**Trigonometry** word is formed from the ancient Greek words “trigonon” and “metron” which mean triangle and measure respectively, thus collectively Trigonometry mean measures of a triangle. Historians believe that in ancient Greece, a mathematician named Hipparchus was the first to introduce the idea of trigonometry by giving the first tables of chords which is the modern-day equivalent to the table of values of trigonometric ratio sine. Other than Greece, roots of this subject are also found in ancient India where Aryabhatta (an Indian mathematician and astronomer) documented the modern intuition of trigonometric ratios.

**Core Trigonometry**(deal with right angle triangles only)**Plane Trigonometry**(deals with all types of 2-dimensional geometry)**Spherical Trigonometry**(deals with all types of 3-dimension geometry)

**Angles:**The measure of space between two intersecting lines are known as angles.**Right-angle Triangle:**A triangle with one of its interior angles being the right angle i.e., 90°, is called right angles triangle.**Pythagoras Theorem:**In right angles triangle, according to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of squares of the other two sides,**Trigonometric Ratios:**Trigonometric Ratios are defined as the ratio of the sides of the right angle triangles. As there are 3 ways to choose two sides out of three and two ways for each chosen pair to arrange in ratio, thus there are 3×2 =6 trigonometric ratios which are defined for each possible pair of sides of the right angle triangle.

- tangent or tan can be defined as the ratio of sin and cos i.e., tan x = sin x/cos x,
- cotangent or cot can be defined as the ratio of cos and sin i.e., cot x = cos x/sin x and
- cosecant or cosec is the inverse of the sin .i.e, cosec x = 1/sinx,
- secant or sec is the inverse of cos i.e., sec x = 1/cos x.

That’s why the only two important trigonometric ratios are sin and cos.

Similarly, cos is defined as the ratio of the base and hypotenuse of the right-angle triangle,

and, trigonometric ratio tan is defined as the ratio of perpendicular to the base of the right-angle triangle.

Other, then sin, cos and tan, sec, cosec and cot are also defined as the ratio of the sides of right angle triangle as follows:

Thus, using these steps, resulting table is formed.

- Trigonometry is very essential for modern-day
**navigation systems**such as GPS or any other similar system. - In most streams of
**engineering,**trigonometry is used extensively for various kinds of analysis and calculations, which helps engineers to make more sound decisions for the construction of various kinds of structures. - Various trigonometric concepts and formulas are used in the
**computer graphics**of the modern age, as computer graphics are created in 3-D environments so all the calculations are done by the graphics processing unit of the computer to deliver the computer graphics as output. - Various
**astronomical calculations**such as the radius of celestial bodies, the distance between objects, etc. involve the use of trigonometry and its different trigonometric ratios. - In
**Physics**, we use trigonometry to understand and evaluate many real-world systems such as the orbits of planets and artificial satellites, the reflection or refraction of light ni various environments, etc.

- sin
^{2}θ + cos^{2}θ = 1 - 1+tan
^{2}θ = sec2 θ - cosec
^{2}θ = 1 + cot^{2}θ

- 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) = (tan A + tan B)/(1 – tan A tan B)
- tan (A-B) = (tan A – tan B)/(1 + tan A tan B)

- sin 2θ = 2 sinθ cosθ
- cos 2θ = cos
^{2}θ – sin^{2}θ = 2 cos^{2 }θ – 1 = 1 – sin^{2}θ - tan 2θ = (2tanθ)/(1 – tan
^{2}θ)

where,

Putting, **-Φ **instead of **Φ **in the above identity, we get

Now, adding and subtracting these two values together we get, values of sin and cos in terms of imaginary power of Euler’s number,

and

Distance = Height / Tan ∝

Let us assume that height is 20m and the angle formed is 45 degrees, then

Distance = 20 / Tan 45°

Since, tan 45° = 1

So, Distance = 20 m

To find: Value of m

To solve m, use the sine ratio.

