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PreCalculus
Section 4-1 Right Triangle Trigonometry
Name:______________________
Period _____
* Trigonometry means triangle measure.
Trigonometric Functions : See key concept on page 220.
Let θ (theta) be an acute angle in a right triangle and opp is the
length of the side opposite θ, adj is the side adjacent and
hyp is the length of the hypotenuse.
The trigonometric functions of θ are:
opp
hyp
sine    sin  
cosecant    csc  
hyp
opp
adj
hyp
cosine    cos  
secant    sec  
hyp
adj
opp
adj
tangent    tan  
cotangent    cot  
adj
opp
opp
hyp
θ
adj
The cosecant, secant and cotangent are just the reciprocals of the sine, cosine, and tangent.
1
1
1
csc 
sec  
cot  
sin 
cos 
tan 
(*Notice that there is one “co-” in each definition.)
From there, you can derive the following: tan  
sin 
cos 
and
cot  
cos 
sin 
Ex. 1: Find the exact values of the six trigonometric functions of θ.
B
33
θ
C
A
56
See page 221: Notice that Trig values for the similar triangles are the same since the ratios of the
corresponding sides are equal.
Ex. 2: If sin  
1
, find the exact values for the five remaining trig functions for the acute angle θ.
3
PreCalculus Ch 4A Notes_Page 1
Memorize special right triangles --- They are hugely important in this chapter!
30  60  90
(
)
45  45  90
(
)
Trigonometric values for special angles
sin 

cos 
tan 
30
45
60
Ex. 3: Find the value of x. Round to the nearest tenth, if necessary.
a.
35˚
b.
x
7
21˚
9
x
Ex. 4: A competitor in a hiking competition must climb up the inclined course as shown to reach the
finish line. Determine the distance in feet that the competitor must hike to reach the finish line.
(Hint: 1 mile = 5280 feet)
When a trig value of an acute angle is known, we can use inverse trig functions to find the measure of the
angle. See key concept on page 223: Inverse Trigonometric Functions
 Inverse Sine: if sin θ = x, then sin-1 x = θ
 Inverse Cosine: if cos θ = x, then cos-1 x = θ
 Inverse Tangent: if tan θ = x, then tan-1 x = θ
Ex. 5: Use a trig function to find the measure of θ. Round to the nearest degree, if necessary.
a.
b.
12
15.7 12
θ
5
θ
PreCalculus Ch 4A Notes_Page 2
An angle of elevation is formed by a horizontal line and an observer’s line of sight to an object above.
An angle of depression is formed by a horizontal line and an observer’s line of sight to an object below.
angle of
depression
line of sight
angle of elevation
Ex. 6: The chair lift at a ski resort rises at an angle 20.75˚ while traveling up the side of a mountain and
attains a vertical height of 1200 feet when it reaches the top. How far does the chair lift travel up the side
of the mountain?
Ex. 7: A sightseer on vacation looks down into a deep canyon using binoculars. The angles of depression
to the far bank and near bank of the river below are 61˚ and 63˚, respectively. If the canyon is 1250 feet
deep, how wide is the river?
Ex. 8: Solve each triangle (Find the measures of all sides and angles). Round side lengths to the nearest
tenth and angle measures to the nearest degree.
a.
H
b.
f
C
28
5
A
B
41.4˚
G
h
9
F
PreCalculus Ch 4A Notes_Page 3
PreCalculus
Section 4-2 Degrees and Radians
Name:____________________
*Angles in Standard Position: page 231
In a coordinate plane, an angle can be formed by fixing one ray,
called the initial side, and rotating the other ray,
called the terminal side, about the vertex.
An angle is in standard position if its vertex is at origin
and its initial side lies on the positive x-axis.
 counter clockwise : (+) measure
 clockwise : (-) measure
* Degree measures can also be expressed using a decimal form or a degree-minute-second (DMS)
form where each degree is subdivided into 60 minutes(’) and each minute is subdivided into 60
seconds (”): Conversion factor is 1˚ = 60’ = 3600”
Ex. 1: Write each decimal degree measure in DMS form and each DMS measure in decimal form to the
nearest thousandths.
a. 329.125˚
b. 35˚12’7”
Measuring angles in degrees is appropriate when applying trigonometry to solve many real-world
problems, but a degree isn’t always sensible, since a degree has no relationship to any linear measure
(inch to degree has no meaning). For this reason, we use radians.
See Key concept on page 232: Radian Measure
 Words: The measure θ in radians of a central angle of a circle is equal to the
ratio of the length of the intercepted arc s to the radius r of the circle.
s
 Symbols:   , where θ is measured in radians (rad)
r
 Example: 1 radian angle is an angle whose arc length equals to the radius
(1 radian is about 57.3°)
θ = 1 radian
when s = r
*Notice that as long as the arc length and radius are measured using the same linear units, the ratio
s:r is unitless. Therefore, the word radian or rad is usually omitted when writing the radian measure of
an angle. (you must use ˚ for degrees, otherwise it’s implied that the unit is radian)
See the top of page 233 and the key concept.
* Degree/Radian Conversion Rules
 radians
1. To convert a degree measure to radians, multiply by
180
180
2. To convert a radian measure to degrees, multiply by
 radians
PreCalculus Ch 4A Notes_Page 4
Ex. 2: Write each degree measure in radians as a multiple of  and each radian measure in degrees.
2
3
a. 135˚
b. -30˚
c.
d. 
3
4
*Memorize the most common angles’ conversion:
30 

