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HONORS PRE-CALCULUS - CHAPTER 4 ASSIGNMENTS/SCHEDULE
SECTION
ASSIGNMENT
1.) ________
DATE
4.3
p. 308, #1-15 odds, 43-45 all, 59-64 all
2.) ________
4.1
p. 290, #4-76 multiples of 4
3.) ________
4.2
p. 299, # 1-35 odds and #43-51 odds
4.) ________
4.2
p. 299 #37-41 all
5.) ________
4.1-4.2
Review Packet
6.) ________
4.1-4.2
Quiz
7.) ________
4.3
p. 309, #33-42 all
8.) ________
4.3
p. 310, #65-70 all
9.) ________
4.4
10.) ________
4.4
p. 319, # 59-64 all, 81-86 all, 88, 92
11.) ________
Review
Worksheet 4.3-4.4
12.) ________
4.3-4.4
Quiz
13.) ________
4.5
p. 328, #1-14 all, 27 g(x), 28 all, 29 f(x), 30 f(x), 32 f(x)
14.) ________
4.5
p. 328, # 35-49 odds
15.) ________
4.5
Worksheet
16.) ________
Quiz 4.5
Quiz
17.) ________
4.6
p. 339, # 7-10 all, 19, 20, 23, 24, 30
18.) ________
4.6
p. 339, # 12, 13, 15, 18, 28, 29
19.) ________
4.6
Worksheet
20.) ________
Quiz 4.6
Quiz
21.) ________
4.7
p. 349, # 1-34 all (read section as well)
22.) ________
4.7
p. 349, # 37-68 all
23.) ________
4.8
p. 359, # 1-13 odds, 15-35 odds (read section)
24.) ________
Review
Problems tba
25.) ________
Need extra practice?
Log on to
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TEST CHAPTER 4 (2 day test)
Honors Pre–Calculus – Chapter 4
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1
4.3 Right Triangle Trigonometry
SOHCAHTOA
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
▪ Evaluate the 6 trig functions for  in the following triangles. (Hint: find all sides first)
3
θ
4
Honors Pre–Calculus – Chapter 4
2
▪ Evaluate the 6 trig functions for  in the following triangle.
5

13
30o  60o  90o Triangle
ll  sl 3
hyp  2gsl
▪ Find all sides of a 30o  60o  90o triangle if the ll = 10.
45o  45o  90o
hyp  l 2
▪ find all sides of an isosceles right triangle if the leg is 4 5
Honors Pre–Calculus – Chapter 4
3
▪ Given cos  
5
find the remaining trig functions of 
7
▪ Use a calculator to evaluate
a.
b. cot 57.2
sin12
▪ The angle of elevation of to the top of the Empire State Building in New York is found to be 11o from the ground at a
distance of 1 mile from the base of the building. Use this information to find the height of the Empire State Building.
Honors Pre–Calculus – Chapter 4
4
4.1 Radian and Degree Measure
Trigonometry
→derived from the Greek language “measurement of triangles”
→Initially dealt with relationships among the side and angles of triangles – used in development of astronomy,
navigation and surveying.
→With the development of Calculus – viewed as the classic trig relationships as functions with the set of real
numbers
→trig expanded to rotations, vibrations, sound waves, light rays, planetary orbits, vibrating strings,
pendulums and orbits of atomic particles.
Angle→ determined by rotating a ray about its endpoint.
→initial side – starting position of the ray
→terminal side – position of the ray after rotation
→vertex – endpoint of the ray
⬄Standard Position
→positive angles (counterclockwise)
Honors Pre–Calculus – Chapter 4
→negative angles (clockwise)
5
→coterminal angles
Radian Measure
Measure of an angle → determined by the amount of rotation from the initial side to the terminal side
→degrees or radians
→central angle
Radian → measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle
What does a radian look like?
Arc length (s) = radius (r) when θ = 1 radian
Because C  2 r , it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc
length of s  2 r , so there are 2π ≈ 6.28 radius lengths in a full circle

s (arc length)
r (radius )
⬄Using the fact that one full revolution is 2π, find radian measures for each of the following:
→ ½ revolution
→ ¼ revolution
→Use the circle to identify where the angles
1
revolution
3
   
