5.2: Identifying Trigonometric Ratios
Learning Objectives
- Identify the different parts of right triangles.
- Identify complementary angles in right triangles.
- Become familiar with the basic trigonometric ratios of sine, cosine, and tangent.
What is Trigonometry?
Trigonometry is the study of triangles and the relationships between their side lengths and the angles in between their sides.
The word Trigonometry has two parts: “trig” means triangle and “metry” means measure
Where is Trigonometry used?
There are many applications of trigonometry. Of particular value is the technique of triangulation, which is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. GPSs (global positioning systems) would not be possible without trigonometry. Other fields which make use of trigonometry include:
- astronomy (and navigation - on the oceans, in aircraft, and in space)
- music theory and acoustics
- analysis of financial markets
- electronics
- probability theory and statistics
- medical imaging (CAT scans and ultrasound)
- pharmacy, chemistry, and biology
- number theory (and cryptology)
- land surveying and geodesy
- architecture
- various types of engineering (electrical, mechanical, and civil)
- computer graphics
- cartography (the study of maps)
In your own words, Trigonometry is:
What are the three most interesting applications of trigonometry for you?
Parts of a Triangle
In trigonometry, there are a number of different labels attributed to different sides of a right triangle. They are usually in relation to a specific angle. The hypotenuse of a triangle is always the same, but the terms adjacent and opposite depend on which angle you are referencing.
A side adjacent to an angle is the leg of the triangle that helps form the angle.
A side opposite to an angle is the leg of the triangle that does not help form the angle.
What does the word adjacent mean?
Adjacent means “next to,” so an adjacent side is next to the angle in question.
Examine the picture below: Segment \begin{align*}AB\end{align*} is next to, or adjacent to, angle \begin{align*}B\end{align*}. Notice that it is also the leg of the triangle that helps to form angle \begin{align*}B\end{align*}.
In the triangle shown above, segment \begin{align*}\overline{AB}\end{align*} is adjacent to \begin{align*}\angle B \end{align*}, and segment \begin{align*}\overline{AC}\end{align*} is opposite to \begin{align*}\angle B \end{align*}.
Similarly, \begin{align*}\overline{AC}\end{align*} is adjacent to \begin{align*}\angle C \end{align*}, and \begin{align*}\overline{AB}\end{align*} is opposite to \begin{align*}\angle C \end{align*}.
The hypotenuse is always \begin{align*}\overline{BC}\end{align*}.
In the picture above,
Segment \begin{align*}AC\end{align*} is __________ angle \begin{align*}B\end{align*} and __________ angle \begin{align*}C\end{align*}.
Segment \begin{align*}AB\end{align*} is __________ angle \begin{align*}B\end{align*} and __________ angle \begin{align*}C\end{align*}.
Because angle \begin{align*}A\end{align*} is \begin{align*}90^\circ\end{align*}, Segment \begin{align*}CB\end{align*} is always the __________.
Example 1
Examine the triangle in the diagram below.
Identify which leg is adjacent to \begin{align*}\angle R \end{align*}, opposite to \begin{align*}\angle R \end{align*}, and the hypotenuse.
The first part of the question asks you to identify the leg adjacent to \begin{align*}\angle R \end{align*}. Since an adjacent leg is the one that helps to form the angle and is not the hypotenuse, it must be \begin{align*}\overline{QR}\end{align*}.
The next part of the question asks you to identify the leg opposite \begin{align*}\angle R \end{align*}.
Since an opposite leg is the leg that does not help to form the angle, it must be \begin{align*}\overline{QS}\end{align*}.
The hypotenuse is always opposite the right angle, so in this triangle the hypotenuse is segment \begin{align*}\overline{RS}\end{align*}.
