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. The three sides of a right-angled triangle are,
The side that is perpendicular to the angle ‘A’ is called perpendicular.
The neighboring side of angle ‘A’ is called the base.
The hypotenuse is the side of the right angle that is opposite the right angle.


By using specific formulas, the trigonometric functions are determined. They are as follows:
Sin A = P / H
Cos A = B / H
Tan A = P/ B
Cosec A = H / P
Sec A = H / B
Cot A = B / P
Here, P, B, and h are the perpendicular, base, and height of the right-angled triangle.
 

Trigonometry originated with the ancient Egyptians and Babylonians for practical purposes, such as architecture and astronomy. However, the 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.
   

In trigonometry, the most crucial triangle is the right-angled triangle. This triangle always has:
 

• One right angle 90°
• Two smaller angles that are less than 90°
• Right-angled triangle: The right-angled triangle is one of the types of triangles whose one angle is 90 degrees.

Let’s understand the three sides of the 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.
• Pythagoras' theorem: This is a fundamental rule that helps us solve a triangle.
• It is given as: (Hypotenuse)2=(Opposite side)2 + (Adjacent side)2

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.


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.
 

   \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
 

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, we flip the sine fraction. It is how we compare the 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, we flip the cosine fraction. It is how we compare the 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. It is how we compare the adjacent side to the opposite side.

\( \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \;.\; \cot(\theta) = \frac{1}{\tan(\theta)} \)

• Odd functions: In trigonometry, we call a function f(x) odd if the function satisfies the following property: f(−x) = −f(x). This means that flipping the input will also cause a flip in the output.
• sin(-x)= -sin(x)

The sine function flips its sign when the angle is negative, as it is odd.

• tan(-x)= sin(-x)cos(-x) = -tan(x)

As we know, tan x is defined as sin(x)cos(x). The sine function is negative, and the cos function remains the same. The negative sine function affects the tangent, so 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.
 

Even functions: We call a function f(x) even if it satisfies the property f(−x) = f(x). Here, the flipping of the input doesn’t change the output.

• 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 values. 


 

Let's learn about how to create a trigonometric ratios table.


Step 1: Write the first five whole numbers 
Let’s start by writing the first five whole numbers. Imagine we are counting like a game of hopscotch, with some spaces in between each number so we can move to the next position.

0 | 1 | 2 | 3 | 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 \({4 \over 4}\), so 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

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

Simplifying this we get,

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

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

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° = 0/1 = 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° = 1/0 = Not defined.

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

1. Pythagoras trigonometric identities 

\( \begin{aligned} \sin^2 \theta + \cos^2 \theta &= 1 \\ 1 + \tan^2 \theta &= \sec^2 \theta \\ \csc^2 \theta &= 1 + \cot^2 \theta \end{aligned} \)

2. Double angle identities

\( \begin{aligned} \sin(2\theta) &= 2 \sin \theta \cos \theta \\ \cos(2\theta) &= \cos^2 \theta - \sin^2 \theta \\ \tan(2\theta) &= \frac{2 \tan \theta}{1 - \tan^2 \theta} \end{aligned} \)
 

3. Product - Sum identities

\( \begin{aligned} \sin A + \sin B &= 2 \, \sin\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right) \\ \sin A - \sin B &= 2 \, \cos\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right) \\ \cos A + \cos B &= 2 \, \cos\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right) \\ \cos A - \cos B &= -2 \, \sin\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right) \end{aligned} \)

4. Sum and difference identities 

\( \begin{aligned} \sin(A+B) &= \sin A \cos B + \cos A \sin B \\ \sin(A-B) &= \sin A \cos B - \cos A \sin B \\ \cos(A+B) &= \cos A \cos B - \sin A \sin B \\ \cos(A-B) &= \cos A \cos B + \sin A \sin B \\ \tan(A+B) &= \frac{\tan A + \tan B}{1 - \tan A \tan B} \\ \tan(A-B) &= \frac{\tan A - \tan B}{1 + \tan A \tan B} \end{aligned} \)

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

• Precision: Architects apply trigonometry to design buildings and structures, ensuring perfect angles and measurements.
  
   
• Navigating Air and Sea: Pilots and sailors use trigonometry to navigate safely.
  
   
• Engineering with Precision: Engineers rely on trigonometry to design machines and stable structures.
  
   
• Beyond Earth and Into Sports: Trigonometry plays a vital role in both astronomy and sports, enabling accurate calculations and analysis.
   

Trigonometry is all about angles and sides of a right angle 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/hypotenuse
               CAH: cosine= adjacent/hypotenuse
               TOA: tangent= opposite/adjacent


2. You can also create a fun sentence/phrase to memorize, “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.


5. Use color coding for different ratios and functions.
 

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