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Trigonometry

Have you ever tried to find the height of a lighthouse without climbing it? Trigonometry makes it possible. By studying the relation between angles and sides of triangles, we can solve real-life and mathematical problems.

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What is 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 perpendicular 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. 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 perpendicular, base, and height of the right-angled triangle respectively.

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History of Trigonometry

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.
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Concepts of Trigonometry

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

 

  • Angles: Measured in degrees or radians.

 

  • Trigonometric ratios: Sine, cosine, tangent, cotangent, secant, and cosecant.

 

  • Heights and distances: Trigonometry and its real-life applications.

 

  • Graphs of trigonometric functions: Helps us understand how sine, cosine, and tangent behave.

 

  • Trigonometric identities: These are basically formulas like sin2\(\theta\) + cos2\(\theta\) = 1.

 

  • Pythagoras’ theorem: It gives us the relation between the three sides of a right-angled triangle.

 

  • 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.

 

 

Quadratic Equation Algebraic Expressions
Geometry Ratio
Dot Product Arithmetic Progression
Logarithm Linear Graph
Inverse Function Complex Numbers
Matrix Multiplication Geometric Sequence
Linear Equations Identity Matrix
Vector Equations Scalar Triple Product
Graphing Linear Equations Polar Form of Complex Numbers
Geometric Probability  
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Trigonometric Ratios

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\(\theta\) = \(opposite\over adjacent\)        

 

tan\(\theta\) = \(sin\over cos\)
 

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Trigonometric Functions

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.


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

 

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Even and Odd Trigonometric Functions

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, flipping 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. 

 

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Trigonometry Ratios for Different Angles

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

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 | 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 \({4 \over 4}\), 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.

 

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Trigonometric Formulas

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

 

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

 

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 

 

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

 

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Real-life Applications of Trigonometry

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.
     

1. Designing with Precision:

1. Designing with Precision:

Architects use trigonometry to design buildings and structures. They calculate angles and lengths to ensure everything fits together properly.

2. Navigating the Skies and Seas:

2. Navigating the Skies and Seas:

Pilots and sailors rely on trigonometry for navigation.

3. Engineering with Accuracy:

3. Engineering with Accuracy:

Trigonometry is important for engineers to design machines and structures.

4. Exploring the Universe and Sports:

4. Exploring the Universe and Sports:

Trigonometric calculations are also used in astronomy and sports.

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Tips and Tricks to Remember Trigonometry

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/hypotenuse
CAH: cosine = adjacent/hypotenuse
TOA: tangent = 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.


5. Use color coding for different ratios and functions.
 

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Common Mistakes and How to Avoid Them in Trigonometry

Hey kids! Trigonometry can be a little tricky sometimes, and we tend to make mistakes. Don't worry, it happens to everyone. Let's get to know some common mistakes that students often make in trigonometry and how to avoid them.
 

Mistake 1

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 Getting confused between trigonometric ratios
 

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Students get confused and merge sine, cosine, and tangent ratios. They sometimes use sine when they have to use cosine. To avoid this, students can remember the SOH-CAH-TOA. 
 

Mistake 2

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Not remembering the unit circle

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Students sometimes ignore the unit circle and its values. They forget that the sine and cosine values are common for angles like 0, 30, 45, 60, and 90 degrees. This can be easily avoided by memorizing the values.
 

Mistake 3

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Not placing the right angle correctly

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Students sometimes forget how to locate the right angle in a triangle. This may lead to incorrect calculations. To avoid this, they should always label the triangles clearly.    
 

Mistake 4

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Confusing degrees with radians

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Students often mix up degrees and radians when solving trigonometric problems. For example, using sin 30 assuming it is in radians instead of degrees will give the wrong value. Always check the unit of the angle before calculating

Mistake 5

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Forgetting the Pythagorean identity

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Students sometimes forget or misuse the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), leading to wrong calculations in problems involving sine or cosine. Always remember the Pythagorean identity and use it carefully to find the unknown trigonometric values.

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Solved Examples of Trigonometry

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Problem 1

Find Sin 30° + cos 60°.

