Download An Introduction to Trigonometry

An Introduction to Trigonometry
First of all, let’s check out the right angled triangle below.
A
The LETTERS A, B & C indicate the angles and the letters a, b & c indicate the sides.
It is important to note that side BC is opposite angle A. That is why BC = a.
b
Similarly, AC = b and AB = c.
c
This notation is used for easy reference.
B
C
a
Now we have to discuss the special names given to the sides of right triangles. These names are used to
define the formulas that will be used.
Consider the triangle below: we can use any letters to denote the angles.
X
y ( hypotenuse )
z
( opposite )
The side opposite the right angle is always labelled the hypotenuse.
If angle Z is the reference angle, then side XY = z is labelled the opposite.
Y
x
( adjacent )
Z
Side YZ = x is labelled the adjacent.
If angle X is the reference angle, then the hypotenuse remains the same, but side XY = z is labelled the
adjacent and side YZ = x is labelled the opposite.
I’ll let you sketch that situation to help you understand.
The right angle = 90 can never be used as the reference angle for this definition.
Now, we will get down to business.
There are 3 primary trig ratios and 3 reciprocal trig ratios. We will deal with the primary trig ratios.
Their names are sine, cosine and tangent. The ratios are always associated with an angle.
Here are the basic definitions:
sin A 
opposite
hypotenuse
cos A 
adjacent
hypotenuse
tan A 
opposite
adjacent
There is an acronym to remember the ratios.....It is SOH CAH TOA. I’ll let you figure that out!!!!!!
We will use these definitions shortly but let’s look at calculator use first.
Use your scientific calculator to calculate the value: sin 45 .
Just a few comments.......all calculators have 3 settings: degrees, radians and gradients. They are just 3
ways to measure angles. We will only deal with degrees.
There are many types of calculators......but basically there are the old school ones and those called DAL
which means Direct Algebraic Logic.
Calculate sin 45 ........make sure your calculator is on the degree setting.
For old school calculators...enter 45 then press sin. For DAL press sin45.
Both will give the answer 0.7071 + several more digits. It’s that simple!!!!
Now we will use right triangles to calculate unknown sides and angles...........
In each triangle below, calculate the indicated side:
1)
First we must choose the reference angle....in this case 61 .
Now, determine the names of the sides x and 23.5.
23.5
x is adjacent to 61 and 23.5 is the hypotenuse.
This requires the trig ratio cosine.....CAH.
61
x
The ratio is cos A 
Using simple substitution, we have cos 61 
adjacent
hypotenuse
x
. Now use a calculator to determine the trig value.....
23.5
This gives us: 0.48480962 
x
23.5
Next step: 0.48480962  23.5  x
 x  11.39302608
And finally x  11.4 ( x has the same accuracy as side lengths in the question)
Now, let’s do another one. Also you can use variables other than x.
Reference angle..... 28
28
m is the opposite side and 13.84 is the adjacent side.
tan 28 
m
13.84
13.84
0.531709431 
m
13.84
0.53170943113.84  m
m
m  7.36
Now, we will look at one last example to determine an unknown side.
sin 42 
a
0.669130606 
a
7.9
7.9
a
42
7.9
a
0.669130606a  7.9 using the Cross Product Rule.
a
7.9
0.669130606
 a  11.8
The extra challenge here was that the variable ended up in the denominator.
Using the Cross Product Rule that difficulty was overcome.
Now, let’s look at some right triangles that only have the right angle given, along with any 2 side lengths.
Remember: the Pythagorean Theorem can be used to calculate the 3rd side length.
We will focus on calculating the unknown values for the remaining 2 angles.
There are several possibilities........we will consider one of them. I will let you work on the others.
Calculate angle A in the example below.
12.7
9.2
A
Reference angle is A.......the angle we want to calculate.
9.2 is the opposite side and 12.7 is the hypotenuse........the trig ratio required is sine.
sin A 
 sin A 
oposite
hypotenuse
9.2
12.7
sin A  0.724409448 Carry at least 4 decimal places in this step. Fewer will affect the accuracy.
Now, on your calculator press” 2nd” or” INV” or “shift” then press sin.
These buttons are usually found on the top left.
 A  46.41974309
A  46 round off to the nearest whole number.
This gives you an intro to trig...SOH CAH TOA...for right triangles.
On the next page, there are questions to practice your Soh Cah Toa techniques.
Calculate the indicated quantity
1)
Calculate the value of m.
11.4
27
m
2) In XYZ : X  90 , Z  63 and y  4.2 . Calculate the value of z.
3) In ABC : A  57 , B  90 and b  19.4 . Calculate the value of a.
4)
Calculate the value of A.
6.1
A
13.9
5) In EFG : F  90 , f  9.3 and g  2.5 . Calculate the value of G.
6) In PQR : Q  90 , q  9.5 and r  4.9 . Calculate the value of P.
7) In ABC : B  90 , C  49 and c  3.8 . Calculate the value of b.
8) In XYZ : Y  90 , Z  23 and x  8.1. Calculate the value of y.
9) In PQR : P  50 , R  90 and p  5.1 . Calculate the value of q.
Answers
1) 10.2
2) 8.2
3) 16.3
4) 24
6) 59
7) 5.0
8) 8.8
9) 4.3
5) 16