6
45 

4
60 

3
90 

2
Coterminal angles are angles that have the same initial and terminal sides but have different measures.
See figures and key concept on page 234.
Coterminal Angles:
 Degrees: If α is the degree measure of an angle, then all angles measuring α + 360˚n, where n is
an integer , are coterminal with α.
 Radians: If α is the radian measure of an angle, then all angles measuring α + 2  n, where n is
an integer, are coterminal with α.
Ex. 3: Identify all angles that are coterminal with the given angle. Then find one smallest positive
coterminal angle and one smallest negative coterminal angle. Then draw the angles.

a. 80˚
b. 
4
*Sector : A region of a circle that is bound by two radii and an arc of the circle
*Arc length and area of a sector
The arc length s and area A of a sector with radius r and
central angle  (measured in radians) are as follows.
Arc length: s = r
1
Area : A  r 2
2

) , which already incorporates the degree-radian conversion.
When θ is in degrees, you can use s  r (
180
PreCalculus Ch 4A Notes_Page 5
Ex. 4: Find the length of the intercepted arc and area of a sector in each circle with the given central
angle measure and radius. Round to the nearest tenth.

, r  4 in.
a.
b. 125˚, r = 7 cm
You Try: 75˚, r = 5 m
3
Linear speed is the rate at which an object moves along a circular path (total distance/time).
Angular speed is the rate at which an object rotates about a fixed point (total angular measurement/time).
Suppose an object moves at a constant speed along a circular path of radius r.
 If θ is the angle of rotation (in radians) through which the object moves during time t,

then the angular speed ω (omega) of the object is given by   .
t
 If s is the arc length traveled by the object during time t,
s
r

 r
then the object’s linear speed v is v 
t
t
Ex. 5: A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at
331/3 revolutions per minute (rpm).
a. Find the angular speed, in radians per minute, of a record as it plays. Round the nearest tenth.
b. Find the linear speed at the outer edge of the record as it spins, in centimeters per second.
You Try: A wheel has a diameter of 30 inches and it spins at 500 revolutions per minute.
a. Find the angular speed, in radians per minute.
b. Find the linear speed at the outer edge of the wheel, in miles per hour.
PreCalculus Ch 4A Notes_Page 6
PreCalculus
Section 4-3 Trigonometric Functions of the Unit Circle
Name:______________________
* General Definitions of Trigonometric Functions
Let θ be any angle in standard position and point P (x, y) be a point on the terminal side of θ.
Let r represent the nonzero distance from P to the origin. That is, r  x 2  y 2 (r is always positive.)
Then the trig function of θ are as follows:
y
r
sin  
csc  
(y  0)
r
y
x
r
cos  
sec  
(x  0)
r
x
y
x
tan  
(x  0)
cot  
(y  0)
x
y
Ex. 1: Let (-4, 3) be a point on the terminal side of an angle θ in standard position. Find the exact values
of the six trigonometric functions of θ.
When the terminal side of an angle θ that is in standard position lies on the one of the coordinate axes,

