Introduction To Trigonometry functions
There are six functions that are the central part of trigonometry. There are three major ones that you are required to understand completely:
o Sine (sin)
o Cosine (cos)
o Tangent (tan)
The other three are not used as often and can be derived from the three primary functions. Due to the fact that they can be readily derived, calculators and spreadsheets do not normally have them.
o Secant (sec)
o Cosecant (csc)
o Cotangent (cot)
All six functions have three-letter abbreviations (shown in parentheses above).
Definitions of the six functions
Consider the right triangle above. For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle. The only difference between the six functions is which pair of sides we make use of.
In the table below:
o a is the length of the side adjacent to the angle (x) in question.
o o is the length of the side opposite the angle.
o h is the length of the hypotenuse.
“x” stands for the measure of the angle in either degrees or radians.
| Sine | sin x=o/h | |
| Cosine | cos x=a/h | The three primary functions |
| Tangent | tan x=o/a |
| Cosecant | csc x=h/o | csc x=1/sin x | |
| Secant | sec x=h/a | Observe how each is the reciprocal of sin, cos or tan. | sec x=1/cos x |
| Cotangent | cot x=a/o | cot x=1/tan x |
For instance, in the figure above, the cosine of x is the side adjacent to x (labeled a), over the hypotenuse (labeled h): cos x =a/h If a=12cm, and h=24cm, then cos x = 0.5 (12 over 24).
Soh Cah Toa
These 9 letters are a memory aid to remember the ratios for the three primary functions – sin, cos and tan. Pronounced a bit like “soaka towa”.
The ratios are constant
Due to the fact that the functions are a ratio of two side lengths, they at all times produce the same result for a given angle, irrespective of the size of the triangle.
In the figure above, if you drag the point C. The triangle will adjust to keep the angle C at 30°. Observe the way the ratio of the opposite side to the hypotenuse does not alter, even though their lengths do. In view of that, the sine of 30° does not differ either. It is at all times 0.5.
Remember: When you apply a trignometric function to a given angle, it at all times yields the same result. For instance tan 60° is at all times 1.732.
Using a calculator
The majority of calculators have buttons to to press and obtain the value of the sin, cos and tan of an angle. Ensure you set the calculator to degrees or radians mode depending on what units you are making use of.
Inverse functions:
For each of the six functions, there is an inverse function that functions in reverse. The inverse function has the letters ‘ARC’ in front of it.
For instance, the inverse function of COS is ARCCOS. While COS tells you the cosine of an angle, ARCCOS tells you what angle has a given cosine. On calculators and spreadsheets, the inverse functions are occasionally written as acos(x) or cos-1(x).
Trigonometry functions of large and/or negative angles
The six functions can as well be defined in a rectangular coordinate system. This permits them to go beyond right triangles, to where the angles can have any measure, even beyond 360°, and can be both positive and negative.
Identities – replacing a function with others
Trigonometric identities are merely ways of writing one function with the use of others. For instance, from the table above, we observe that sec x= 1/cos x. This equivalence is known as an identity. If we had an equation with sec x in it, we could replace sec x with
one over cos x if that assists us to get to our final destination. There are a lot of such identities.
Not just right triangles
These functions are defined with the use of a right triangle, but they have uses in other triangles as well. For instance the Law of Sine and the Law of Cosines can be used to solve any triangle – not just right triangles.
Graphing the functions
The functions can be graphed, and a few, particularly the SIN function, produce shapes that commonly occur in nature. For instance see the graph of the SIN function, frequently referred to as a sine wave, on the right. For more information see the:
• Graph of the sine function
• Graph of the cosine function
• Graph of the tangent function
Pure audio tones and radio waves are sine waves in their respective medium.
Derivatives of the trig functions
Each of the functions can be differentiated in calculus. The result is a new function that shows its rate of change (slope) at a particular values of x. These derivative functions are stated in terms of other trig functions.
Sin, Cos and Tan
A right-angled triangle is a triangle in which one of the angles is a right-angle. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The adjacent side is the side which is between the angle in question and the right angle. The opposite side is opposite the angle in question.
In any right angled triangle, for any angle:
The sine of the angle = the length of the opposite side/the length of the hypotenuse
The cosine of the angle = the length of the adjacent side/the length of the hypotenuse
The tangent of the angle = the length of the opposite side/the length of the adjacent side
Therefore, in shorthand notation:
sin = o/h cos = a/h tan = o/a
Frequently remembered by: soh cah toa
Right Triangle
Sine, Cosine and Tangent are all based on a Right-Angled Triangle
Before getting stuck into the functions, it is of a very big help to give a name to each side of a right triangle:
• “Opposite” is opposite to the angle θ
• “Adjacent” is adjacent (next to) to the angle θ
• “Hypotenuse” is the long one
Adjacent is always next to the angle
And Opposite is opposite the angle
Sine, Cosine and Tangent
Sine, Cosine and Tangent are the three major functions in trigonometry.
They are frequently shortened to sin, cos and tan.
To calculate them:
Divide the length of one side by another side
… but which sides?
For a triangle with an angle θ, they are calculated this way:
| Sine Function: | sin(θ) = Opposite / Hypotenuse |
| Cosine Function: | cos(θ) = Adjacent / Hypotenuse |
| Tangent Function: | tan(θ) = Opposite / Adjacent |
In picture form:
Example: What is the sine of 35°?
With The use of this triangle (lengths are only to one decimal place):
sin(35°)= Opposite / Hypotenuse
= 2.8 / 4.9
= 0.57…
How to remember? Think “Sohcahtoa”! It works like this:
| Soh… | Sine = Opposite / Hypotenuse |
| …cah… | Cos = Adjacent / Hypotenuse |
| …toa | Tan = Opposite / Adjacent |
Examples
Example: what are the sine, cosine and tangent of 30° ?
