Trigonometry

The word trigonometry is derived from the Greek words 'trigonon,' meaning triangle, and 'metron,' meaning measure. The principles of trigonometry include the measurement of angles and problems involving angles.

Three basic functions of trigonometry are sine, cosine, and tangent. These functions can be used to build other critical trigonometric functions such as cotangent, secant, and cosecant. It is important to know the three sides of a right-angled triangle to understand trigonometry. The three sides are:
 

• Perpendicular: The side that is opposite to the angle ‘A’.
   
• Base: The neighboring side of angle ‘A’.
   
• Hypotenuse: It is the side opposite to the right angle in a right-angled triangle.

The trigonometric functions are determined by using specific formulas. The trigonometric formulas are as follows:

1. \(\text {Sin A} = {P \over H }\)
    
2. \(\text {Cos A} = {B \over H}\)
    
3. \(\text{Tan A} = {P\over B}\)
    
4. \(\text{Cosec A} = {H \over P}\)
    
5. \(\text{Sec A} = {H \over B}\)
    
6. \(\text{Cot A} = {B \over P}\)

Here, P, B, and H are perpendicular, base, and height of the right-angled triangle respectively.

The ancient Egyptians and Babylonians first developed trigonometry to help with practical tasks like building structures and studying the stars. However, Greek mathematician Hipparchus developed it as a mathematical discipline in the 2nd century BCE, creating the first trigonometric tables using chords.

 

• Ancient Egyptians employed early forms of trigonometry to construct their pyramids.
  
   
• The Greeks shared an equal interest in triangles. Hipparchus, the father of trigonometry, created the first trigonometric tables.
  
   
• The Greeks shared an equal interest in triangles. Hipparchus, the father of trigonometry, created the first trigonometric tables.
  
   
• Indian mathematicians, such as Aryabhata, developed advanced concepts regarding angles.
  
   
• Furthermore, Islamic scholars developed additional ideas, such as using trigonometry to navigate across deserts and seas.

The concepts of trigonometry refer to the important ideas that the subject covers. Let’s take a look at these concepts one by one.

1. Angles: Measured in degrees or radians.
    
2. Trigonometric ratios: Sine, cosine, tangent, cotangent, secant, and cosecant.
    
3. Heights and distances: Trigonometry and its real-life applications.
    
4. Graphs of trigonometric functions: Helps us understand how sine, cosine, and tangent behave.
    
5. Trigonometric identities: These are basically formulas like sin²\(\theta\) + cos²\(\theta\) = 1.
    
6. Pythagoras’ theorem: It gives us the relation between the three sides of a right-angled triangle.
    
7. Right-angled triangles: This is the foundation of trigonometric ratios.  

Let's now understand the three sides of a right-angled triangle.

• Hypotenuse: It is the longest side/arm of the triangle.  

• Opposite side: The side opposite to the angle that we are supposed to find is called the opposite side.

• Adjacent side: The side next to the angle you are focusing on, but it is not the hypotenuse.

The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.

 

Let's meet the three most important ratios:


1. Sine (sin) - The height finder 
 

Sine of an angle is defined by the ratio of length of sides which is opposite to the angle and the hypotenuse.
   
    \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
 

2. Cosine (cos) - The adjacent to the hypotenuse.

The cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. 

 \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
 

3. Tangent (tan) - The angle decider

Tangent of an angle is defined by the ratio of the length of sides which is opposite to the angle and the side which is adjacent to the angle.
 

 \(​ tan \space\theta = {opposite \over adjacent }\)        

 \(​ tan \space\theta = {sin \space \theta\over cos \space \theta}\)
 

Trigonometric functions are six basic functions where the input is the angle of a right triangle, and the output is a numerical value.


1. Cosecant (Cosec) - The reverse of sine 
It is the reciprocal of sine function. It is defined as the ratio of hypotenuse to the opposite side.
 

