unit 7 trigonometric functions mid checkpoint apply, algebra homework help

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Unit 7: Mid Checkpoint Apply
Lessons
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– Unit 7: Mid Checkpoint Apply
Unit 7: Trigonometric Functions Mid Checkpoint Apply
Interactive Tools
Unit 7: Trigonometric Functions Mid Checkpoint Apply
Lessons 1-3
Algebra 2
Please complete the following questions. It is important that you show all work you did to solve the problems when you submit
your work. This includes any calculations, diagrams, or graphs that helped you solve it.
1. OPEN ENDED Draw and label an example of an angle with negative measure in standard position. Then find an angle with
positive measure that is coterminal with this angle.
2. Writing in Math Use the information on page 779 to explain how trigonometry is used in building construction. Include an
explanation as to why the ratio of vertical rise to horizontal run on an entrance ramp is the tangent of the angle the ramp makes
with the horizontal.
3. Writing in Math Use the information on page 796 to explain how you can model the position of riders on a skycoaster.
Your Answer …
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American
High School
100
778 – 779 / 1100
ACTIVITY
The legs of a 45°-45-90″ triangle, 2 and b, are equal in measure. Use a
spreadsheet to investigate the dimensions of 450-450-90 triangles. What
pattems do you observe in the ratios of the side measures of these triangles?
-SQRT(AZ2+82-2)-B2/A2 -B2/C2 /
C-A2/C2]
25
45
1
C
Main Ideas
. Find valus
trigonomencions
fur skues
. Sehr problems
inhing right ringles
New Vocabulary
Donnery
trigonometric lunctions
functions
sine
cosine
tangent
angit
GET READY for the Lesson
The Americans with Disabilities
Act (ADA) provides regulations
Q D-0 @
designed to make public buildings
accessible to all. Under this act,
the slope of an entrance ramp
designed for those with mobility
disabilities
must not exceed a ratio
of 1 to 12. This means that for
12
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 1
45-45-90 Triangles
OX
OAB с D ΕΙ
A
16DY s.
Dica
2 1 1 1.414213581 0.70710678 0.70710678
3 2 2 282842712 1 070710678 0.70710678
4 3 3 1.24264089 1 0.7071 0.7071067
5 4 4 5.66885425 1 0.70710678 0.70710674
6 SS 2.07106791 10.70710678 0.70710678
11
HOLM Sheet1 Sheet2.Sht:
III
seca
cotangent
sahe a right triangle
angle af den
angle of depression
les
Reading Math
Trigonometry
The word iniganamEyE
derived from Greek
wond-g meaning
triangle andmete meaning
measurement
Trigonometric Values The tongent 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 tulemuse ԱրմՖllu
Consider right triangle ABC in which the
measure of acute angle A is identified by the
Greek letter theter, The sides of the triangle adating
are the hypotermee, the leg opposite e, and the
leg adjacent to
Using these sides, you can define six trigonometric functions: sine,
cosine, tangent, cosecant, secant, and cotangent. These functions are
abbreviated sin, cos, tan,,, and conspectively.
C
*
The spreadsheet shows the formula that will calculate the length of side c. The
formula uses the Pythagorean Theorem in the form = VA+ Since 150-45-90
triangles share the same angle measures, the triangles listed in the spreadsheet are all
similar triangles. Notice that all of the ratio of side to side e are 1. All of the ratios
of side to side cand of side to side care approximately 0.71.
MODEL AND ANALYZE
For Exercises 1-3, use the spreadsheet for 30-60-90 triangles.
If the measure of one kg of a right triangle and the measure of the
hypotenuse are in a ratio of 1 to 2, then the acute angles of the
triangle measure and 60
30-60-90 Triangles
06 X
OA
C
1 a
21
2
3 2
8
54
65
7
NON Sheet Sheet2 Sheet
D
B
b
E
bic
F
ac
הה
43
1 4
8
B
10
sin – Opp
COS-ed
KEY CONCEPT
Trigonometric Functions
if is the measure of an acute angle of a right triangle, opp is the measure
of the leg opposite , adj is the measure of the leg adjacent to 0, and hyp is
the measure of the hypotenuse, then the following are true.
tan – Opp
hyp
hyp
adj
hyp
SC ) = bor
hyp
ad
co
оро
Notice that the sine, cosine, and tangent functions are reciprocals of the
cosecant, secant, and cotangent functions, respectively. Thus, the following
are also true
CSC
sec –
cot –
1
tan
sece adi
1
1. Copy and complete the spreadsheet above
2. Describe the relationship among the 30-60-90 triangles with the dimensions given
3. What patterns do you observe in the ratios of the side measures of these triangles
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UI UUMI nnyius
796 – 797 / 1100
the ratios
American
High School
lan –
100
Main Ideas
– Find values of
trigonometric functie
formal angles
– Use referenceanges
hond values of
trigonometric
functions.
.)
– or -13
esce –
sec8 -!
cot 8 –
13
– OE 12
01-12
GET READY for the Lesson
A skycoaster consists of a large anh
from which two steel cables hang
and are attached to riders suited
together in a harness. A third cables,
coming from a larger tower behind
the arch, is attached with a ricord.
Riders are hosted to the top of the
larger tower, pull the ripoord, 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 of
the riders from the center of the arch is given by 8 = 0.2 cos (1.61).
where is the time in seconds after leaving the bottom of their swing
New Vocabulary
quadrant de
referente angle
CHECK Your Progress
1. Find the exact values of the six trigonometric functions of if the
terminal side of contains the point (-8.-15).
If the terminal side of angle Blies on one of the axes, is called a quadrantal
angle. The quadrantal angles are 0,90,180″, and 27r. Notice that for these
angles either or y is equal to 0. Since division by zero is undefined, two of
the trigonometric values are undefined for each quadrantal angle.
KEY CONCEPT
0-ror Oradians 0-90 or radians
Quadrantal Angles
0 – 180° or radians 0-270″ or radians
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 ! > in the
quation 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.
(-200
0-11
KEY CONCEPT Trigonometric Functions, in Standard Position
Leto be an angle in standard position and let
Pixy) be a point on the terminal side of e. Using
the Pythagorean Theorem, the distance from the
origin to P is given by r= v*+y. The trigonometric
functions of an angle in standard position may be
defined as follows.
sin
COS 0 – tan 6 = +0
cster+0 sec = *#06# = x +0
EXAMPLE Quadrantal Angles
Find the values of the six trigonometric functions for
an angle in standard position that measures 270°.
When – 270,- andy-
sin –
COS) – 2
tan –
-To-1 – or
or indefined
csc 8
sec
coti-
0
-7 or 0
= -1
= or undefined
EXAMPLE Evaluate Trigonometric Functions for a Given Point
O Find the exact values of the six trigonometric
functions of if the terminal side of contains the
point 65,- 12).
From the coordinates, you know that r – 5 and
y=-12. Use the lythagorean Theorem to find r.
-12
CHECK Your Progress
2. Find the values of the six trigonometric functions for an angle in
standard position that measures 150
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