Transcript Chapter 5 Ppt

Chapter 5
The Trigonometric Functions
Gettin’ Triggy wit’ it!
Section 5.1 – Angles and Degree Measure
Learning Targets:
 I can find the number of degrees in a given number of
rotations.
 I can identify angles that are co-terminal with a given
angle.
 I can identify a reference angle for a given angle
measure.
Anatomy of an Angle
Initial Side
Terminal Side
Vertex
Standard Position
Quadrantal angle
Example 1:
Give the angle measure represented by each rotation
listed below:
a) 5.5 rotations clockwise:
5.5 x -360
-1980
b) 3.3 rotations counterclockwise:
3.3 x 360
1180
Definition
Two angles in standard position are called coterminal
angles if they have the same terminal side.
Draw a pair of coterminal angles below.
48 degrees
408 degrees
Example 2:
Identify all angles that are coterminal with each angle.
Then, find one positive angle and one negative angle
that are coterminal with the angle.
a) 45o 405o
-315o
b) 225o
585o
-135o
Example 3:
If each angle is in standard position, determine a
coterminal angle that is between 0o and 360o. State
the quadrant in which the terminal side lies.
775 - 360 - 360
55o
a) 775o
b) -1297o
-1297 + 360 + 360 + 360 + 360
143o
Definition
If a is a nonquadrantal angle in standard position, its
reference angle is defined as the acute angle formed by
the terminal side of the given angle and the x-axis.
Example 4:
Find the measure of the reference angle for each angle.
a) 120o 60o
b) -135o
45o
Warm up!
1. Find 9.5 rotations clockwise.
9.5 x -360
-3420
2. Find all angles coterminal to 86o. Then find one pos and neg.
86 + or - 360n
446, -274
3. If the angle -777o is in standard position, determine a coterminal
angle that is between 0o and 360o. -777/360 = -2.15..
-777 + 720 = -57
303
4. Find the measure of the reference angle for each angle below:
a) 312o 48
b) -195o
15
5. What is the difference between coterminal angles and
reference angles? CO-TERMINAL (Same ending position)
“reference” can be used to simplify problems.
5.2 – Trig Functions
Objectives:
 Find the values of trigonometric ratios for acute
angles of right triangles.
Anatomy of a Triangle
Example 1:
20
12
27 3 3
5
sin  
13
12
cos  
13
5
tan  
12
13
csc  
5
13
sec  
12
12
cot  
5
sin  
4
5
csc  
5
4
cos  
3
5
sec  
5
3
tan  
4
3
cot  
3
4
33 3
sinsin   6
2
22 3
csc
csc 
33
3
cos  1
cos
26
sec  2
3 3
tan   3
3
1 3
cot
  
cot
33
Example 2:
If cos x = ¾, find sec x.
4
3
If sc x = 1.345, find sin x.
b. If csc x = 1.345, find sin x.
Start here with 2nd and 3rd hour
C  2r
r 1
parts 330
x
:

whole 360 2
330(2 )
x
360
330( )
x
180


180
2
parts 3
x
:

whole 2 360
2
(360)
3
x
2
2
(180)
3
x


180

30-60-90 Triangle:
3060
Pythagorean Thm
x 2 b 2  (2 x) 2
x b  4x
2
2
x???
3
2
b 2  3x 2
b  3x
2x
2x
60
x
2
bx 3
60
90
2x
45-45-90 Triangle:
x
90
Pythagorean Thm
90
x2  x2 c2
x
x
???
x 2
2x2  c2
2x2  c
90
90
x
x 2 c
Example 3:
Find x and y using the rules from the special
right triangles above.
30
x  4.5
x
60°
2x
x 3
9
60
4.5 3
y
x
90
Example 4:
Find x and y using the rules from the special
right triangles above.
30
2x
1.5 x
60°
3
2x
x 3
30°
y
1.5 3
x 3
1.5 3
60
x
90
Warm-up
Find x in the triangle below
Hi.
30-60-90 Triangle:
3060
Pythagorean Thm
x 2 b 2  (2 x) 2
x b  4x
2
2
x???
3
2
b 2  3x 2
b  3x
2x
2x
60
x
2
bx 3
60
90
2x
45-45-90 Triangle:
x
90
Pythagorean Thm
90
x2  x2 c2
x
x
???
x 2
2x2  c2
2x2  c
90
90
x
x 2 c
Unit Circle 
NOTE: stop for radians until after 6.1 mini lesson
Section 6.1 (mini-lesson)
Example 1:
a. Convert 330o to radian measure.
315 x