\( cosec(\theta) = \frac{\text{Hypotenuse}}{\text{Opposite}} \)

\(cosec(\theta) = \frac{1}{\sin(\theta)} \)



2. Secant (sec) - The reverse of cosine
Secant is the reciprocal of cosine function. It gives the ratio of hypotenuse to the adjacent side.

\( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}}\)

\(\sec(\theta) = \frac{1}{\cos(\theta)} \)
 

3. Cotangent (cot) - The reverse of tangent
Cotangent is the reciprocal of tangent function. It is how we compare the adjacent side to the opposite side.

\( \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \)

\(\cot(\theta) = \frac{1}{\tan(\theta)} \)

The following table represents the trigonometric functions and its relations to the ratio of sides:

The trigonometric functions are classified into even and odd function on the basis of their symmetrical relations. Let's understand each functions individually.

Odd Functions
In trigonometry, a function f(x) is said to be odd if it satisfies the following property: f(−x) = −f(x). This means that flipping the input will also cause a flip in the output.

The following trigonometric functions are odd function:
 

• \(sin(-x)= -sin(x)\)
  
  The sine function flips its sign when the angle is negative, as it is odd.

• \(tan(-x) = \frac{sin(-x)}{cos(-x)} = -tan(x)\)
  
  As we know, tan x is defined as \(sin(x) \over cos(x)\). Since, the sine function is negative, and the cos function remains the same, the negative sine function affects the tangent. Hence, the tan function is also negative.

• \(cosec(-x)= -cosec(x)\)
  
  Here, the cosecant is the reciprocal of sine. As the sine function changes sign, the cosecant function also changes sign.

• 
  \(cot(-x)= -cot(x)\)

  
  Since, cot is the reciprocal of tan function, its sign varies corresponding to the sign of tan.

Even Functions
A function f(x) is said to be an even function if it satisfies the property: f(−x) = f(x). Here, flipping the input doesn’t change the output.

The even trigonometric functions are given below:
 

• \(cos(-x)= cos(x)\)
  
  In the cosine function, the value of x remains positive for negative angles.

• \(sec(-x)= sec(x)\)
  
  The secant function is the reciprocal of cosine, hence it is positive. 

Trigonometric ratios can be applied to common angles. The table given below shows the value of trigonometric ratios at certain angles. 


First, let's learn about how to create a trigonometric ratio table.


Step 1: Write the first five whole numbers 

Let’s start by writing the first five whole numbers. Imagine we are counting in a game of hopscotch, with some spaces in between each number so we can move to the next position.

\(0 \space |\space 1\space | \space2\space | \space3\space |\space 4\)


Step 2: Divide each number by 4

Now, we divide each number by 4. Here, the number is divided by 4 to make it easier for students to understand. 

\( 0 \;\big|\; \tfrac{1}{4} \;\big|\; \tfrac{2}{4} \;\big|\; \tfrac{3}{4} \;\big|\; \tfrac{4}{4} \)
 

When we simplify \({2 \over 4}\) and \({4 \over 4}\), we write it as:

\( 0 \;\big|\; \tfrac{1}{4} \;\big|\; \tfrac{1}{2} \;\big|\; \tfrac{3}{4} \;\big|\; 1 \) 


Step 3: Take the square root of each resulting number

\( 0 \;\big|\; \sqrt{\tfrac{1}{4}} \;\big|\; \sqrt{\tfrac{1}{2}} \;\big|\; \sqrt{\tfrac{3}{4}} \;\big|\; 1 \)


Step 4: Sine values for angles 0°, 30°, 45°, 60°, and 90°

Simplify each values:
 

\( \sin(\theta) : \; 0 \;\big|\; \tfrac{1}{2} \;\big|\; \tfrac{1}{\sqrt{2}} \;\big|\; \tfrac{\sqrt{3}}{2} \;\big|\; 1 \)

Step 5: Reverse the order for cosine values
 

Now, let's have some fun and reverse the order of sine values to find the cosine values. 
 

