Download 4.8 Solving Problems with Trigonometry Pre-Calculus

4.8 Solving Problems with Trigonometry
Pre-Calculus
Learning Targets
1. Set up and solve application problems involving right triangle trigonometry.
2. Use overlapping right triangles to solve word problems including the use of indirect measurement.
Next we will apply what we know about the trigonometric functions, and their inverses, to solve real world application
problems. The most important part of these types of questions is an accurate, detailed picture.
Angle of Elevation: The acute angle measured from a horizontal
line UP to an object.
Angle of Depression: The acute angle measured from a horizontal
line DOWN to an object.
Angle of Depression
Angle of Elevation
Example 1: From a boat on the lake, the angle of elevation to the top of a cliff is 20.7°. If the base of the cliff is 1394 feet
from the boat, how high is the cliff (to the nearest foot)?
Example 2: The bearings of two points on the shore from a boat are 115° and 123°. Assume the two points are 855 feet
apart. How far is the boat from the nearest point on shore if the shore is straight and runs north-south?
Unit 5–1
4.1 Angles and Their Measures
Pre-Calculus
Learning Targets
1. Draw an angle in Radians by thinking in radians (without converting to degrees).
2. List the reference angle of the radian angle.
3. Convert Degrees → Radians and vice versa.
4. Apply arc length formulas in Radians when given a radian angle or a degree angle.
5. Convert linear speed to angular speed & vice versa
6. Relate definitions of bearing and heading to real world problems.
7. Convert DMS to decimal degrees and vice versa
There are multiple ways to measure angles: degrees, revolutions, bearings and radians
 Degrees: From Geometry: straight angle = 180 degrees and a circle = 360 degrees
y
 Revolutions: 1 revolution = 1 full turn around a circle
Arc length = 1
Radius
 Radians: An angle of 1 radian is defined to be the
angle at the center of a circle which spans an arc of
length equal to the radius of that circle.
1 radian
x
Standard Position: For degree angles, revolutions and radian angles, we draw the angles with their initial side
along the positive x-axis of the coordinate axes. The terminal (end) side of the angle is then measured in a
counterclockwise direction. NOTE: Bearings work differently and will be covered in class.
Reference Angles: The acute angle between the terminal side of an angle and the x-axis.
Example 1: Draw the following Radian angles. Then list their reference angle.
a) 
2
c) 4
3
b) 11
6
Unit 5–2
d) 2.9
Example 2: Convert from degrees to radians or radians to degrees:
b) 7 radians
8
a) 200°
ArcLength: The length of a portion of the circumference of a circle is S = θ r where S is the arc length, θ is the
angle measured in radians, and r is the radius of the circle.
Example 3: Find the length of an arc. Express the answer in terms of π
.
a) θ =
5π
6
; r = 4 cm
b) θ = 125°; r = 1.5 mm
Angular and Linear Motion
 Angular Speed: How fast something is spinning.
 Linear Speed: How fast something is travelling in one direction.
Example 4: A turntable rotates at 50 revolutions per minute. What is its angular speed in radians per
second?
Unit 5–3
4.2 Trig. Functions of Acute Angles
Pre-Calculus
Learning Targets
1. Know and apply the six trigonometric ratios
2. Solve right triangles using the six trig. ratios
3. Know the ratios of the sides of the 30-60-90 special right triangle
4. Know the ratios of the sides of the 45-45-90 special right triangle
5. Apply the ratios of the special right triangles to real life application questions.
Next let’s put our angles inside triangles…specifically right triangles.
 Three Basic Trigonometric Ratios
