4.3 Right Triangle Trigonometry
Trigonometric Functions can be looked at from two perspectives. One is the
“point in the plane” perspective and the other is from a right triangle
perspective.
Right Triangle Description
When  is an angle between 0 and 90, we can use right triangles.
Relative to the angle  , the three sides of a triangle are the:
_______________(the side opposite the right angle)
________________(the side opposite the angle  )
________________ ( the side adjacent (next to) the angle  )
hypotenuse
Opposite

Adjacent
Using the length of these sides, you can form six ratios that define the six
trigonometric functions of the acute angle  .
sine
cosecant
cosine
secant
tangent
cotangent
Abbreviations:
Consider a right triangle containing an angle  ( is an acute angle of a right
triangle.) The trig functions are ratios based on the sides of a right triangle.
The values of the six trigonometric functions of the angle  are given by:
sin  
opposite
hypotenuse
cos =
adjacent
hypotenuse
tan  
opposite
adjacent
csc 
hypotenuse
opposite
sec 
hypotenuse
adjacent
cot  
adjacent
opposite
NOTE: The functions in the second row are the reciprocals of the corresponding
functions in the first row.
NOTE: Angles can be measured in degrees or radians. Sin 30 denotes the sine
of an angle of 30 radians BUT when angles are measured in degrees you
must use the degree symbol. Sin 30 means the sine of an angle of 30
degrees.
sin 30
 .988
sin 30
Degree symbol is ESSENTIAL to avoid error

1
2
Example: Evaluate the six trigonometric functions at the angle 
4

3
By the Pythagorean Theorem, hyp   (opp )2  (adj )2 , therefore
2
sin  
opp

hyp
csc  
hyp
opp
cos =
adj
=
hyp

sec 
hyp
adj
tan  
opp

adj

cot  
adj
opp

Special Angles:
Radians:
Degrees:
0

6

4

3
0
30
45
60

2

90
180
2
360
Evaluating Trigonometric Functions of 45
Construct a right triangle having 45 as one of its acute angles. Let the length
of the adjacent side be 1. From geometry, you know the other acute angle is also
45 . So, the triangle is isosceles and the length of the opposite side is also 1.
Using the Pythagorean Theorem you find the length of the hypotenuse to be 2 .
1
45
2
45
1
Find the exact values of sin 45 , cos 45 and tan 45
sin 45 
opp

hyp

cos 45 
adj

hyp

tan 45 
opp

adj
Radians:
sin

4

opp

hyp

cos

4

adj

hyp

tan

4

opp

adj


Evaluating Trigonometric Functions of 30 and 60
30
2
3
60
1
Remember from geometry, that in a 30, 60, 90 triangle, the hypotenuse is
twice the length of the shorter leg and the angle opposite 30 is half of the
hypotenuse.
Evaluate sin 60, cos 60 , sin 30 and cos 30
Degrees:
sin 60 

Radians:
sin
Degrees:
sin30 
Radians:
sin

6

3

opp

hyp
cos 60 
opp

hyp
cos
opp

hyp
opp

hyp

3

adj

hyp
cos30 
cos

6

adj

hyp
adj

hyp
adj

hyp
The angles 30, 45 and 60 (
  
, , ) occur frequently in trigonometry.
6 4 3
Therefore you should know the trigonometric functions of these special angles.
Sines, Cosines and Tangents of Special Angles
sin30  sin
sin 45  sin
sin 60  sin

6

4

3

1
2

2
2

3
2
cos30  cos

6
cos 45  cos

4
cos 60  cos
Note from above that sin 30 
3
2



3

tan30  sin

6
2
2
tan 45  tan
1
2
tan 60  tan


4

3
3
3
1
 3
1
 cos 60 . This occurs because 30 and 60
2
are complementary. Cofunctions of complementary angles are equal.
That is, if  is an acute angle, the following relationships are true.
sin(90   )  cos
cos(90   )  sin 
tan(90   )  cot 
cot(90   )  tan 
sec(90   )  csc
csc(90   )  sec
The following relationships between trigonometric functions (identities) are used
many times in trigonometry and in calculus. You should learn them well!
Fundamental Trigonometric Identities
Reciprocal Identities
sin  
csc  
1
cos =
csc
1
sec 
sin
1
sec
1
cos
Quotient Identities
tan  
sin 
cos 
tan  
cot  
1
cot
1
tan
Pythagorean Identies
cot  
sin2   cos2   1
cos
sin 
1  tan2   sec2 
1  cot2   csc2 
Note that
 
NOT cos  
2
sin2  represents  sin   NOT sin  2 ;
2
cos2  represents  cos  
Example 4 & 5 Text pgs 306 & 307
Make sure your calculator is in the correct mode – radians or degrees.
Evaluate sec (5 40’ 12”).
Convert to decimal form: 5 40’ 12” =
Use calculator to evaluate sec _______
sec (5 40’ 12”) = _____________=
1
 1.00492
cos5.67
2
Solving Right Triangles
Solving a triangle means finding the lengths of all three sides and the
measures of all three angles when only some of these quantities are
given.
The right triangle description of a trigonometric function relates three
quantities: the angle  and two sides of the right triangle. When two of
these three quantities are known, then the third can always be found.
Example: Find the lengths of sides b and c.
17
c
sin 75 
opposite
c

hypotenuse 17
75
b
Ex.: Find the measure of angles  and  in the right angle shown.


6.4
Many applications use the angle between the horizontal and some other line (for
instance, the line of sight from an observer to a distant object). This angle is
called the angle of elevation or the angle of depression, depending on whether
the line is above or below the horizontal.
Angle of Elevation
Horizontal---------------------------------------
Angle of Depression
Example: A wire is to be stretched from the top of a 10-meter high building
to a point on the ground. From the top of the building the angle of
depression to the ground point is 22. How long must the wire be?
Try this: A person standing on the edge of a bank of a canal observes a lamp
post on the edge of the other bank of the canal. A person’s eye level is 152 cm
above the ground ( 5 ft). The angle of elevation from eye level to the top of
the lamp post is 12, and the angle of depression from eye level to the bottom
of the lamp post is 7. How wide is the canal? How high is the lamp post?
Examples 7 – 9 on pages 308 & 309