The standard 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √3:
Now we know the lengths, we can calculate the functions:
| Sine | sin(30°) = 1 / 2 = 0.5 |
| Cosine | cos(30°) = 1.732 / 2 = 0.866… |
| Tangent | tan(30°) = 1 / 1.732 = 0.577… |
(Use your calculator and cross check them!)
Example: what are the sine, cosine and tangent of 45° ?
The standard 45° triangle has two sides of 1 and a hypotenuse of √2:
| Sine | sin(45°) = 1 / 1.414 = 0.707… |
| Cosine | cos(45°) = 1 / 1.414 = 0.707… |
| Tangent | tan(45°) = 1 / 1 = 1 |
Why?
Why are these functions very significant?
• Due to the fact that they allow us work out angles when we know sides
• And they allow us work out sides when we know angles
Example: Use the sine function to calculate “d” in the above diagram
We know:
• The cable makes a 39° angle with the seabed
• The cable has a 30 meter length.
And we want to know “d” (the distance down).
| Begin with: | sin 39° = opposite/hypotenuse |
| sin 39° = d/30 | |
| Swap Sides: | d/30 = sin 39° |
| Make use of a calculator to find sin 39°: | d/30 = 0.6293… |
| Multiply both sides by 30: | d = 0.6293… x 30 |
| d = 18.88 to 2 decimal places. |
The depth “d” is 18.88 m
Inverse Sine, Cosine, Tangent
Sine, Cosine and Tangent are all based on a Right-Angled Triangle and they are very similar functions therefore we will look at the Sine Function and then Inverse Sine to learn what it is all about.
Sine Function
The Sine of angle θ is:
• the length of the side Opposite angle θ
• divided by the length of the Hypotenuse
Or more simply:
sin(θ) = Opposite / Hypotenuse
Example: What is the sine of 35°?
With the use of this triangle (lengths are only to one decimal place):
sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57…
The Sine Function can assist us to solve questions like this:
Example: Use the sine function to find “d”
We know
• The angle the cable makes with the seabed is 39°
• The cable’s length is 30 m.
And we want to know “d” (the distance down).
| Begin with: | sin 39° = opposite/hypotenuse |
| sin 39° = d/30 | |
| Swap Sides: | d/30 = sin 39° |
| Make use of a calculator to find sin 39°: | d/30 = 0.6293… |
| Multiply both sides by 30: | d = 0.6293… x 30 |
| d = 18.88 to 2 decimal places. |
The depth “d” is 18.88 m
Inverse Sine
But sometimes it is the angle that we are required to find.
This is where “Inverse Sine” comes in.
It answers the question “what angle has sine equal to opposite/hypotenuse?”
The symbol for inverse sine is sin-1
Example: Find the angle “a”
We are given:
• The distance down is 18.88 m.
• The cable’s length is 30 m.
And we want to know the angle “a”
| Begin with: | sin a° = opposite/hypotenuse |
| sin a° = 18.88/30 | |
| Calculate 18.88/30: | sin a° = 0.6293… |
| What angle has sine equal to 0.6293…? The Inverse Sine will tell us. |
|
| Inverse Sine: | a° = sin-1(0.6293…) |
| Make use of your calculator to find sin-1(0.6293…): | a° = 39.0° (to 1 decimal place) |
The angle “a” is 39.0°
They Are Like Forward and Backwards!
• The Sine function sin takes an angle and offers us the ratio “opposite/hypotenuse”
• Inverse Sine sin-1 takes the ratio “opposite/hypotenuse” and offers us the angle.
Example:
| Sine Function: | sin(30°) = 0.5 |
| Inverse Sine: | sin-1(0.5) = 30° |
More Than One Angle!
Inverse Sine merely shows you one angle … but there are more angles that could work.
Example: Here are two angles where opposite/hypotenuse = 0.5
In fact there are infinitely a lot of angles, because you can keep adding (or subtracting) 360°:
Remember this, due to this there are times when you in reality would require one of the other angles!
Summary
The Sine of angle θ is:
sin(θ) = Opposite / Hypotenuse
And Inverse Sine is :
sin-1 (Opposite / Hypotenuse) = θ
What About “cos” and “tan” … ?
Precisely, the same idea.
COSINE
The Cosine of angle θ is:
cos(θ) = Adjacent / Hypotenuse
And Inverse Cosine is :
cos-1 (Adjacent / Hypotenuse) = θ
Example: Find the size of angle a°
cos a° = Adjacent / Hypotenuse
cos a° = 6,750/8,100 = 0.8333…
a° = cos-1 (0.8333…) = 33.6° (to 1 decimal place)
TANGENT
The Tangent of angle θ is:
tan(θ) = Opposite / Adjacent
Therefore Inverse Tangent is :
tan-1 (Opposite / Adjacent) = θ
Example: Find the size of angle x° in the above diagram
tan x° = Opposite / Adjacent
tan x° = 300/400 = 0.75
x° = tan-1 (0.75) = 36.9° (correct to 1 decimal place)
Other Names
Sometimes sin-1 is called asin or arcsin.
Likewise cos-1 is called acos or arccos
And tan-1 is called atan or arctan.
Examples:
• arcsin(y) is the same as sin-1(y)
• atan(θ) is the same as tan-1(θ)
• etc.
Less Common Functions
There are 3 other functions where we divide one side by another, but they are not used very frequently.
They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan:
| Secant Function: | sec(θ) = Hypotenuse / Adjacent | (=1/cos) |
| Cosecant Function: | csc(θ) = Hypotenuse / Opposite | (=1/sin) |
| Cotangent Function: | cot(θ) = Adjacent / Opposite | (=1/tan) |
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