, , , ,  , 2 fall on a circle
6 4 3 2
→Which angles are acute?
Honors Pre–Calculus – Chapter 4
→
→Which angles are obtuse?
6
Coterminal angles → angles with the same initial and terminal sides.
→ Add or subtract 2π to find coterminal angles
⬄Find a coterminal angle for each of the following
▪
13
6
▪
Complementary Angles → angles whose sum is
3
4
▪ 

2
Supplementary Angles → angles whose sum is π
⬄Find the complementary and supplementary angles of
▪
2
5
▪
4
5
Conversions between degrees and radians
Degrees to Radians → multiply by
Radians to Degrees→ multiply by

180
180

WHEN NO UNIT OF MEASURE IS SPECIFIED, RADIAN MEASURE IS IMPLIED!!
Honors Pre–Calculus – Chapter 4
7
2
3
Convert from degrees to radians
▪ 135°
▪ 540°
▪ -270°
Convert from radians to degrees
▪ 

2
Honors Pre–Calculus – Chapter 4
▪
9
2
▪2
8

Conversion between decimal degrees and degrees, minutes, seconds D M ' S "
1'  1minute 
1
1
60

1"  1second 
1
1
3600

▪ Write the angle of 64 32' 47" in decimal form
Honors Pre–Calculus – Chapter 4
9

4.2 Trigonometric Functions: The Unit Circle
Unit Circle
x2  y 2  1
→Imagine the real number line wrapped around the circle, then:
Counterclockwise wrapping – positive numbers
Clockwise wrapping – negative numbers
→As the number line is wrapped around the circle, each point t corresponds to a point (x, y) on the circle

→
2
0→
π→
3
→
2
2π→
→t also corresponds to a central angle θ. Since the arc length formula is s = rθ (and r = 1, since it is a unit circle),
t=θ
Definitions of Trig Functions – Let t be a real number and let  x, y  be the point on the unit circle corresponding to t.
sin t  y
(sine)
csct 
1
y
(cosecant), y ≠ 0
cost  x
(cosine)
sec t 
1
x
(secant), x ≠ 0
(tangent)
cot t 
x
y
(contangent), y ≠ 0
tan t 
y
x
Honors Pre–Calculus – Chapter 4
10
Unit circle divided into 8 equal arcs
Unit circle divided into 12 equal arcs
** Fill these into the Unit Circle given.
⬄Evaluate the six trig functions at each real number. Remember the coordinates are written (cos, sin)
▪t=

6
Honors Pre–Calculus – Chapter 4
▪t= 
11
5
4
Graph each of the following angles
1.
4
3
4.

7.

13
3
2.

23
2
5.
12
3
20
3
8.

Honors Pre–Calculus – Chapter 4
35
6
12
3.
27
3
6.

9.
9
2
14
6
Domain and Period of Sine and Cosine
Recall the unit circle
Remember that the coordinates for each point is  cos ,sin   . You can determine the domain and range of each
function. The domain is all reals (because you can find the sin or cos of any real angle.
To determine the range, since 1  x  1 and  1  y  1 , then 1  sin   1 and  1  cos   1
If you add 2π (one full revolution) to each value of t, then:
sin  t  2   sin t
cos  t  2   cos t
FUNCTIONS THAT BEHAVE IN SUCH A REPETITIVE (OR CYCLICAL) MANNER ARE CALLED PERIODIC.
⬄Graph y  sin x and y  cos x on your calculator. Be sure the mode is set to “Radian”. Check that the domain and
range are consistent with what we determined.
***Conclusion – Sine and Cosine have a period of 2π
 13
 6
▪sin 

=

Honors Pre–Calculus – Chapter 4
 5 
=
 2 
▪sin  
13
 7
 3
▪cos 

=

Even and Odd Trig Functions
Even function - If f ( x)  f ( x) (symmetric to the x-axis - when you substitute positive or negative x, the
function (y) is the same)
→Cosine and Secant
Odd function – If f ( x)   f ( x) (symmetric to the origin - when you substitute –x, the function (y) is negative)
→Sine, Cosecant, Tangent, Cotangent
▪If sin( x) 
4
, find sin(  x) and csc( x)
5
▪If cos( x) 
3
, find cos(  x) and sec( x)
4
WHEN USING A CALCULATOR TO EVALUATE TRIG FUNCTIONS – BE SURE TO SET THE MODE TO “RADIANS”.
To evaluate cosecant, secant and cotangent, use x 1 with the respective reciprocal function.
 