side opposite θ
side adjacent to θ
side opposite θ
, cos θ
, tan θ
=
=
hypotenuse
hypotenuse
side adjacent to θ
sin θ
o Remember the ratios are used on an acute angle.
o
We memorized _______________.
 The Reciprocal Ratios: __________________, _________________ and ___________________
csc θ =
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
Example 1: Using the triangle at the right, find all six
trigonometric functions of the angle θ.
29
θ
21
Example 2: Given tan θ =
5
, find the remaining trigonometric functions.
12
Unit 5–4
Special Right Triangles:
A
#1: 45 – 45 – 90 Right Triangle (MEMORIZE THESE RATIOS)
a) If AC = 1, and m A  45 , solve for the remaining parts of the triangle.
B
C
b) Find the sine, cosine and tangent values of 45°.
A
#2: 30 -60 – 90 Right Triangle (MEMORIZE THESE RATIOS)
a)  ABC is equilateral, so each angle is ____________.
b) Draw the altitude of the triangle from A to BC .
Call the point of intersection D.
c) Therefore, m BAD  ________ .
d) Suppose AB = 2. Solve for the remaining parts of  BAD .
B
C
e) Find the sine, cosine and tangent values of 30° and 60°.
Example 4: A ladder is extended to reach the top floor of an 84 foot tall burning building. The fire fighters see
someone who needs rescuing in a window 8 feet below the roof. How far should the ladder be extended to
reach the window if the ladder must be placed at the optimum operating angle of 60°?
Unit 5–5
4.3 Trig. Extended: The Circular Functions
Pre-Calculus
Learning Targets
1. Graph Radian and Degree angles in standard position
2. Find reference angles for Radian and Degree angles in standard position
3. Identify positive and negative angles that are Coterminal with a given angle.
4. Find the exact value of the six trig. ratios of an angle in standard position
5. Find the exact value of the six trig. ratios of quadrantal angles
6. Find the exact value of the six trig. ratios of non-quadrantal angles.
Day 1
Day 2
DAY 1:
SohCahToa works well for acute angles; but what if we need angles 90° or larger?
What if we need negative angles?
Standard Position: By placing angles in standard position, we can extend the terminal sides past the first
quadrant. Remember each of the following facts…
 Positive Angles are measured counterclockwise.
 Negative Angles are measured clockwise.
 Reference Angles are still the acute angle measured to the x-axis (just as in lesson 4.1)
Coterminal Angles: Angles with the same initial and terminal sides, but different measures. The measures
differ by integral multiple(s) of 2π or 360° .
Example1: Find and draw one positive and one negative angle that is co terminal with the given angle.
a) –210°
b)
13
4
Unit 5–6
Trigonometric Functions – Redefined: Let θ be any angle in standard position and let P( x, y ) be any point on
the terminal side of the angle. Then r is the distance from the origin to P( x, y ) or the radius of a circle and
=
r
x 2 + y 2 . The six Trigonometric Ratios are…
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
→ Notice if r = 1, then we have a “Unit Circle” which is a circle with radius 1.
sin θ =
cos θ =
tan θ =
Example 2: Find the six trigonometric ratios of θ whose terminal side passes through the given point.
a) (3, 4)
b) (3, −4)
A Shortcut for the Signs of Trigonometric Functions:
Now we know the trig functions when given a point. We can also predict the _________ of the trig functions if
we know the ______________. But how do we find the trig functions when given an angle?
Unit 5–7
DAY 2:
Quadrantal Angles: Any angle in standard position whose terminal side is on the x-axis or the y-axis.
Example 3: Find the exact value of each given trig function using the given angle.
a) sin π
b) cos (–360°)
 3π 
c) csc  
 2 
 −11π 
d) sec 

