Name ________________________
Worksheet 5.5
Triangle Trigonometry
Consider a right triangle with one of the acute angles of the triangle labeled  , as in the diagram
below:
h
r

s
The following trigonometric functions, in terms of  , relate the sides of any right triangle:
(hint: for those of you who like acronyms, use SOH-CAH-TOA)
sin  
cos 
tan  
You can use your calculator to determine the measure of an angle by using the “inverse” trig
keys ( sin 1 x, cos 1 x, tan 1 x ).
For example, to solve the following equation for the angle in the first quadrant whose sine value
is ¾, we would perform the following steps:
sin  
3
3
 sin 1  sin    sin 1      48.590
4
4
Suppose you were asked to solve for the missing angle,  , given the following triangle:
Find  :
6

9
Exercise:
The angle of elevation to the top of the Empire State
Building in New York is found to be 11 degrees from the ground at a distance of 1 mile from the
base of the building. Using this information, find the height of the Empire State Building. Sketch
a diagram of this scenario.
Area of a Triangle
A 
In the past, you probably learned that the area of a triangle is given by:
1
bh
2
Question: What important relationship HAS to be true about the base and height to calculate
area?
It has to be true that every triangle has an area (obviously)!
What may not be so obvious is how to calculate the base and height when these measures are not
given about a triangle…
F
Consider the following triangle:
e
d
D
f
E
We will now generate a general formula that will
enable us to determine the area of a triangle:
Although every triangle has three bases, for the sake of consistency, let’s consider f as the base
of the triangle for this exercise…
Draw in the height (to the base f); let’s call it h.
A 
So, the formula for the area of DEF (in terms of base and height) is:
Suppose the following information is given about DEF : e,
(create your own tick marks for these parts)
Write a mathematical equation for h, using e and
_____________
(hint: use a trig function in your equation)
1
( f )(h)
2
D
h
D:
Now, rewrite the formula for the area of DEF (in terms of f, e,
D ):
A  ____________
This is a general formula for the area of any triangle!
As you will find out soon, given two sides and an included angle of a triangle uniquely defines a
triangle. This means that any two sides and the included angle suffice to determine the area of a
triangle. Given the triangle above, write the three distinct formulas for the area of the triangle:
A  ____________
Law of Sines
A  ____________
A  ____________
While we know that SSS, SAS, ASA, & AAS all are manners by which we can prove the
uniqueness of a triangle, unfortunately these theorems/postulates do not give us an explicit way
to determine the measures of the missing sides and/or angles.
Fortunately, there are ways of finding the missing pieces…for now we will only consider the
scenarios where we know two angles and a side, ASA and AAS.
The method we will use to find the missing pieces is called the Law of Sines!
Let’s derive the “formula” for the Law of Sines…
Consider this triangle:
F
e
D
d
f
E
Below, express the area of the triangle in the three distinct ways you did earlier:
(using two sides and the included angle)
A  ____________
A  ____________
A  ____________
Since all the above equations all provide the area of the same triangle, then all the equations can
be set equal to each other…
Set all the above equations equal to each other:
=
=
Now, divide all three sides of the equation by the expression, ( 12 def ):
=
=
Simplify the above equations and rewrite the simplified version:
=
=
This general formula is the Law of Sines!
The Law of Sines allows us to solve for unknown parts of triangles…but this only guarantees a
solution to solving a triangles when given ASA or AAS.
Exercise: Draw a diagram to get started…
1) Solve the following triangle (the missing sides and angles): A  23 , B  110 , c  50 .
2) Given that the area of a triangle is A  12 ab sin  , where  is the included angle
between any two sides a and b in a triangle, calculate the area of the shaded region
below.
61
12
9
8
42
10
SSA (The Ambiguous Case)
As we’ve seen in the past, SSA cannot be used to show that two triangles are congruent because
two possible triangles might exist (in fact, no triangles might exist).