 
1
 . Since csc is the reciprocal of sin. Type in sin   , enter, then x .
8
8
 
 
⬄One the calculator, evaluate csc 
Evaluate each on the calculator
 2 

 3 
▪ sec 
 4 

 3 
 11 

 6 
▪ csc  
 11 

 6 
▪ csc  
▪ cot 
Find the exact value
 2 

 3 
▪ sec 
Honors Pre–Calculus – Chapter 4
 4 

 3 
▪ cot 
14
4.3 Right Triangle Trigonometry
Look at the unit circle – derive the coordinates using the formulas for 30o  60o  90o and 45o  45o  90o . Remember
that the coordinates are  cos ,sin  
sin 30 
cos 30 
tan 30 
sin 60 
cos 60 
tan 60 
sin 45 
cos 45 
tan 45 
What relationships are notable?
Cofunctions
sin  & cos 
sec  & csc 
COFUNCTIONS OF COMPLEMENTARY ANGLES ARE EQUAL.
Honors Pre–Calculus – Chapter 4
15
tan  & cot 
Trigonometric Identities
Reciprocal Identities
1
csc 
1
csc  
sin 
sin  
1
sec 
1
sec  
cos 
cos  
1
cot 
1
cot  
tan 
tan  
Quotient Identities
tan  
sin 
cos 
cot  
cos 
sin 
Pythagorean Identities
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
WHERE DO THESE COME FROM??????
Quotient Identities – SOHCAHTOA
Pythagorean Identities - opp 2  adj 2  hyp 2
1. Divide by hyp 2
Honors Pre–Calculus – Chapter 4
2. Divide sin 2   cos 2   1 by sin 2 
16
3. Divide sin 2   cos 2   1 by cos 2 
Ex. Use trigonometric identities to transform one side of the equation into the other
1. sin  sec   tan 
2. tan 2   sec 2   1
3. sin   cot  cos   csc 
4.
sec   cos 
 tan 
sin 
6.
cot  sec 
1
csc 
5.
sin  sec 
1
tan 
Honors Pre–Calculus – Chapter 4
17
▪ Let θ be an acute angle such that sin   .6 Use the trig identities to find
▪ cos
▪ tan 
▪ Let θ be an acute angle such that tan   3 Use the trig identities to find
▪ cot 
▪ sec
Solving Right Triangles – Process of finding all of the sides and angles of a right triangle.
▪ A 96 ft. tree casts a shadow that is 120 ft long. What is the angle of elevation of the sun?
▪ A person is 200 yards from a river. Rather than walking directly to the river, the person walks 400 yards along a
straight path to the river’s edge. Find the acute angle θ between this path and the river’s edge
Honors Pre–Calculus – Chapter 4
18
▪ Solve the triangle
4 10
4
Honors Pre–Calculus – Chapter 4
19
4.4 Trigonometric Functions of Any Angle
In the previous section, the definitions of trig functions were restricted to acute angles.
Definitions of Trigonometric Functions of Any Angle
Let θ be an angle in standard position with  x, y  a point on the terminal side of θ and r 
y
r
y
tan  
x
r
sec  
x
sin  
x
r
x
cot  
y
r
csc  
y
cos  
▪ Let  3, 4  be a point on the terminal side of θ. Find sin  , cos  and tan 
Signs of sin  , cos  and tan  . (REMEMBER: the coordinate is  cos ,sin  
sin  
sin  
cos  
cos  
tan  
tan  
sin  
sin  
cos  
cos  
tan  
tan  
Honors Pre–Calculus – Chapter 4
20
x2  y 2  0
▪ Given tan   
5
and cos  0 , find sin  and cos
4
▪ Given csc  4 and cot   0 , find the six trig functions
Reference Angles – corresponding acute angles for which values of trig functions of angles greater than 90° (or less than
0°) can be determined.
→ Let θ be an angle in standard position. The reference angle is the acute angle  ' formed by the terminal side
of θ and the horizontal axis.
▪ Find the reference angle θ’
▪ θ=300°
Honors Pre–Calculus – Chapter 4
▪ θ=2.3
21
▪ θ=-135
▪ Identify the quadrant for each angle and find the reference angle θ’
5
6

