RIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY
The word Trigonometry can be broken into the parts Tri, gon, and metry, which means "Three angle measurement," or equivalently "Triangle measurement." Throughout this unit, we will learn new ways of finding missing sides and angles of triangles which we would be unable to find using the Pythagorean Theorem alone.
The basic trigonometric theorems and definitions will be found in this portion of the text, along with a few examples, but the reader will frequently be directed to refer to detailed "tutorials" that have numerous examples, explorations, and exercises to complete for a more thorough understanding of each topic.
One comment should be made about our notation for angle measurement. In our study of Geometry, it was standard to discuss the measure of angle A with the notation mA . It is a generally accepted practice in higher level mathematics to omit the measure symbol (although there is variation from text to text), so if we are discussing the measure of a 20o angle, for example, we will use the notation A = 20D rather than mA = 20D .
Special Right Triangles
In Trigonometry, we frequently deal with angle measures that are multiples of 30o, 45o, and 60o. Because of this fact, there are two special right triangles which are useful to us as we begin our study of Trigonometry. These triangles are named by the measures of their angles, and are known as 45o-45o-90o triangles and 30o-60o-90o triangles. A diagram of each triangle is shown below:
45o leg
hypotenuse
45o leg
30o longer
hypotenuse
leg
60o shorter leg
Tutorial: For a more detailed exploration of this section along with additional examples and exercises, see the tutorial entitled "Special Right Triangles."
The theorems relating to special right triangles can be found below, along with examples of each.
Right Triangle Trigonometry
Special Right Triangles
Theorem: In a 45o-45o-90o triangle, the legs are congruent, and the length of the hypotenuse is 2 times the length of either leg.
Examples Find x and y by using the theorem above. Write answers in simplest radical form.
1.
x
y
45o
7
Solution: The legs of the triangle are congruent, so x = 7. The
hypotenuse is 2 times the length of either leg, so y = 7 2.
2.
x
13 2
45o
y
3.
45o
7
x
y
Solution:
The hypotenuse is 2 times the length of either leg, so
the length of the hypotenuse is x 2. We are given that the length of the hypotenuse is 13 2 , so x 2 = 13 2 , and we obtain x = 13. Since the legs of the triangle are congruent, x = y, so y = 13.
Solution:
The hypotenuse is 2 times the length of either leg, so
the length of the hypotenuse is x 2. We are given that
the length of the hypotenuse is 7, so x 2 = 7 , and we obtain x = 7 . Rationalizing the denominator,
2
x=
7
2
2 2
=
72 2
. Since
the
legs
of
the
triangle
are
congruent,
x=
y,
so y =
7
2 2
.
Theorem: In a 30o-60o-90o triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg.
Right Triangle Trigonometry
Special Right Triangles
Examples Find x and y by using the theorem above. Write answers in simplest radical form.
1.
30o x y
60o
6
Solution: The length of the shorter leg is 6. Since the length of the hypotenuse is twice the length of the shorter leg, x = 2 6 = 12. The length of the longer leg is 3 times
the length of the shorter leg, so y = 6 3.
2.
Solution:
y
The length of the shorter leg is x. Since the length of the
8 3 longer leg is 3 times the length of the shorter leg, the
60o
length of the longer leg is x 3. We are given that the
x
length of the longer leg is 8 3 , so x 3 = 8 3, and
therefore x = 8 . The length of the hypotenuse is twice
the length of the shorter leg, so y = 2x = 2 8 = 16.
3.
9
Solution:
y The length of the shorter leg is y. Since the length of the
30o
hypotenuse is twice the length of the shorter leg,
x
9 = 2 y , so
y=
9 2
= 4.5 .
Since the length of the longer
leg is 3 times the length of the shorter leg,
x= y
3
=
9 2
3 = 4.5
3.
4.
60o
y
x
30o
12
Solution: The length of the shorter leg is x. Since the length of the
longer leg is 3 times the length of the shorter leg, the
length of the longer leg is x 3. We are given that the
length of the longer leg is 12, so x 3 = 12. Solving for
x and rationalizing the denominator, we obtain
x = 12 = 12
3
3
3 3
=
12 3 3
=
4
3. The length of the
hypotenuse is twice the length of the shorter leg, so
y = 2x = 2 4 3 = 8 3.
Right Triangle Trigonometry
Special Right Triangles
Trigonometric Ratios
There are three basic trigonometric ratios which form the foundation of trigonometry; they are known as the sine, cosine and tangent ratios. This section will introduce us to these ratios, and the following sections will help us to use these ratios to find missing sides and angles of right triangles.
Tutorial: For a more detailed exploration of this section along with additional examples and exercises, see the tutorial entitled "Trigonometric Ratios."
The three basic trigonometric ratios are defined in the table below. The symbol , pronounced "theta", is a Greek letter which is commonly used in Trigonometry to represent an angle, and is used in the following definitions. Treat it as you would any other variable.
If is an acute angle of a right triangle, then:
Trigonometric Function
The sine of
The cosine of
The tangent of
Abbreviation
Ratio of the Following Lengths
=
sin( )
=
The leg opposite angle
The hypotenuse
=
cos( )
=
The leg adjacent to angle
The hypotenuse
=
tan( )
=
The leg opposite angle
The leg adjacent to angle
*Note: A useful mnemonic (in very abbreviated form) for remembering the above chart is:
SOH-CAH-TOA
SOH stands for sin( ), Opposite, Hypotenuse:
sin(
) = Opposite Hypotenuse
CAH stands for cos( ), Adjacent, Hypotenuse:
cos( ) =
Adjacent Hypotenuse
TOA stands for tan( ), Opposite, Adjacent:
tan( )
=
Opposite Adjacent
Right Triangle Trigonometry
Trigonometric Ratios
Example Find the sine, cosine, and tangent ratios for each of the acute angles in the following triangle.
13 R
Q Solution:
We first find the missing length of side RS. Solving the equation (RS)2 +122 = 132 , we obtain 12 RS = 5.
We then find the three basic trigonometric ratios for angle R:
S
sin R = The leg opposite angle R = 12
The hypotenuse
13
cos R = The leg adjacent to angle R = 5
The hypotenuse
13
tan R = The leg opposite angle R = 12 The leg adjacent to angle R 5
We then find the three basic trigonometric ratios for angle Q:
sin Q = The leg opposite angle Q = 5
The hypotenuse
13
cos Q = The leg adjacent to angle Q = 12
The hypotenuse
13
tan Q = The leg opposite angle Q = 5 The leg adjacent to angle Q 12
Right Triangle Trigonometry
Trigonometric Ratios
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