In order to understand trigonometry, we must first understand the general theory behind it.
As in any right angled triangle, the longest side is called the hypotenuse. It is also the side that is directly across from the 90 degree angle. In this case, the hypotenuse is Side X.
What, then, are the other two sides called?
Let's first look at angle Q. We know that the hypotenuse is Side X, but what are sides Y and Z?
Just like Side X is the side opposite the perpendicular angle, Side Y is also the side opposite Angle Q. Therefore, Side Y is opposite Angle Q.
Side Z is what we call the side adjacent to Angle Q.
Now, if we look at Angle W, we know that the hypotenuse, Side X, remains the same. However, this time, Side Y is the adjacent, and Side Z is the opposite.
One must be careful when writing which side is which, and must pay careful attention to which angle they are referring.
With respect to Angle W
X is the hypotenuse,
Y is the adjacent,
Z is the opposite
In trigonometry, we use a few common symbols to represent angles. They are certain letters from the Greek alphabet, such as
Ø -> "Phi"
Now that we have understood some terminology, we can continue to some more theory.
Trigonometric Ratios are an important part in trigonometry. They can be used to determine any missing side lengths or angles in a right-angled triangle.
Trigonometric Ratios compare specific angles to sides, in a specific order.
The three Primary Trigonometric Ratios are called
* Sine
* Cosine, and
* Tangent
Cosine compares an angle to its adjacent and hypotenuse. Cosine Ø = adjacent/hypotenuse. Again using the above triangle, we get Cosine Q = Z/X.
Finally, Tangent compares an angle to its opposite and adjacent sides. Tangent Ø = opposite/adjacent. In the above triangle, we get Tangent Q = Y/Z.
There is also a shorter, more common, way of writing Sine, Cosine and Tangent. Sine = Sin (pronounced 'Sine'); Cosine = Cos (pronounced 'Cose'); Tangent = Tan.
Your calculator uses these abbreviated forms.
In addition, there is a trick to remembering which trig ratio goes with each set of information. It is called SOH CAH TOA.
Let's break it down into parts:
CAH
Cos = Ajd/Hyp
TOA
Tan = Opp/Adj
Example 1: Write all of the primary trigonometric ratios for the following triangle.
First, look at all of the sides, and how they correspond with the angle, Ø.
The side with the length, 5, is the hypoteneuse, as it is the longest side, and it is directly across from the 90 degree angle.
The side with a length of 4 is directly opposite angle Ø, so it is the opposite.
Finally, the side with a length of 3 is beside angle Ø, so it is the adjacent.
Therefore, we can state:
sin Ø = opp/hyp
sin Ø = 4/5
cos Ø = adj/hyp
cos Ø = 3/5
tan Ø = opp/adj
tan Ø = 4/3
These are the three primary ratios.
Example 2: Solve the following expressions, correct to two decimal places.
a) sin 30
b) cos 45
c) tan 15
a) sin 30
Note: You will need your calculator to solve these expressions. Depending on your calculator, you will either need to type in sin first, or you will need to type in the angle first. Also: make sure that you are working in Degrees (there should be a little D, or a DEG icon at the top of your calculator's screen.)
ie) punch in SIN then 30, or punch in 30 then SIN
sin 30 = 0.5
b) cos 45 = 0.71
c) tan 15 = 0.27
Practice Questions
Page 496 #4, 6ad, 11b, 15
