🏆 GRANDMASTER BIKRAM SUTRADHAR

Table of Contents

📐 TRIGONOMETRY
MASTERCLASS

All Important Formulas • Derivations • Concepts • Smart Methods

Understand the Formula → Derive the Formula → Apply the Formula

A complete student-friendly Trigonometry guide for school mathematics, board examinations and competitive-exam preparation.

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🌟 Welcome to the Trigonometry Masterclass

Welcome students!

I am GrandMaster Bikram Sutradhar, and in this masterclass we are going to build Trigonometry from the foundation to the most important identities and formulas.

But remember one thing: I don’t want you to memorize hundreds of formulas blindly.

My approach is simple:

1️⃣ Understand

Understand where the formula comes from.

2️⃣ Derive

Learn the mathematical reason behind it.

3️⃣ Apply

Use it in questions and examinations.

🎯 GrandMaster Rule:
Don’t ask only, “What is the formula?”Ask: “Why is this formula true?”

🗺️ Complete Trigonometry Roadmap

01. Angle Measurement

Degrees, radians and conversion.

02. Six Trigonometric Ratios

sin, cos, tan, cot, sec and cosec.

03. Reciprocal Relations

Important reciprocal identities.

04. Quotient Relations

tan and cot relationships.

05. Pythagorean Identities

The three fundamental identities.

06. Standard Angles

0°, 30°, 45°, 60°, 90°.

07. Allied Angles

90°, 180°, 270° and 360° transformations.

08. Sum & Difference

sin(A±B), cos(A±B), tan(A±B).

09. Double Angle

sin2A, cos2A and tan2A.

10. Half Angle

sin(A/2), cos(A/2), tan(A/2).

11. Triple Angle

sin3A, cos3A and tan3A.

12. Product-Sum

Convert products into sums.

13. Sum-Product

Convert sums into products.

14. Power Reduction

sin²x and cos²x transformations.

15. t = tan(x/2)

Universal half-angle substitution.

16. Triangle Formulae

Sine rule, cosine rule and area.

17. General Solutions

Solutions of trigonometric equations.

18. Inverse Trigonometry

Important identities and principal values.

📏 1. Angle Measurement

Degree-Radian Conversion

180° = π radians

1° = π/180 radians

1 radian = 180°/π

Derivation:

A complete revolution is 360°. The same complete revolution is 2π radians.

360° = 2π

Divide by 2:

180° = π

📐 2. Six Fundamental Trigonometric Ratios

For a right-angled triangle, relative to an acute angle θ:

Ratio	Formula
sin θ	Perpendicular / Hypotenuse
cos θ	Base / Hypotenuse
tan θ	Perpendicular / Base
cot θ	Base / Perpendicular
sec θ	Hypotenuse / Base
cosec θ	Hypotenuse / Perpendicular

🧠 Memory Trick:SOH – CAH – TOA

Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent

🔄 3. Reciprocal Identities

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Example:Since

sin θ = P/H

and

cosec θ = H/P

therefore

sin θ × cosec θ = 1

Hence

cosec θ = 1/sin θ

➗ 4. Quotient Identities

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

Derivation:

sin θ / cos θ = (P/H)/(B/H)

Cancel H:

= P/B

But

P/B = tan θ

Therefore:

tan θ = sin θ/cos θ

🔥 5. Fundamental Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Derivation of sin²θ + cos²θ = 1

From Pythagoras:

P² + B² = H²

Divide everything by H²:

P²/H² + B²/H² = 1

Therefore:

sin²θ + cos²θ = 1

Derivation of 1 + tan²θ = sec²θ

Start with:

sin²θ + cos²θ = 1

Divide by cos²θ:

tan²θ + 1 = sec²θ

Therefore:

1 + tan²θ = sec²θ

Derivation of 1 + cot²θ = cosec²θ

Divide

sin²θ + cos²θ = 1

by sin²θ:

1 + cot²θ = cosec²θ

⭐ 6. Standard Trigonometric Values

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	Undefined
cosec θ	Undefined	2	√2	2/√3	1
sec θ	1	2/√3	√2	2	Undefined
cot θ	Undefined	√3	1	1/√3	0

💡 Fast Memory Method:For sine:√0/2, √1/2, √2/2, √3/2, √4/2
Then cosine is simply the reverse order.

🧭 7. Signs in Four Quadrants

Quadrant I

All positive.

Quadrant II

Only sin and cosec are positive.

Quadrant III

Only tan and cot are positive.

Quadrant IV

Only cos and sec are positive.

