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Name: _________________________
Worksheet: 4.1
Quick Review of Pythagorean Theorem and Right Triangle Trig
The Pythagorean Theorem still exists:
Pythagorean Theorem: Given sides a, b, and c of a right triangle, with side c being the
longest side or hypotenuse, the following result holds: a 2  b 2  c 2 .
Exercises
1. Find the missing side length. Diagrams are not to scale. Leave your answers in
radical form (i.e. leave any square roots in your answer – don’t approximate with
your calculator).
a.
b.
c.
5
10
12
10
10
10
d.
e.
5
4
5
4
2. What (if anything) is wrong with these diagrams? In other words, could these be the
actual lengths of a triangle? If not, explain briefly.
4
a.
b.
7
12
4
4
5
11
c.
d.
7
5
11
7
5
3. Some right triangles have all three sides that are integers (what lucky triangles!).
a. Verify that right triangles with sides 3-4-5 AND 5-12-13 satisfy the Pythagorean
Theorem.
b. Show that the only right triangle in which the lengths of the sides are consecutive
integers is the 3-4-5 right triangle.
Hint: If the smallest integer side is n, what is the next side? The longest side?
Use these expressions in the Pythagorean Theorem and solve for n.
Solving for Missing Lengths and Angles
Now that we know how to use the Pythagorean Theorem to find the lengths of the sides,
we turn our attention to the relationship between the sides and the angles. To do this we
introduce sine, cosine, and tangent. Of course there are three sides to a right triangle and
three corresponding angles, one of which is 90 degrees, or a right angle. Given either of
the other two angles, we can refer to the sides corresponding to that angle as opposite,
adjacent, and hypotenuse, that is the side opposite the angle is the opposite, the side next
to the angle that is not the hypotenuse is the adjacent, and the hypotenuse is the
hypotenuse. Consider the diagram below.
A
hypotenuse
hypotenuse
adjacent
opposite
to A
of B
B
opposite
adjacent
of A
to B
Naming of sides with respect to angle A.
Naming of sides with respect to angle B.
Students are often initially confused by the naming of triangle sides. The hypotenuse is
always the longest side of a right triangle. That means the hypotenuse is always across
from the right angle. The terms adjacent and opposite are relative – that means the terms
depend on which angle you are referencing.
The Trigonometric Ratios
The ratio of two side lengths in a right triangle is of special importance in mathematics.
So much so that three terms are defined for three of these ratios.
Definition - Sine:
The Sine of an angle B is defined to be Opposite / Hypotenuse, and
is denoted by sin(B).
Definition - Cosine: The Cosine of an angle B is defined to be Adjacent / Hypotenuse,
and is denoted by cos(B).
Definition - Tangent: The Tangent of an angle B is defined to be Opposite / Adjacent,
and is denoted by tan(B).
Before proceeding to exercises, it is worth mentioning a mnemonic that generations of
young mathematicians have used to remember the three definitions above, namely:
SOH-CAH-TOA,
S = sine, C = cosine, T = tangent, O = Opposite, H = Hypotenuse and A = Adjacent.
sin B 
opposite
hypotenuse
cos B 
adjacent
hypotenuse
tan B 
opposite
adjacent
Example 1: Find all three trig ratios with respect to angles x and y below.
6
3
y
x
8
4
10
5
sin(x) = 4/5
sin(y) = 8/10 = 4/5
cos(x) = 3/5
cos(y) = 6/10 = 3/5
tan(x) = 4/3
tan(y) = 8/6 = 4/3
You probably noticed that the sin(x) = sin(y), cos(x) = cos(y), and tan(x) = tan(y). For
two (arbitrary) triangles this usually doesn’t happen. For this to occur the two triangles
must be related. This brings us back to the idea of similarity introduced in the previous
lesson. Remember, similar figures have exactly the same shape, but one is just
proportionally larger than the other. It is an important observation that the corresponding
angles (like x and y in this problem) of similar triangles will always have the same trig
ratios.
Example 2: Use trig ratios to find the measure of angle x.
3
x
 Write down a trig ratio involving x and two of the given sides.
4
5
We could pick sine, cosine, or tangent since we have all three sides. Let’s go
with sine this time: sin(x) = 4/5 = 0.8
 Use a table of values at the back of the chapter (or your calculator) to find the
angle that has a sine closest to 0.8.
The table gives the angle to be around 53º.
Definition – the symbols sin−1(ratio): The inverse sine, denoted sin−1(ratio), is an angle
such that sin(angle) = ratio. For example: If sin(53º) = 0.8, we say sin−1(0.8) = 53º.
Example 3: Evaluate sin−1(0.5).
The symbols sin−1(0.5) represents an angle, A, such that sin(A) = 0.5. Using our
table, we see that sin(30º) = 0.5. Therefore, sin−1(0.5) = 30º.
Example 5: Evaluate tan(35º).
This is easy on your calculator, but by doing so you miss an important observation
about the tangent function.
Way 1:
Way 2:
Alternate Definition – Tangent:
tan( angle ) 
sin( angle )
cos( angle )
.
Solving Triangles using Trigonometry
With surprisingly little information, we can solve a right triangle for all of its sides and
angles. With at least this much information, we can solve any right triangle:


Any two sides or
One side and one of the acute angles.
The first two Exercises will lead you through how to work each type of problem.
Exercises
48
1. Solving with an angle and a side.
9
x
a. In relation to the 48º angle, is side X the opposite / adjacent / or hypotenuse?
b. In relation to the 48º angle, is side of length 9 units the opposite / adjacent / or
hypotenuse?
c. Fill in the blank with the appropriate trig function:
_______(48º) = X/9
d. Solve your equation from part C, using the table to find the appropriate decimal
trig value.
e. Find the remaining side using the Pythagorean Theorem.
f. Since the three angles of a triangle must add up to 180º, find the measure of
the missing angle?
8
2. Solving with two sides.
a. Circle: In relation to xº, the 8 is (opposite / adjacent /
hypotenuse) and the 15 is
(opposite / adjacent / hypotenuse).
x
15
b. Fill in the blanks with the appropriate trig function:
______(xº) = 8/15, or
______−1(8/15) = xº and now look at your trig table.
c. Use the Pythagorean Theorem to find the missing side.
d. Find the measure of the remaining angle.
3. Write an appropriate trig ratio and solve for the indicated unknown variable(s).
a.
12
b.
8
x
40
50
x
c.
d.
x
e.
20
8
x
y
30
20
30
x
3
4. Solve each of the following triangles. That means find all the missing angles and
sides.
a.
b.
10
12
15
10
c.
38
d.
145
120
85
5. The distance between A and B is 10 feet. Find the length of X.
Hint: Substitution might come in handy.
X
62
38
B
10
A