\( \: \; 1 \;\big|\; \frac{\sqrt{3}}{2} \;\big|\; \frac{1}{\sqrt{2}} \;\big|\; \frac{1}{2} \;\big|\; 0 \)


Step 6: Find the Tangent values

Now we find the tangent values, which are simply the sine that is divided by the cosine.
 

• \( \tan 0^\circ = {0\over 1} = 0\)
   
• \( \tan 30^\circ = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \)
   
• \( \tan 45^\circ = \frac{1/\sqrt{2}}{1/\sqrt{2}} = 1 \)

   

• \( \tan 60^\circ = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \)
   
• \( \tan 90^\circ = 1/0 = \text {Not defined.}\)

The following chart represents the values of sine, cosine and tangent function for basic angles in a tabular form.
 

Trigonometry formulas are used to solve trigonometric problems. These problems may include trigonometric ratios like sin, cos, tan, sec, cosec, and cot. Memorizing these mathematic formulas in trigonometry will help students solve problems easily.

1. Pythagoras trigonometric identities 
 

• \(\sin^2 \theta + \cos^2 \theta = 1\)
   
• \(1 + \tan^2 \theta = \sec^2 \theta\)
   
• \(csc^2 \theta = 1 + \cot^2 \theta \)

2. Double angle identities

• \(sin(2\theta) = 2 \sin \theta \cos \theta \)
   
• \(cos(2\theta) = \cos^2 \theta - \sin^2 \theta \)
   
• \(tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\)
   

3. Product - Sum identities

• \(\sin A + \sin B = 2 \, \sin\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right)\)
   
• \(\sin A - \sin B = 2 \, \cos\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right) \)
   
• \(\cos A + \cos B = 2 \, \cos\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right)\)
   
• \(cos A - \cos B = -2 \, \sin\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right)\)

4. Sum and difference identities 

• \( \sin(A+B) = \sin A \cos B + \cos A \sin B\)
   
• \(\sin(A-B) = \sin A \cos B - \cos A \sin B\)
   
• \(\\cos(A+B) = \cos A \cos B - \sin A \sin B\)
   
• \(\\cos(A-B) = \cos A \cos B + \sin A \sin B\)
   
• \(\\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
   
• \(\\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

Trigonometry is all about angles and sides of a right-angled triangle. Kids can easily get confused by it. To master trigonometry, you can follow certain tips and tricks. In this section, let’s learn a few trigonometry tips and tricks.


1. SOH-CAH-TOA helps you remember the basics of sine, cosine, and tangent ratios:
             

• SOH: \(sine = {opposite \over {hypotenuse}}\)
• CAH: \(cosine = \frac {adjacent} {hypotenuse}\)
• TOA: \(tangent = \frac {opposite} {adjacent}\)

2. You can also create a fun sentence/phrase to memorize, like “Silly Owls Help Cats And Turtles”.


3. You can also make a trigonometric ratio triangle.

 

4. Draw a triangle and label its sides as opposite, adjacent, and hypotenuse to avoid confusion between sides.


5. Use color coding for different ratios and functions.
 

Trigonometry is a field of math that deals with the relationship between angles and sides of triangles. It has many practical applications, from measuring heights and distances to understanding sound and light waves. Trigonometry is applied in various fields like physics, engineering, computer science, etc.

Here are some fun and relatable real-life applications of trigonometry:

 

1. Precision: Architects apply trigonometry to design buildings and structures, ensuring perfect angles and measurements.
   
    
2. Navigating Air and Sea: Pilots and sailors use trigonometry to navigate safely.
   
    
3. Engineering with Precision: Engineers rely on trigonometry to design machines and stable structures.
   
    
4. Beyond Earth and Into Sports: Trigonometry plays a vital role in both astronomy and sports, enabling accurate calculations and analysis.
   
    
5. Computer Graphics: Trigonometry is used to model movements of objects in 2-D plane. They are used for tracking motion in animations, graphic and video games.

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