🧠 Memory: All Students Take Calculus

A → AllS → SinT → TanC → Cos

🔁 8. Allied Angle Identities

90° − θ

sin(90°−θ)=cosθ

cos(90°−θ)=sinθ

tan(90°−θ)=cotθ

cot(90°−θ)=tanθ

sec(90°−θ)=cosecθ

cosec(90°−θ)=secθ

180° − θ

sin(180°−θ)=sinθ

cos(180°−θ)=−cosθ

tan(180°−θ)=−tanθ

180° + θ

sin(180°+θ)=−sinθ

cos(180°+θ)=−cosθ

tan(180°+θ)=tanθ

360° − θ

sin(360°−θ)=−sinθ

cos(360°−θ)=cosθ

tan(360°−θ)=−tanθ

🚀 9. Compound Angle Formulas

Sine

sin(A+B)=sinA cosB + cosA sinB

sin(A−B)=sinA cosB − cosA sinB

Cosine

cos(A+B)=cosA cosB − sinA sinB

cos(A−B)=cosA cosB + sinA sinB

Tangent

tan(A+B)=(tanA+tanB)/(1−tanA tanB)

tan(A−B)=(tanA−tanB)/(1+tanA tanB)

Deriving tan(A+B)

Start with:

tan(A+B)=sin(A+B)/cos(A+B)

Substitute:

= [sinA cosB + cosA sinB] / [cosA cosB − sinA sinB]

Divide numerator and denominator by cosA cosB:

= (tanA + tanB)/(1−tanA tanB)

Therefore:

tan(A+B)= (tanA+tanB)/(1−tanA tanB)

⚡ 10. Double Angle Formulas

sin2A = 2sinA cosA

cos2A = cos²A − sin²A

cos2A = 2cos²A − 1

cos2A = 1 − 2sin²A

tan2A = 2tanA/(1−tan²A)

Derivation of sin2A:

sin(A+A)

Using the addition formula:

sinA cosA + cosA sinA

Therefore:

sin2A=2sinAcosA

Derivation of cos2A:

cos(A+A)

Using cosine addition:

cosAcosA − sinAsinA

Therefore:

cos2A=cos²A−sin²A

🌓 11. Half Angle Formulas

sin²(A/2) = (1−cosA)/2

cos²(A/2) = (1+cosA)/2

tan²(A/2) = (1−cosA)/(1+cosA)

tan(A/2)=sinA/(1+cosA)

tan(A/2)=(1−cosA)/sinA

From:

cos2x=1−2sin²x

Put:

x=A/2

Then:

cosA=1−2sin²(A/2)

Therefore:

sin²(A/2)=(1−cosA)/2

Similarly:

cos²(A/2)=(1+cosA)/2

🔥 12. Triple Angle Formulas

sin3A = 3sinA − 4sin³A

cos3A = 4cos³A − 3cosA

tan3A = (3tanA−tan³A)/(1−3tan²A)

Derivation of sin3A:Start with:

sin3A = sin(2A+A)

Therefore:

= sin2A cosA + cos2A sinA

Use:

sin2A=2sinAcosA

and

cos2A=1−2sin²A

After simplification:

sin3A=3sinA−4sin³A

🔋 13. Power Reduction Formulas

sin²A=(1−cos2A)/2

cos²A=(1+cos2A)/2

sinA cosA=(sin2A)/2

🎯 Whenever you see sin²A or cos²A, think about the double-angle formula.

🔄 14. Product-to-Sum Formulas

2sinA cosB = sin(A+B)+sin(A−B)

2cosA sinB = sin(A+B)−sin(A−B)

2cosA cosB = cos(A+B)+cos(A−B)

2sinA sinB = cos(A−B)−cos(A+B)

🧠 These are particularly useful when a product of two trigonometric functions needs to be converted into a sum or difference.

🔁 15. Sum-to-Product Formulas

sinA + sinB = 2sin[(A+B)/2] cos[(A−B)/2]

sinA − sinB = 2cos[(A+B)/2] sin[(A−B)/2]

cosA + cosB = 2cos[(A+B)/2] cos[(A−B)/2]

cosA − cosB = −2sin[(A+B)/2] sin[(A−B)/2]

🧠 16. Universal Substitution: t = tan(x/2)

One extremely useful substitution is:

t = tan(x/2)

Then:

sin x = 2t/(1+t²)

cos x = (1−t²)/(1+t²)

tan x = 2t/(1−t²)

Since:

t=tan(x/2)

Using:

sinx = 2sin(x/2)cos(x/2)

and dividing by cos²(x/2):

sinx = 2tan(x/2)/(1+tan²(x/2))

Therefore:

sinx=2t/(1+t²)

Similarly:

cosx=(1−t²)/(1+t²)

🔺 17. Trigonometry in a Triangle

Sine Rule

a/sinA = b/sinB = c/sinC = 2R

where R is the circumradius.