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\(sin30^∘+cos60^∘=1\)
 

Explanation

We know that,

sin 30° = 1/2, cos 60° = 1/2


Add them,
 

sin 30° + cos 60° 

 

Simplify the values

 

1/2 + 1/2 = 1

 

\(sin30^∘+cos60^∘=1\)
 

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Problem 2

If tan θ = 1, find θ in degrees.

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θ = 45°

Explanation

We know that,

tan θ = 1

So, tan \(45^∘\) = 1

Therefore,
 

θ = 45°

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Problem 3

Prove that sin² 45° + cos² 45° = 1

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\(sin^245^∘+cos^245^∘=1\)


 

Explanation

We know that,
 

sin 45° = 1/2, cos 45° = 1/2

 

By using the Pythagorean identity.


sin2 45° + cos2 45°

 

Substituting the values,
 

= (1/√2)2 + (1/√2)2  

 

= 1/2 + 1/2 = 1


\(sin^245^∘+cos^245^∘=1\)

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Problem 4

If sin A = 5/13, find sec A.

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\( \cos A = \frac{12}{13}, \quad \sec A = \frac{13}{12} \)

Explanation

We know that,

sin2 A + cos2 A = 1


cos2A = 1 – (5/13)2


= 1 – (25/169)

Converting 1 to have the same denominator,

1 = 169/169

 

So the equation becomes,

= (169/169) - (25/169)


= 144/169


cos A = 12/13


\( \cos A = \frac{12}{13}, \quad \sec A = \frac{13}{12} \)

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Problem 5

Prove that 1 + tan²θ = sec² θ

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\(1+tan^2θ=sec^2θ\)

Explanation

 We know that,

tan θ = sin θ / cos θ 

Square tan θ 

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

Add 1

1 + tan2 θ = 1 + sinθ / cos2 θ

Write with a common denominator


\( 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} \)

 

Now, use Pythagorean identity


\( \sin^2 \theta + \cos^2 \theta = 1 \)

 

\( \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta \)


Therefore, we get \(1+tan^2θ=sec^2θ\)

 

 

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FAQs on Trigonometry

1.Name the six trigonometric ratios of trigonometry.

The six trigonometric ratios are sine, cosine, tangent, cotangent, cosecant, and secant.
 

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2.What is Cot 0?

There is no value found for cot 0, it is undefined (or) infinity.
 

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3.What is Soh Cah Toa?

 It is a mnemonic that helps us remember the definition of trigonometric functions like sine, cosine, and tangent.
 

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4.How do we find the hypotenuse?

We find the hypotenuse by using the Pythagoras theorem. Add the square of the other two sides, and then take the square root.
 

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5.What are the basics of trigonometry?

The study of relationships between the sides and angles of triangles, particularly right-angled triangles.

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6.Who is the father of trigonometry?

Hipparchus is considered the father of trigonometry.

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7.Can trigonometry be used in real life?

Yes, trigonometry can be used in real-life situations, such as engineering, astronomy, physics, and computer graphics.

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8.How to use trigonometry in medicine?

Trigonometry is used in medicine for medical imaging, such as CT and MRI scans. These scans help reconstruct 3D images from 2D data, enabling doctors to understand the electrical axis and rhythm of the heart. It is also used in ultrasound to calculate the trajectories of needles.

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9.Which engineering field uses trigonometry the most?

Civil engineering, mechanical engineering, and electrical engineering utilize it more for tasks, such as structural analysis, motion dynamics, circuit analysis, and surveying.

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10.How do chemists use trigonometry?

Chemists use trigonometry to calculate bond lengths and bond angles within molecules, model their 3D shapes, and find molecular properties. They also use it to understand the intersections between them.

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11.How do pilots use trigonometry?

Pilots use it to calculate flight paths, descent angles, and wind correction angles. These calculations are based on triangles formed by the aircraft’s position, altitude, distance, and wind vector.

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Explore More Math Topics

From Numbers to Geometry and beyond, you can explore all the important Math topics by selecting from the list below:
 

Numbers Multiplication Tables
Geometry Algebra
Calculus Measurement
Commercial Math Data
Math Formulas Math Questions
Math Calculators Math Worksheets
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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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