3
, 2 , ...)
the angle is called a quadrantal angle. (example: 0 , 90 , 180 , 270 , 360 , ... or 0, ,  ,
2
2
* Use the unit circle to find the trig functions of any quadrantal angle!
The circle x2 + y2 = 1, which has center (0, 0) and radius 1, is called the unit circle.
sin  =
cos  =
tan  =
Ex. 2: Find the exact value of each trig function, if defined. If not defined, write undefined.
7
a. cos 
b. tan 450˚
c. cot
2
PreCalculus Ch 4A Notes_Page 7
*Reference Angle Rules: See key concept on page 244
How to find the trig functions of angles that are neither acute nor quadrantal.
--- Use θ’ (theta prime), called a reference angle, to find the trig values of any angle θ.
Let  be an angle in standard position. The reference angle for  is the acute angle ’ formed by the
terminal side of  and the x-axis. The relationship between  and ’ is shown below for nonquadrantal

angles  such that 90° <  < 360° (
<  < 2).
2
Quadrant II
Quadrant III
Quadrant IV
Degrees: (90° <  < 180°)
Radians: (

<  < )
2
Degrees: (180° <  < 270°)
Radians: (<  <
3
)
2
Degrees: (270° <  < 360°)
Radians: (
3
<  <2 )
2
To find a reference angle outside the interval 0    360 or 0    2 , first find a corresponding
coterminal angle in this interval.
Ex. 3: Sketch each angle. Then find its reference angle.
3
a. -150˚
b.
4
Evaluating Trig Functions of any Angle:
Step 1 Find the reference angle  ' .
'.
Step 2
Evaluate the trigonometric functions for
Step 3
Determine the sign of the trigonometric function value
from the quadrant in which  lies.
Look at the paragraph under the key concept about the Quadrant signs
on page 245.
Also see study tip for memorizing the trig values of 30˚, 45˚ and 60˚ angles.
PreCalculus Ch 4A Notes_Page 8
Ex. 4: Find the exact value of each expression using the reference angle. Show the diagram.
4
15
a. sin
b. tan 150˚
c. sec
3
4
Ex. 5: Let sec  
29
, where sin θ > 0. Find the exact values of the remaining five trig functions of θ.
5
You Try: a. Find the exact value of cot
b. Let sin  
10
using the reference angle. Show the diagram.
3
5
, where cot θ < 0. Find the exact values of the remaining five trig functions of θ.
7
PreCalculus Ch 4A Notes_Page 9
A unit circle is a circle of radius 1, centered at the origin.
Trig functions of the unit circle: Let t be any real number
on a number line and let P(x,y) be the point on t when
the number line is wrapped onto the unit circle.
Then the trig functions of t are as follows:
sin t  y
csc t 
1
(y  0)
y
cos t  x
sec t 
1
(x  0)
x
y
(x  0)
x
x
cot t 
(y  0)
y
tan t 
Therefore, the coordinates of P corresponding to the angle t can be written as P(cos t, sin t).
* 16-point Unit Circle
PreCalculus Ch 4A Notes_Page 10
Ex. 7: Find the exact value of each expression using the unit circle. If undefined, write undefined.
7

4
a. sin
b. cos
c. tan
d. sec270˚
6
3
3
Read page 249 to see why the domain of the sine and cosine functions is (-∞, ∞). The values of the sine
and cosine function lie in the interval [-1,1] and repeat for every integer multiple of 2π on the number
line. Functions with values that repeat at regular intervals are called periodic functions.
Periodic functions: y = f(t) is periodic if there exists a positive real number c such that f(t + c)= f(t) for
all values of t in the domain of t. The smallest number c for which f is periodic is called the period of t.
Ex. 8: Find the exact value of each expression.
9
a. cos
b. sin (-300˚)
4
c. tan
Remember that f is even if f(-x)=f(x) and f is odd if f(-x) = -f(x).
You can use the unit circle to show that the cosine function is even and
that the sine and tangent functions are odd.
cos (-t) = cos t
sin (–t) = - sin t
PreCalculus Ch 4A Notes_Page 11
tan (–t) = - tan t
29
6