Cosine Rule

a²=b²+c²−2bc cosA

Similarly:

b²=c²+a²−2ca cosB

c²=a²+b²−2ab cosC

Area Formula

Δ = 1/2 bc sinA

Also:

Δ = 1/2 ca sinB

Δ = 1/2 ab sinC

📚 18. Important Triangle Relations

A+B+C=180°

sin(A+B)=sinC

because:

A+B=180°−C

Therefore:

sin(A+B)=sin(180°−C)=sinC

Half-Angle Triangle Relations

sin(A/2)=√[(s−b)(s−c)/(bc)]

cos(A/2)=√[s(s−a)/(bc)]

tan(A/2)=r/(s−a)

where s=(a+b+c)/2 is the semiperimeter and r is the inradius.

🎯 19. General Solutions of Basic Trigonometric Equations

sin x = sin α

x = nπ + (−1)ⁿ α

where n is any integer.Equivalent form:

x=2nπ+α

x=(2n+1)π−α

—

cos x = cos α

x=2nπ±α

—

tan x = tan α

x=nπ+α

where n ∈ ℤ.

🔙 20. Inverse Trigonometric Functions

sin⁻¹x + cos⁻¹x = π/2

tan⁻¹x + cot⁻¹x = π/2

Important Addition Formula

tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)]

with the appropriate principal-value adjustment depending on x and y.

Important Difference Formula

tan⁻¹x − tan⁻¹y = tan⁻¹[(x−y)/(1+xy)]

again subject to principal-value considerations.

🏆 GrandMaster’s Quick Revision Formula Sheet

Topic	Formula
Basic Identity	sin²x + cos²x = 1
Tan Identity	1 + tan²x = sec²x
Cot Identity	1 + cot²x = cosec²x
Quotient	tanx = sinx/cosx
Double Sine	sin2x = 2sinx cosx
Double Cosine	cos2x = 1−2sin²x
Double Cosine	cos2x = 2cos²x−1
Double Tangent	tan2x = 2tanx/(1−tan²x)
Triple Sine	sin3x = 3sinx−4sin³x
Triple Cosine	cos3x = 4cos³x−3cosx
Sum Sine	sin(A+B)=sinAcosB+cosAsinB
Difference Sine	sin(A−B)=sinAcosB−cosAsinB
Sum Cosine	cos(A+B)=cosAcosB−sinAsinB
Difference Cosine	cos(A−B)=cosAcosB+sinAsinB
Sum Tangent	tan(A+B)=(tanA+tanB)/(1−tanAtanB)
Difference Tangent	tan(A−B)=(tanA−tanB)/(1+tanAtanB)
Half Angle	sin²(x/2)=(1−cosx)/2
Half Angle	cos²(x/2)=(1+cosx)/2
Triangle	a/sinA=b/sinB=c/sinC=2R
Cosine Rule	a²=b²+c²−2bc cosA

🧠 GrandMaster Smart Method

Students, when you see a trigonometric question, don’t immediately start manipulating every term.

STEP 1

Look for a fundamental identity.

STEP 2

Convert tan/cot into sin/cos if useful.

STEP 3

Look for a double-angle pattern.

STEP 4

Check whether numerator and denominator have a common identity.

STEP 5

Use standard-angle values only when required.

STEP 6

Check the final result.

🔥 Ranker Thinking:
Don’t calculate everything.First ask: “What structure is hidden inside this expression?”

🔥 30-SECOND TRIGONOMETRY CHALLENGE

Pause the article and try this yourself.

If tan θ = 3/4, find sin θ and cos θ.

Since:tanθ = P/B = 3/4Take:P = 3, B = 4Then by Pythagoras:H = √(3²+4²) = 5Therefore: sinθ = 3/5 and cosθ = 4/5

📖 How Students Should Study Trigonometry

Day 1

Learn six ratios and reciprocal identities.

Day 2

Master Pythagorean identities.

Day 3

Memorize standard values through understanding.

Day 4

Practice allied angles and quadrant signs.

Day 5

Learn compound-angle identities.

Day 6

Master double, half and triple angles.

Day 7

Practice mixed problems and timed questions.

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❓ Frequently Asked Questions

What is the most important Trigonometry identity?

The fundamental identity sin²θ + cos²θ = 1 is the foundation from which many other identities can be derived.

How can I remember Trigonometric formulas?

Instead of memorizing every formula separately, learn the basic identities and derive related formulas from them.

How can I learn Trigonometry faster?

Build the concepts first, memorize standard values, understand compound-angle identities and then solve progressively harder questions.

Are shortcuts enough for competitive examinations?

Shortcuts can be useful for suitable questions, but conceptual understanding, accuracy and regular practice remain essential.

Can beginners use this guide?

Yes. Start with the six ratios, standard values and fundamental identities before moving to compound and multiple-angle formulas.

How can I contact bAstronautWay?

Call or WhatsApp: 9863002294.