5
6
15
4
23
3

25
6
5
4
37
3

38
3

AN ANGLE AND IT’S REFERENCE ANGLE ALWAYS HAVE THE SAME SIN, COS, TAN, CSC, SEC AND COT, EXCEPT POSSIBLY
IN SIGN – WHICH CAN BE DETERMINED BY THE QUADRANT IN WHICH θ LIES.
Evaluating Trig Functions of Any Angle
1. Determine the function value for the associated reference angle θ’
2. Depending on the quadrant in which θ lies, affix the appropriate sign to the function value.
▪ Evaluate each trig function
▪ cos
4
3
Honors Pre–Calculus – Chapter 4

▪ tan 210

22
▪ csc
11
4
▪ Let θ be an angle in Quadrant II such that sin  

▪ Use a calculator to evaluate cot 410
1
, find cos and tan 
3

▪ Use a calculator to evaluate sin  4 
▪ Use a calculator to solve tan   4.812
Find 
▪ sin   
2
2
Honors Pre–Calculus – Chapter 4
▪ cot   0
▪ csc  2
23
4.5 Graphs of Sine and Cosine Functions
Graph y  sin x , using the coordinates from the unit circle.
▪ What is the range? ____________________
▪ The graph begins to repeat after how many radians? Period is ____________
▪ X-intercepts? __________________________________
▪ Maximum and Minimum points? _____________________________
**KEY POINTS – INTERCEPTS, MAXIMUM AND MINIMUM POINTS
Graph y  cos x , using the coordinates from the unit circle.
▪ What is the range? ____________________
▪ The graph begins to repeat after how many radians? Period is ____________
▪ Key Points

Honors Pre–Calculus – Chapter 4
,

,

,
24

,

,

Amplitude and Period
Amplitude – half the distance between the maximum and minimum values of the function. For y  a sin x and
y  a cos x the amplitude is a
Period – Radians (or degrees) it takes for a function to repeat. The period of y  a sin bx and y  a cos bx is
⬄ Find the amplitude and period of the following:
▪ y  sin
x
2
▪ y  cos 2 x
▪ y  3sin 2 x
Honors Pre–Calculus – Chapter 4
▪ y  2 cos 3 x
25
2
b
Transformations and Translations of Sine and Cosine - using y  a sin  bx  c   d and y  a cos  bx  c   d
Amplitude → a→ scaling factor (vertically)→ range of the function is a  y  a
→ If a  1 , curve has a vertical stretch(skinny)
▪ y  3sin x
→ If a  1 , curve has a vertical shrink (fat)
▪ y
1
cos x
3
→if a is negative – graph reflects on the x - axis
▪ y   sin x
Period →
2
→ measure of how long before the function repeats itself → affects the horizontal factor
b
→If 0  b  1 , period is greater than 2π (horizontal stretch)
▪ y  sin
x
2
→If b > 1, period is less than 2π (horizontal shrink)
▪ y  cos 3x
→If b is negative, use the identity sin   x    sin x (because it is odd) and cos   x   cos x (because
it is even)
Honors Pre–Calculus – Chapter 4
26
Translations →Horizontal Shift – Using y  a sin  bx  c   d and y  a cos  bx  c   d the graph completes one cycle
from bx  c  0 and bx  c  2 . (NOTE: bx  c represents an angle). Therefore the interval for one cycle is
To find left endpoint for one cycle of the graph set bx  c  0
To find the right endpoint for one cycle of the graph set bx  c  2
NOTE: Phase Shift- solve bx  c  0 for x. If x is positive, phase shift is right. If x is negative, phase shift is left.
Cycle Interval – represents the left and right endpoint for one full revolution of the trig function without the vertical
shift
Translations → Vertical Shift
d > 0 → graph shifts d units up
d < 0 → graph shifts d units down
To Graph Any Sine or Cosine Function
1. Find the Amplitude: a
2. Find the Period:
2
b
3. Find the cycle interval: 0  bx  c  2
4. Find the x coordinates of the key points: add
period
starting with the left interval of the cycle interval
4
5. Find the y-coordinates: multiply the original y value of the key points by the amplitude and add the vertical
shift.
Note: original y values for sine are 0, 1, 0, -1, 0
original y values for cosine are 1, 0, -1, 0, 1
⬄Sketch y 
1