Sin 72.3° = m/315

0.953 = m/315

m= 315 x 0.953

m=300.195 mtr

The man is 300.195 mtr above the ground.
**Question 3:** A man is observing a pole of height 55 foot. According to his measurement, pole cast a 23 feet long shadow. Can you help him to know the angle of elevation of the sun from the tip of shadow?
**Solution:** Let x be the angle of elevation of the sun, then

tan x = 55/23 = 2.391

x = tan^{-1}(2.391)

or x = 67.30 degrees
**Question 4:** A ladder is leaning against a wall. The angle between the ladder and the ground is 45 degrees and the length of the ladder is 10 meters. How far is the ladder from the wall?
**Solution:** Let the distance between the ladder and the wall be x meters.

Here, ladder, wall and ground together makes a right angle triangle, where for given angle,

Length of ladder = hypotenous = 10 meter,

Distance between wall and laddar = base = x meter

Using trigonometric ratio cos, we get

⇒ cos(45°) = = x/10

⇒ cos(45°) = 1

⇒1/√2 = x/10

⇒ x = 10/√2 = 5√2 meters
**Therefore, the ladder is 5√2 meters away from the wall.**

Now, sin (30°) = perpendicular/hypotenous = a/10

⇒ a = 10 × sin(30°) = 5 cm [sin(30°) = 1/2]

and cos(30°) = b/10

⇒ b = 10 × cos(30°) = 10 × √(3)/2 ≈ 8.66 cm

⇒ LHS = [cos

⇒ LHS = 1/(cosx sinx) [Using cos

⇒ LHS = (1/cosx) × (1/sinx)

⇒ LHS = secx × cosecx = RHS [ 1/cosx = sec x and 1/sinx = cosec x]

And, all the distances here in the question make a right angle triangle, with a base of 10 meters and height h meter.

As tan θ = Perpendicular/Base

⇒ tan(60°) = h / 10

⇒ h = 10 tan(60°)

Using the values of tan(60°) = √3, we get:

h = 10√3 ≈ 17.32 m

As we know, cos

π/2 = π/4, which is not true.

Basic Concepts, Methods, Examples, Math Tricks

Basic Concepts, Formulas and Examples

Basic Concepts, Formulas and Examples

Basic Concepts, Examples, Methods, Shortcuts, Math Tricks

Basic Concepts, Formulas and Examples

Subscribe us for Quantitative Aptitude

Top 16 Psychology and Mental Health Quotes

Top 16 Inspiration and Motivation Quotes

Subscribe us for Daily Inspiration Quotes

Age Problems – Present Age, Current Age, Average Age Average Problems

Average Problems – Average Squares, Natural Numbers, Even Numbers Blood Relations

Blood Relations – Father, Mother, Son, Daughter, Brother, Sister, Uncle, Aunt Boats and Streams

Boats and Streams – Speed Boat, Speed Stream, Upstream, Downstream Cause and Effect

Cause and Effect – Reasoning, Statements, Independent Causes Classification

Classification – Choose the Odd One, three of the following four, alike in a certain way Coding-Decoding

Coding and Decoding – Alphabetical Order, Series, First and Third letters Comparison

Comparison – Sequence, Age, Elder, Younger, Taller, Shorter and Marks Compound Interest

Compound Interest – Interest Payable, Amount, Simple Interest Coordinate Geometry

Coordinate Geometry – Coordinates of a Point, x1, y1, x2, y2, x3, y3, Formulas Cube and Dice

Cube and Dice – Faces of a Cube, Faces of a Dice, Number of Cubes, Sides Painted Data Interpretation

Data Interpretation – Qualitative and Quantitative DI Methods Direction Sense Test