180 
180x  315
7
x
4
trick 
180

2
b. Change 3 radians to degree measure.
2 180

3

60 o
NOTE: stop for radians until after 6.1 mini lesson
5.3 – Trig and the Unit Circle
Objectives:
 I can find the values of the six trig functions using the
unit circle.
 I can find the values of the six trig functions of an
angle in standard position given a point on its
terminal side.
Example 1:
 Using the unit circle, find the six trigonometric values
for a
(cos, sin)
135 angle
2
2
sin x = _______
csc x = ______
2
2
cos x = _______
sec x = ______

1
tan x = _______
2
(
2 2
,
)
2 2
 2
1
cot x = ______
45 0
6.1 (Mini-Lesson)
1/ 2
3
2

2
2
3

2
Warm- Up
Find the sin, cos and tan of  5
6
(no need to write…. Just think)
Tuesday – Watch video and do
few examples below
http://www.youtube.com/watch?v=X1E7I7_
r3Cw
Warm-Up
Suppose x is an angle in standard position whose
terminal side lies in Quadrant IV. If cos x =.5 , find the
values of the remaining five trig functions of x. (Hint:
Sketch a picture).
Unit Circle time!!!!!!
5.4 – Word Problems and Trig
Objectives:
 Use trigonometry to find the measures of the sides of
right triangles.
Example 1:
The circus has arrived and the roustabouts must put up the main tent in a
field near town. A tab is located on the side of the tent 40 feet above the
ground. A rope is tied to the tent at this point and then the rope is placed
around a stake on the ground.
a. If the angle that a rope makes with the level ground is 53, how long is the
rope?
Picture:
Work:
b. What is the distance between the bottom of the tent and the stake?
Example 2:
A regular pentagon is inscribed in a circle with
diameter 8.34 centimeters. The apothem of a
regular polygon is the measure of a line
segment from the center of the polygon to the
midpoint of one of its sides. Find the apothem
for the pentagon.
Definition: An angle of elevation is the angle
between a horizontal line and the line of sight
from an observer to an object at a higher level.
Definition: An angle of depression is the angle
between a horizontal line and the line of sight
from the observer to an object at a lower level.
Example 3: (skip it)
On May 18, 1980, Mt. Saint Helens erupted with such
force that the top of the mountain was blown off. To
determine the new height at the summit, a surveyor
measured the angle of elevation to the top of the
volcano to be 37. The surveyor then moved 1000 feet
closer to the volcano and measured the angle of
elevation to be 40. Determine the new height of Mt.
Saint Helens.
Picture:
Work:
YOU TRY IT!!
The chair lift at a ski resort rises at an angle of 20.75 and
attains a vertical height of 1200 feet.
How far does the chair lift travel up the side of the
mountain?
You try it…answers…..
Quiz tomorrow
Warm-up
 The Ponce de Leon lighthouse in St. Augustine, FL, is
the second tallest brick tower in the U.S. It was built
in 1887 and rises 175 feet above sea level. How far
from the shore is a motorboat if the angle of
depression from the top of the lighthouse is 13
degrees?
Ad Math students – please read
 Mama ain’t happy.
 It is clear to me that you have not been been
checking on Moodle. The answers are numbered
differently, but they are there.
 Do not blame this on the snow days.
 Many of you did well on the quiz. Many of you did
not.
 I am disappointed.
 Come in. Grab your quiz. And immediately start
figuring out what you did wrong.
 DO.NOT. Do anything else. Period.
5.5 – Inverse Functions
Objectives:
 Evaluate inverse trig functions.
 Find missing angle measurements.
 Solve right triangles.
Inverse trig functions
Discussion:
in x = ½ can be solved by using the Arcsin function:
arcsin ½ = x which is read “x is the angle whose sine is ½ .
How many solutions does arcsin ½ = x have?
Similarly, there is an arccosine and arctangent function (arcos
and arctan). We use these functions to ________.
Example 1:
Solve each equation:
 a.
3
sin x 
2
x  60 and 120
o
o
b.
 2
cos x 
2
x  135o and 225o
c.
3
tan x 
3
x  30o and 210o
Example 2:
Evaluate each expression. Assume
that all angles are in Quadrant I.
a. tan  tan 116 
b. cos arcsin 2 
1