sin( x  )
2
3
Amplitude ______
Key Points

,
Honors Pre–Calculus – Chapter 4
Period _________

,

Cycle Interval ____________
,
27

,

,

⬄Sketch y  3cos(2 x  4 )
Amplitude ______
Key Points

,
Period _________

,

Cycle Interval ____________
,

,

,

⬄Graph y  2  3cos 2 x
Amplitude ______
Key Points

,
Honors Pre–Calculus – Chapter 4
Period _________

,

Cycle Interval ____________
,
28

,

,

4.6 GRAPHS OF OTHER TRIG FUNCTIONS
⇨Graph y  tan x
Where is tangent = 0?
Where is tangent undefined?
What are the values in between?
Period ________
Cycle Interval _____________ Asymptote ___________________
TO GRAPH TANGENT y  a tan(bx  c)  d
1. Find the period ⇨

b
2. Find the asymptote ⇨ solve 

2
 angle 

2
▪ In general, add n  period to an asymptote
3. Determine the direction of the tangent graph
▪ if a is positive – graph goes down on the left and up on the right
▪ if a is negative – graph goes up on the left and down on the right\
4. The x- intercept is the midpoint of the two asymptotes. If there is a vertical shift, move the midpoint up or down
accordingly.
5. Use the a value to find the other 2 points on the branch of the graph
6. Graph at least two more branches, identify the general asymptote and at least two more asymptotes on the
graph
▪ Graph y  tan
x
2
Period ________
Honors Pre–Calculus – Chapter 4
Cycle Interval _____________ Asymptote ___________________
29
▪ Graph y  3 tan 2 x
Period ________
Cycle Interval _____________ Asymptote ___________________


▪ Graph y  2 tan  x 

 1
4
Period ________
Honors Pre–Calculus – Chapter 4
Cycle Interval _____________ Asymptote ___________________
30
Graph y  cot x
Where is cotangent = 0?
Where is cotangent undefined?
What are the values in between?
Period ________
Cycle Interval _____________ Asymptote ___________________
TO GRAPH COTANGET y  a cot(bx  c)  d
1. Find the period ⇨

b
2. Find the asymptote ⇨ solve 0  angle  
▪In general, add n  period to an asymptote
3. Determine the direction of the cotangent graph
▪ if a is positive – graph goes up on the left and down on the right
▪ if a is negative – graph goes down on the left and up on the right
4. The x- intercept is the midpoint of the two asymptotes. If there is a vertical shift, move the midpoint up or down
accordingly.
5. Use the a value to find the other 2 points on the branch of the graph
6. Graph at least two more branches, identify the general asymptote and at least two more asymptotes on the graph
▪ Graph y  2 cot
x
3
Period ________
Honors Pre–Calculus – Chapter 4
Cycle Interval _____________ Asymptote ___________________
31


▪ Graph y  3cot  2 x 

2
2
Period ________
Cycle Interval _____________ Asymptote ___________________


▪ Graph y  2 cot  x 

3
4
Period ________
Honors Pre–Calculus – Chapter 4
Cycle Interval _____________ Asymptote ___________________
32
Graphs of the Reciprocal Functions
csc x 
1
sin x
sec x 
1
cos x
→What values will each function be undefined? What does that indicate (graphically)?
Steps to sketch a graph of the secant or cosecant function:
1. Sketch the reciprocal function
2. The x intercepts (before the vertical shift) of sine and cosine become vertical asymptotes
3. The maximum point on the sin/ cos curve is the minimum point for the csc/ sec curve. The
minimum point on the sin/ cos curve is the maximum point on the csc/ sec curve
4. Draw at least two periods of the curve using the asymptotes and the minimum or maximum point
▪ Graph y  2 csc( x 

4
)
For the reciprocal function:
Amplitude ______
Key Points

Period _________
,

,
Cycle Interval ____________

,

,

For the given function:
Asymptotes_________________
Honors Pre–Calculus – Chapter 4
Minimum/Maximum Coordinates______________
33
,

▪ Graph y  sec 2 x
For the reciprocal function:
Amplitude ______
Key Points

Period _________

,
,
Cycle Interval ____________

,

,

,

,

For the given function:
Asymptotes_________________
▪ Graph y  sec( x 

4
Minimum/Maximum Coordinates______________
)
For the reciprocal function:
Amplitude ______
Key Points

Period _________
,

,
Cycle Interval ____________

,

,

For the given function:
Asymptotes_________________
Honors Pre–Calculus – Chapter 4
Minimum/Maximum Coordinates______________
34
4.7 Inverse Trig Functions
Recall the graph of y  sin x
→Does it pass the Horizontal Line test? What does this indicate?
→Suppose we restrict the domain to the interval 

