
of a certain skycoaster, the angle 8 of
the riders from the center of the arch is given by 0 = 0.2 cos (1.6t),
where t is the time in seconds after leaving the bottom of their swing.
quadrantal angle
reference angle
Trigonometric Functions and General Angles In Lesson 13-1, you found
values of trigonometric functions whose domains were the set of all
acute angles, angles between 0 and, of a right triangle. For t> 0 in the
equation above, you must find the cosine of an angle greater than In
this lesson, we will extend the domain of trigonometric functions to
include angles of any measure.
KEY CONCEPT
Trigonometric Functions, 0 in Standard Position
Let e be an angle in standard position and let
P(x, y) be a point on the terminal side of 8. Using
the Pythagorean Theorem, the distance r from the
origin to P is given by V x2 + y2. The trigonometric
functions of an angle in standard position may be
defined as follows.
r=
sin =
COS A =
0 = ☆
tan =
X+ 0
CSC 0 = $. y 0
sec 0 = 5.x #0 cot 9 = v=0