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TRIGONOMETRY
CHAPTER 2: ACUTE ANGLES AND RIGHT TRIANGLES
2.1 TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES
 The figure below shows right triangle ABC. Angle A is an acute angle
in standard position. The legs of the right triangle ABC are x and y,
and the hypotenuse is r. The side of length x is called the side
adjacent to angle A and the side of length y is called the side
opposite angle A.
y
B
r
y
x
A
x
C
 Right-Triangle-Based Definitions of Trigonometric Functions
y side opposite
o sin A= 
r
hypotenuse
r
hypotenuse
o csc A= 
, y not equal to 0
y side opposite
x side adjacent
o cos A= 
r
hypotenuse
r
hypotenuse
o sec A= 
, x not equal to 0
x side adjacent
y side opposite
o tan A= 
, x not equal to 0
x side adjacent
x side adjacent
o cot A= 
, y not equal to 0
y side opposite
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 Cofunctions
o Assume side a is opposite angle A, side b is opposite angle B,
and side c (the hypotenuse) is opposite angle C (which is always
the 90 degree angle).
a
sin A   cos B
c
a
o tan A   cot B
b
c
sec A   csc B
b
b
A
C
a
c
B
o The sum of angles in a triangle is 180°. Since angle C is always
90°, angles A and B must sum to 90°. Therefore we have,
A  B  90°  B  90°  A
which gives us
sin A  cos B  cos  90°  A
o Similar results, called the cofunction identities are true for
the other trigonometric functions.
sin A  cos  90°  A 
csc A  sec  90°  A 
cos A  sin  90°  A 
sec A  csc  90°  A 
tan A  cot  90°  A 
cot A  tan  90°  A 
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 Trigonometric Functions of Special Angles
o Special Angles
o 30-60-90 Triangle
 This type of triangle is formed by bisecting one of the
angles. In our example we have chosen the each side of
the equilateral triangle to be 2.
 The hypotenuse (the longest side opposite the 90 degree
angle) is always twice as long as the shortest side
(opposite the 30 degree angle)
 The middle side (opposite the 60 degree angle) is always
3 times as long as the shortest side
30°
2
3
60°



cos 30 

sec30 =

tan 30 
side adjacent
3

hypotenuse
2
hypotenuse
2 2 3


side adjacent
3
3
side opposite 1
3


side adjacent
3
3
side adjacent
3

 3
side opposite 1
NOW YOUR TURN!!! WHAT ARE THE SIX
TRIGONOMETRIC FUNCTIONS FOR THE 60° ANGLE?


1
side opposite 1
sin 30 =

hypotenuse
2
hypotenuse
2
csc30 =
 2
side opposite 1
cot 30 
3
o 45-45-90 Triangle
 The hypotenuse (opposite the 90 degree angle) always
has a length that is 2 times as long as the length of
either of the shorter sides (which are equal in length).
In this case, we choose the sides to be 1.
2
1
1

sin 45 =
side opposite
1
2


hypotenuse
2
2

csc 45 =
hypotenuse
2

 2
side opposite
1

cos 45 
hypotenuse
2

 2
side adjacent
1
side opposite 1
 tan 45 
 1
side adjacent 1
side adjacent 1
 1
 cot 45 
side opposite 1
o Function Values of Special Angles
sin 
cos 
tan 
cot 
sec 
1
2 3
3
3
3
2
3
2
3


30°
side adjacent
1
2


hypotenuse
2
2
sec 45 =
45°
2
2
2
2
60°
3
2
1
2
1
3
1
3
3
2
2
csc 
2
2
2 3
3
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2.2 TRIGONOMETRIC FUNCTIONS OF NON-ACUTE ANGLES
 Reference Angles
o A reference angle for an angle  , written   , is the positive
acute angle made by the terminal side of the angle  and the xaxis.
 In quadrant I,  and   are the same.
 NEVER use the y-axis to find the reference angle.
ALWAYS use the x-axis!!!
5
 Special Angles as Reference Angles

Each quadrant has a 30°, 45°, and 60° reference angle. The
trigonometric values will be as we discussed for the special acute
angles, except the signs change depending on the quadrant.
o For example: Suppose we have an angle of 240° and we need
to find the trigonometric functions without using a
calculator. A 240° angle has a reference angle of
240° - 180° = 60°. Now we would just put down the
trigonometric functions for 60° and change the signs as
necessary.
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side opposite  3
3


hypotenuse
2
2

sin 240 =

hypotenuse
2
2
2 3



side opposite  3
3
3
side adjacent 1
1
cos 240 


hypotenuse
2
2
hypotenuse
2
sec 240 =

 2
side adjacent 1


csc 240 =

tan 240 
side opposite  3

 3
side adjacent
1

cot 240 
side adjacent
1
3


side opposite  3
3
2.3
FINDING TRIGONOMETRIC FUNCTION VALUES USING A
CALCULATOR
 See Demonstration
2.4 SOLVING RIGHT TRIANGLES
 Significant Digits
o A significant digit is obtained by actual measurement.
 Calculation with Significant Digits
 Adding and Subtracting
o Round the answer so that the last digit you
keep is in the right-most column in which all
the numbers have significant digits.
 Multiplying and Dividing
o Round the answer to the least number of
significant digits found in any of the given
numbers.
 Powers and Roots
o Round the answer so that it has the same
number of significant digits as the number
whose power or root you are finding.
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o Significant Digits for Angles
NUMBER OF
SIGNIFICANT
DIGITS
2
3
4
5
ANGLE MEASURE TO THE NEAREST:
Degree
Ten minutes, or nearest tenth of a degree
Minute, or nearest hundredth of a degree
Tenth of a minute, or nearest thousandth of a degree
 Angles of Elevation or Depression
o Both the angle of elevation and the angle of depression are
measured between the line of sight and the horizontal.
Angle of Elevation
Horizontal
Horizontal
Angle of Depression
 Problem Solving
Step 1: Draw a sketch, and label it with the given information. Assign
variables to any unknown quantities that need to be found.
Step 2: Use the sketch to write an equation relating the given
quantities to the variable.
Step 3: Solve the equation, and CHECK THAT YOUR ANSWER
MAKES SENSE!!!
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2.5 FURTHER APPLICATIONS OF RIGHT TRIANGLES
 Bearing
o Bearing is an important idea in navigation. There are two
methods for expressing bearing.
 Single Angle Given: When a single angle is given, it is
understood that the bearing is measured in a clockwise
direction from due north.
N
N
50°
270°

The second method starts with a north-south line and
uses an acute angle to show the direction, either east or
west, from this line.
N
N 37° E
S
S 83° W
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