Complementary Angles in Right Triangles
Recall that in all triangles, the sum of the measures of all angles must be \begin{align*} 180^\circ\end{align*}. Since a right angle has a measure of \begin{align*} 90^\circ\end{align*}, the remaining two angles in a right triangle must be complementary. Remember, complementary angles have a sum of \begin{align*}90^\circ\end{align*}. This means that if you know the measure of one of the smaller angles in a right triangle, you can easily find the measure of the other. Subtract the known angle from \begin{align*}90^\circ\end{align*} and you’ll have the measure of the other angle.
Reading Check:
1. What is the sum of all three angles in a triangle?
\begin{align*}{\;}\end{align*}
2. What is the sum of any two complementary angles?
\begin{align*}{\;}\end{align*}
3. What is the study of triangles called?
\begin{align*}{\;}\end{align*}
Example 2
What is the measure of \begin{align*}\angle N \end{align*} in the triangle below?
To find \begin{align*}m\angle N \end{align*}, you can subtract the measure of \begin{align*}\angle N \end{align*} from \begin{align*}90^\circ\end{align*}.
\begin{align*}m\angle N +m\angle O & =90\\ m\angle N & =90-m\angle O \\ m\angle N & =90-27\\ m\angle N & =63\end{align*}
So, the measure of \begin{align*} \angle N \end{align*} is \begin{align*}63^\circ\end{align*} since \begin{align*} \angle N \end{align*} and \begin{align*} \angle O \end{align*} are complementary.
If angle \begin{align*}N\end{align*} and angle \begin{align*}O\end{align*} are complementary, that means they add up to ____________.
Trigonometric Ratios
The fundamentals of trigonometry are the trigonometric functions. There are three basic trigonometric functions: sine, cosine and tangent. These are abbreviated to: sin, cos, and tan:
\begin{align*}\text{sine} & = \sin\\ \text{cosine} & = \cos\\ \text{tangent} & = \tan\end{align*}
These functions are defined from a right-angled triangle.
Consider a right-angled triangle:
\begin{align*}\theta\end{align*} is a Greek letter.
In trigonometry, it is very common to see the letter \begin{align*}\theta\end{align*} used as a variable to represent an angle.
You can think of \begin{align*}\theta\end{align*} just like \begin{align*}x - a\end{align*} variable that stands for a number. In the case of an angle, \begin{align*}\theta\end{align*} stands for a number in degrees.
In the right-angled triangle, we refer to the lengths of the three sides according to how they are placed in relation to the angle \begin{align*}\theta\end{align*}.
The Greek letter _____ is used to represent an angle in trigonometry.
The side opposite to \begin{align*}\theta \end{align*} is labeled opposite, the side next to \begin{align*}\theta\end{align*} is labeled adjacent and the side opposite the right-angle is labeled the hypotenuse.
We define:
\begin{align*}\sin \theta & = \frac {opposite}{hypotenuse}\\ \cos \theta & = \frac{adjacent}{hypotenuse}\\ \tan \theta & = \frac{opposite}{adjacent}\end{align*}
These functions relate the lengths of the sides of a triangle to its interior angles.
One way to remember the definitions is to use the first letter of each word. Notice the bold letters in the definitions above spell out:
SOHCAHTOA
This stands for the trigonometric functions and which sides correspond to each of them:
Sine is Opposite over Hypotenuse (SOH)
Cosine is Adjacent over Hypotenuse (CAH)
Tan is Opposite over Adjacent (TOA)
IMPORTANT: The definitions of opposite, adjacent and hypotenuse only make sense when you are working with right-angled triangles! Always check to make sure your triangle has a right-angle before you use them; otherwise you will get the wrong answer.
Graphic Organizer: SOHCAHTOA
Trigonometric Function | Trig Function Abbreviation | Letters stand for: | Ratio of Side Lengths | Draw a picture | Write yourself some notes to help you remember! |
---|---|---|---|---|---|
Sine |
S O H |
\begin{align*}sin \ \theta =\end{align*} _____________ | |||
Cosine |
C A H |
\begin{align*}cos \ \theta =\end{align*} _____________ | |||
Tangent |
T O A |
\begin{align*}tan \ \theta =\end{align*} _____________ |
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