Direction Sense Test – East, West, North and South Embedded Figures

Embedded Figures – Question Figure, Answer Figure Figure Series

Figure Series – Question Figure, Answer Figures Fractions

Fractions – Numerator, Denominator, Types of Fractions Friday Math's Puzzle

Friday Math's Puzzle – Bodmas Rule, Simplification, Math Tricks LCM and HCF

LCM and HCF – Basic Concepts, Examples, Methods, Shortcuts, Math Tricks Mensa Reasoning Puzzles

Mensa Reasoning Puzzles – How to Solve and Reasoning Concept Mensuration

Mensuration – Triangle, Cylinder, Volume, Total Surface Area Mirror Images

Mirror Image – Lateral Inversion, Types of Mirror Images, Original Image Mixtures and Alligations

Mixtures Alligations – Ratio, Quantity, Litres of Milk and Water Number Problems

Number Problems – Fraction, Natural and Prime Numbers Number Series

Number Series – What Number Should Come Next? Order, Ranking and Position

Order, Ranking, Position – Left End, Right End, Answer, Explanation Partnerships

Partnerships – Total Profit, Business, Investment Pattern Completion

Pattern Completion – Question Figure, Answer Figure Percentage Problems

Percentage Problems – Total Marks in Examination, Net% Change Permutation and Combination

Permutation and Combination – How many, Letters, Word, Number Pipes and Cistern

Pipes and Cistern – Three Pipes, Fill Tank, Answer and Explanation Probability Problems

Probability – Bag Contains Blue Balls, Red Balls, Yellow Marbles Profit and Loss

Profit and Loss Problems – Selling Price (SP), Cost Price (CP) Quadratic Equations

Quadratic Equation – Algebra (x², y², a2, b2, a3, b3), Formulas Ratio and Proportion

Ratio and Proportion – Ratio of Two Numbers, Answer, Explanation Seating Arrangement

Seating Arrangement – Left, Right, Centre, Books, English, Hindi Simple Interest

Simple Interest (SI) – How Much, Time, Amount, Sum of Money Simplification

Simplification – BODMAS Rule, Division, Multiplication, Addition, Subtraction Square Root

Square Root – Find Square, Perfect Square, Prime Factorization Statement and Arguments

Statement and Arguments – Reasoning, Strong and Weak Argument Syllogism

Syllogism – Statements, Conclusions, Venn Diagram More Aptitude Test Categories

Quantitative Aptitude Made Simpler

Active and Passive Voice – Rules, Sentence Conversion, Verb, Tense Alphabet Test

Alphabet Test – Meaningful English Word, Arrangement, Letters Cloze Test

Cloze Test – Old Pattern and New Pattern of Cloze Test Direct and Indirect Speech

Direct and Indirect Speech – Rules, Examples, Usage in Sentences Error Spotting – Rules and Tips to Solve Spotting Errors

Error Spotting – Rules and Tips to Solve Spotting Errors Fill in the Blanks

Fill in the Blanks – Types, Paragraph, Excerpt Idioms and Phrases

Idioms and Phrases – Common Idiom and Phrase, Meaning and Examples Multiple Meaning Words

Multiple Meaning Words or Homonyms – Examples, Question and Answer One Word Substitution

One Word Substitution – A list of 190+ One-Word Substitutions Para Jumbles

Para Jumbles – Reading, Paragraph Jumble, Right Sequence Phrase Replacement

Phrase Replacement – Grammatically Correct Phrases and Sentences Reading Comprehension

Reading Comprehension – Examples, Question, Answer and Explanation More English Study Material

Exam Preparation for English

वाक्यांश के लिए एक शब्द Top 100 Synonyms in Hindi

हिंदी में महत्वपूर्ण पर्यायवाची शब्द Important questions on Education Psychology in Hindi

शिक्षा मनोविज्ञान: महत्वपूर्ण प्रश्न Important questions on Child Psychology in Hindi

बाल मनोविज्ञान: महत्वपूर्ण प्रश्न More Resources for CTET, TET, UPTET, Teacher Exam

Study Material and Solved Exam Papers