2
x

2
. Does it have an inverse?
This is called the Inverse Sine Function → y  arcsin x or y  sin 1 x .
The arcsine of x is the angle whose sine is x
Graph of y  sin 1 x (Interchange the x and y coordinates of y  sin x )
Domain: -1 ≤ x ≤ 1 ,  1,1
Range: 

2
 y
   
 ,
2 ,  2 2 
⬄Evaluate the Inverse Sine Function (remember: replace “arcsin” or “ sin 1 with “the angle whose sine is”)
1
2
→ arcsin( )
Honors Pre–Calculus – Chapter 4
→ sin 1
3
2
→ sin 1 2
35
Recall the graph of y  cos x
→Does it pass the Horizontal Line test? What does this indicate?
Suppose we restrict the domain to the interval 0  x   . Does it have an inverse?
This is called the Inverse Cosine Function → y  arccos x or y  cos1 x .
The arccosine of x or cos 1 x is the angle whose cosine is x
Graph of y  cos1 x
(Interchange the x and y coordinates of y  cos x )
Domain: -1 ≤ x ≤ 1 ,  1,1
Range: 0  y  
,
0,  
Evaluate. (Remember: replace “arccos” or “ cos 1 with “the angle whose cosine is”)
1
2
 arccos  
Honors Pre–Calculus – Chapter 4



 arccos  
2

2 
 cos1 1
36
 3
 2
 cos 1   
Recall the graph of y  tan x
Inverse Tangent Function can be defined by restricting the domain to ____________________________
Graph of y  tan 1 x
Domain: all real numbers,  ,  
  
, 
 2 2
Range:  
Summary of the Inverse Functions –THE VALUES OF INVERSE TRIG FUNCTIONS ARE ALWAYS IN RADIANS
Function
Domain
Range
y  sin 1 x iff sin y  x
1,1
  
  2 , 2 
y  cos1 x iff cos y  x
1,1
0,  
y  tan 1 x iff tan y  x
 ,  
  
 , 
 2 2
Honors Pre–Calculus – Chapter 4
37
⬄Find the exact values
▪ arccos
2
2
▪ arccos(1)
▪arctan 0
▪arctan(-1)
▪Use a calculator to approximate the value (if possible)
▪arctan(-8.45)
▪arcsin(.2447)
▪arccos(2)
Composition of Functions→ Recall that the composition of a function and it’s inverse = 1. Therefore:
Inverse Trig Properties
1) If 1  x  1 and 

2
 y

2
, then sin(arcsin x)  x and arcsin(sin y )  y
2) If 1  x  1 and 0  y   , then , then cos(arccos x)  x and arccos(cos y )  y
3) If 

2
 y

Honors Pre–Calculus – Chapter 4
2
, then tan(arctan x)  x and arctan(tan y )  y
38
⬄Find the exact value of each
▪ tan(arctan(5)) 


▪ cos cos 1  


2
3

 5 
 
 12  





3
2




 3 
 5 
▪ cos  arcsin     

▪ sin  arctan  
Honors Pre–Calculus – Chapter 4
5
3
▪ arcsin  sin
▪ tan  arccos  



▪ arcsin  sin
39
4.8 Applications and Models
Angle of Elevation → angle from the horizontal upward to an object
Angle of Depression → angle from the horizontal downward to an object
▪ A safety regulation states that the maximum angle of elevation for a recue ladder is 72°. If a fire department’s longest
ladder is 110 feet, what is the maximum safe rescue height?
▪ There is a building with a smokestack on top of it. At a point 200 feet from the base of a building, the angle of
elevation to the bottom of a smokestack is 35°, whereas the angle to the top is 53°. Find the height of the smokestack
alone.
Honors Pre–Calculus – Chapter 4
40
Bearing – In surveying and navigation, directions are generally given in terms of bearings. The bearing measures the
acute angle that a path or line of sight makes with a fixed north-south line.
Identify the bearing of each segment below
Ex. A ship leaves port at 11:00 am and has a bearing of S 38 W . It is sailing at 23 knots per hour.
a.
How many nautical miles south and how many nautical miles west will the ship have travelled by 4 pm?
b.
At 4 pm the ship changes course to due west. Find the ships bearings and distance from the port of departure
at 5 pm.
Honors Pre–Calculus – Chapter 4
41