 2 
Remember that you cannot divide by zero, so sometimes _________________ & ____________________ are
undefined!
Non – Quadrantal Angles: Any angle in standard position whose terminal side is NOT on the x-axis or
the y-axis. To find a trig. ratio for a non-quadrantal angle, we need three things:
1. ____________________: We identify the quadrant of the angle according to its terminal side. The
quadrant tells us the sign of the trigonometric ratio.
2. ____________________________: Reference triangles are right triangles created by drawing a
perpendicular from the terminal side of a non-quadrantal angle to the x-axis. The reference angles we
used in Lesson 4.1 are inside this right triangle and for our purposes here, the reference angle will
always be a multiple of the angles found in the special right triangles from Lesson 4.2.
3. ________________________________: Remember we memorized these ratios (or the sine, cosine and
tangent of these angles) in lesson 4.2.
Unit 5–8
Example 4: Draw the given angle and list the quadrant in which it lies. List the measure of θ ref . Then,
evaluate the indicated trig value.
a) csc ( −60° )
 7π 
b) cos 

 4 
 − 5π 
c) sin 

 6 
d) tan 120°
Now you have all the pieces to solve any puzzle…
Example 5: Find cos θ and cot θ if sin θ =
1
and tan θ < 0.
4
Unit 5–9
Graphing Trig. Functions: Tangents and Reciprocals
Pre-Calculus
Learning Targets
8. Graph the six parent trigonometric functions.
9. Apply transformations to the six parent trigonometric functions.
1

Warm Up: Graph
=
y 2sin  ( x + π )  − 3
4

Tangent Function: f ( x) = tan x
•
Period:
•
Amplitude:
•
Vertical Asymptotes:
•
Key points:
Example 1: Graph at least one period of each function below. . Be sure to label your axes and clearly identify the
asymptotes.
=
a) y
1 x
tan   − 5
2 4


−2 tan  3 x −
b) y =
π

2
Unit 5–10
So, what about the reciprocal functions?? The best way to create their graph is based on the original trig.
functions. Let’s start with y = csc x which is based on _________________________.
From last chapter, we know there are certain quadrantal angles which make the reciprocal function y = csc x
undefined…Answer the following questions…
• When csc x is undefined, sin x = ?
•
Where on the unit circle does sin x = this value?
•
What angles make us land there on the unit circle?
•
Where are those angles on the sine graph provided?
•
Since y = csc x undefined at those x values, what should
we draw on the graph at the x-intercepts of sine?
•
Add them to your graph.
•
Now let’s find some other values…
o What angles make sin x = csc x ?
o Add these points to your graph.
π  1
π 
o Remember that sin   = …what is csc   ? Add this point to your graph.
6 2
6
1
1
3
4
o Try some other values…when sin x = , what is csc x ? When sin x = , what is csc x ?
o What pattern do you see with the y-values of csc x when x is between the asymptotes?
We can repeat this process for the other reciprocal functions…
Graph y = sec x below.
Graph y = cot x below.
Unit 5–11
4.7 Inverse Trigonometric Functions
Pre-Calculus
Learning Targets
1. Use the appropriate notation for inverse trigonometric functions.
2. Graph the inverse Sine, Cosine and Tangent functions.
3. List the correct Domain and Range of the inverse functions.
4. Find an exact solution to an expression involving an inverse sine, cosine or tangent.
5. Find the composition of trig functions and their inverses.
6. Solve equations with inverse trig expressions (Calculator and non)
Inverse Sine
Inverse Cosine
Inverse Tangent
y = sin −1 x
y = cos −1 x
y = tan −1 x
y = arcsin x
y = arccos x
y = arctan x
D:
D:
D:
R:
R:
R:
Example 1: Find the exact value (in radians).
a) cos-1 0
b) sin-1 0
d) arctan 1
e) cos −1  
− 3

 2 
g) arcsin 



1
(
)
c) arcsin  −
1
2
f) tan −1 − 3
− 3

 2 
h) arccos 
Unit 5–12


2
Compositions with Inverse Functions
 Work from the inside out.
 Remember domain and range restrictions.
Example 2: Evaluate each expression.
(
)
a) sin  arctan − 3 

  5π
b) cos −1  cos 
  3



Example 3: Find the algebraic expression equivalent to the given expression.
(
a) sin cos −1 x
)
(
b) cot sin −1 2x
Unit 5–13
)