Trigonometry can be used to determine when specific SSA given information falls into the
ambiguous case and can allow you to solve for all the information of each of the two possible
triangles.
B
a
c
A
h
b
C
The following picture is
convenient to use when
thinking about SSA. The given SSA information is angle A, side c, and side a. We will focus on
side a and see what happens when it gets bigger or smaller. h represents the shortest distance
from point B to the opposite side of the triangle, and is therefore h = c sin A (much like in our
triangle area exercise)
The Cases: 0/1/2 Triangles
Case 1 – No Triangle
If a  h (where h = c sin A), then no triangles can be formed since a cannot reach where the
opposite side needs to be. Sketch a diagram of this scenario:
Case 2 – One (right) Triangle
If a  h (again, where h = c sin A), then exactly one and only one triangle can be formed, a right
triangle with hypotenuse c. Sketch a diagram of this scenario:
Case 3 – Two Triangles
If h  a  c (again, where h = c sin A), then a can either lie to the left of h or to the right of h and
still form a triangle – therefore we have two options as to the shape of this triangle. Note that a
still has to be less than the length of c or else one of the triangles cannot be formed. Sketch a
diagram for EACH of the two triangle scenarios:
Scenario 1
Scenario 2
Case 4 – One (non-right) Triangle
If a  c , then one and only one triangle can always be formed. Sketch a diagram of this
scenario:
Solving Triangles in the Ambiguous Case
If you are given SSA in a triangle, use the four cases to determine how many triangles you will
be solving. If only one triangle exists, use the Law of Sines to solve the triangle. If it is the
scenario where two triangles are possible, still use the Law of Sines, but it will only give you one
of the two triangles and you must construct and solve the remaining triangle. If no triangles are
possible, obviously you have nothing to solve for and you are done!
NOTE: be very careful when you sketch your diagrams to use ONLY the given information,
which of course, will not necessarily involve sides a & c, and angle A!
1) Find the area of an isosc. triangle with one side of length 3 and two sides of length 5.
2) Find the area of a triangle with mA  46 , b  5 , c  8 .
3) If a triangle has an area of 8 cm squared, and b  4 and c  5 , find mA ?
4) Given: mB  34 , mC  56 , a  12 . Find b.
B
5) ABDE is a rectangle. mAB  2 and mBD  8 . Find mACB .
D
C
A
H
E
4 cm
6) Given: mHFG  35 and mHIF  90
Find: mHGI
F
G
10 cm
I
7) Find the area of a quadrilateral ABCD where mA  90 , mB  105 , mAB  6 , mAD  9 , and
mBC  12 .
8) Two surveyors measure the angle with the ground to the top of a mountain. One measures the angle
to be 10 degrees and the other measures 7 degrees. One of the surveyors is 100 meters closer to the
mountain than the other. How tall is the mountain?
9) Using a liquid nitrogen cannon, I fire a softball straight up into the air. I am standing away at a safe
distance of 45 ft, and one second after the softball is fired I see it at an angle of 35 degrees above the
ground. After three seconds I see it at an angle of 42 degrees above the ground. What is the
difference between the softball’s height at these two times?
10) A ship sailing due east sees a lighthouse at an angle of 15 degrees North of East. It travels at 15
miles per hour for two hours and then sees the lighthouse at an angle of 25 degrees North of East.
How close to the lighthouse is the ship at this second point? What is the closest the ship will get to
the lighthouse?
Directions: Solve the triangles (all possibilities). As always, lower-case letters represent sides of a
triangle and upper-case letters represent angles of a triangle. Sketch a reasonable diagram for
each triangle scenario.
11) Given: a  10, b  5, B  30
12) Given: a  7, c  8, A  60
13) Given: c  13, b  12, C  18
14) Given: a  3, b  3, A  85
15) Given:
ABC: A  30 , c  12, B  40
16) Given:
GHI: I  35, g  8, i  5