6
11


3
5
3
Example 3:
If f = 17 and d = 32, find E when D is a right angle.
Example 4: (skip)
A security light is being installed outside a loading dock. The light is
mounted 20 feet above the ground. The light must be placed at an angle
so that it will illuminate the end of the parking lot. If the end of the
parking lot is 100 feet from the loading dock, what should be the angle of
depression of the light?
Example 5:
Solve the triangle (find all missing sides and
angles): A = 33, b = 5.8, and C = 90
a = 23, c = 45, and C = 90o
Warm-up
Solve tan(cos-1 3 ) if theta is in the 4th Quad
5
cos(cos-1
2
) if theta is in the 1st quad
11
sin(tan-1 
7
) if theta is in the second quad
3
5.6
Objectives:
 Solve triangles using the Law of Sines
 Find the area of a triangle if the measures of two
sides and the included angle or the measures of two
angles and a side are given.
Law of Sines: The Law of Sines can
be used to solve triangles that are
NOT right triangles.
sin A sin B sin C


a
b
c
When do you use Law of Sines? ASA and AAS triangles!
Example 1:
Solve the triangle when A = 33, B
= 105, and b = 37.9.
Example 2:
A baseball fan is sitting directly behind home plate in the last row of the
upper deck of US Cellular Field in Chicago. The angle of depression to home
plate is 29.75o and the angle of depression to the pitcher’s mound is 24.25o.
In major league baseball, the distance between home plate and the pitcher’s
mound is 60.5 feet. How far is the fan from home plate?
Area of a Triangle
The area of a triangle is ½ bh.
By manipulating the formula using the Law of Sines, we get a new area
formula:
Area = ½ bc sin A
Example 3:
Find the area of the triangle with a = 4.7,
c = 12.4, and B = 47.
Warm up
The center of the Pentagon in Arlington, Virginia,
is a courtyard in the shape of a regular
pentagon. The courtyard could be inscribed in a
circle with radius of 300 ft. Find the area of the
courtyard. (WORK TOGETHER to break this
apart!!!!)
 Typo on Area problem #6 and #7
5.8
Objectives:
 Solve triangles using the Law of Cosines
 Find the area of a triangle if the measures of three
sides are given.
Law of Cosines: The Law of Cosines
can be used to solve triangles that
are NOT right triangles.
a 2  b 2  c 2  2bc cos A
When do you use Law of Cosines? SSS and SAS triangles!
Example 1:
Solve the triangle when
A = 120, b = 9, and c = 5.
Example 2:
Solve the triangle when a = 24,
b = 40 and c = 18.
Example 3:
For a right-handed golfer, a slice is a shot that curves to the right of its
intended path, and a hook curves off to the left. Suppose a golfer hits the
ball from the seventh tee at the US Women’s Open and the shot is a 160
yard slice 4o from the path straight to the cup. If the tee is 177 yards from the
cup, how far does the ball lie from the cup?
Example 4:
Using the formula: Area = ½ bc sin A, find the area of the triangle if d = 4, e = 7,
and c = 9.
Section 6.1 (mini-lesson)
Example 1:
a. Convert 315o to radian measure.
2
b. Change 3 radians to degree
measure.