Trigonometry Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Trigonometry — Competishun
Trigonometry Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Ratios · Identities · Compound Angles · Equations · Graphs
Trigonometry is one of the most important and scoring chapters in Class 11 Maths. It carries significant weightage in JEE and is a guaranteed source of marks in board exams. The word "trigonometry" comes from the Greek words "trigon" (triangle) and "metron" (measure), meaning the measurement of triangles.[reference:0]
This chapter covers angle measurement (degrees and radians), trigonometric ratios (sin, cos, tan, etc.), fundamental identities, compound angle formulas, multiple and submultiple angles, transformation formulas, trigonometric equations (principal and general solutions), and graphs of trigonometric functions. Understanding these concepts is essential for success in calculus, physics, and engineering.
This page gives you the complete guide to Trigonometry with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.
AnglesDegrees · Radians
IdentitiesPythagorean · Reciprocal
Compound AnglesSum · Difference
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What is Trigonometry?
Definition: Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles.[reference:1] It is derived from the Greek words 'trigon' (triangle) and 'metron' (measure).[reference:2]
Trigonometry is used extensively in mathematics, physics, engineering, astronomy, and architecture.[reference:3] It forms the foundation for calculus and many other advanced topics.
Angle Measurement
Angles are measured in degrees (sexagesimal system) or radians (circular system).[reference:4]
Trigonometric Ratios
The six ratios (sin, cos, tan, cosec, sec, cot) relate the angles to the sides of a right-angled triangle.[reference:5]
Key Insight: Trigonometry is the bridge between geometry and calculus. Mastering the formulas and identities is essential for solving complex problems in JEE and beyond.
Glossary of Trigonometry Terms — Complete A to Z
Before diving deep into each topic, let's understand the key terminology used in this chapter:
| Term | Definition |
|---|---|
| Angle | The measure of rotation of a revolving line with respect to a fixed line.[reference:6] |
| Degree | A unit of angle measurement where 1° = 60′ and 1′ = 60″.[reference:7] |
| Radian | The angle subtended at the centre of a circle by an arc equal in length to the radius.[reference:8] |
| sin θ | Sine of angle θ = Opposite / Hypotenuse.[reference:9] |
| cos θ | Cosine of angle θ = Adjacent / Hypotenuse.[reference:10] |
| tan θ | Tangent of angle θ = Opposite / Adjacent.[reference:11] |
| cosec θ | Cosecant of angle θ = 1/sin θ.[reference:12] |
| sec θ | Secant of angle θ = 1/cos θ.[reference:13] |
| cot θ | Cotangent of angle θ = 1/tan θ.[reference:14] |
| Pythagorean Identity | sin²θ + cos²θ = 1.[reference:15] |
| Compound Angle | An angle formed by the sum or difference of two or more angles.[reference:16] |
| General Solution | A formula that gives all possible solutions of a trigonometric equation.[reference:17] |
| Mastering these terms is essential for understanding Trigonometry. | |
Angle Measurement — Degrees and Radians
Definition: An angle is the measure of rotation of a revolving line with respect to a fixed line. Angles can be positive (anticlockwise) or negative (clockwise).[reference:18]
Two Systems of Angle Measurement
Sexagesimal System
1° = 60′ (minutes)
1′ = 60″ (seconds)
Full rotation = 360°
Circular System
1 radian = angle subtended by an arc equal to the radius[reference:19]
Full rotation = 2π radians
Relation: 1 radian = 180° / π | 1° = π / 180 radians[reference:20]
Important: In calculus and advanced mathematics, radians are always used. Make sure you are comfortable converting between degrees and radians.
Basic Trigonometric Ratios
Definition: The six trigonometric ratios are defined for a right-angled triangle with respect to an angle θ.[reference:21]
| Ratio | Formula | Reciprocal |
|---|---|---|
| sin θ | Opposite / Hypotenuse | cosec θ = 1/sin θ |
| cos θ | Adjacent / Hypotenuse | sec θ = 1/cos θ |
| tan θ | Opposite / Adjacent = sinθ/cosθ | cot θ = 1/tan θ |
| cosec θ | Hypotenuse / Opposite | sin θ = 1/cosec θ |
| sec θ | Hypotenuse / Adjacent | cos θ = 1/sec θ |
| cot θ | Adjacent / Opposite = cosθ/sinθ | tan θ = 1/cot θ |
| These six ratios form the foundation of all trigonometry.[reference:22] | ||
Table of Standard Angles
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
| cosec | ∞ | 2 | √2 | 2/√3 | 1 |
| sec | 1 | 2/√3 | √2 | 2 | ∞ |
| cot | ∞ | √3 | 1 | 1/√3 | 0 |
| These values must be memorised for quick problem solving.[reference:23] | |||||
Fundamental Trigonometric Identities
Definition: Trigonometric identities are equations that are true for all values of the variable for which the functions are defined.[reference:24]
Pythagorean Identities
Identity 1
sin²θ + cos²θ = 1
The most fundamental identity[reference:25]
Identity 2
1 + tan²θ = sec²θ
From dividing by cos²θ[reference:26]
Identity 3
1 + cot²θ = cosec²θ
From dividing by sin²θ[reference:27]
Reciprocal and Quotient Identities
Reciprocal
sinθ·cosecθ = 1
Also: cosθ·secθ = 1, tanθ·cotθ = 1[reference:28]
Quotient
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ[reference:29]
Negative Angle Identities (Even-Odd)
sin(-θ)
= -sinθ
Odd function[reference:30]
cos(-θ)
= cosθ
Even function[reference:31]
tan(-θ)
= -tanθ
Odd function[reference:32]
cosec(-θ)
= -cosecθ
Odd function[reference:33]
sec(-θ)
= secθ
Even function[reference:34]
cot(-θ)
= -cotθ
Odd function[reference:35]
Key Insight: The Pythagorean identities are the most important. They form the basis of all other identities and are used extensively in problem solving.[reference:36]
Signs of Trigonometric Functions in Different Quadrants
Definition: The sign of each trigonometric function depends on the quadrant in which the angle lies.[reference:37]
Quadrant I
sin: +
cos: +
tan: +
All positive
Quadrant II
sin: +
cos: -
tan: -
sin positive
Quadrant III
sin: -
cos: -
tan: +
tan positive
Quadrant IV
sin: -
cos: +
tan: -
cos positive
Memory Aid — ASTC: All (Q1) · Sin (Q2) · Tan (Q3) · Cos (Q4) are positive.[reference:38]
Compound Angle Formulas
Definition: Compound angle formulas give the trigonometric ratios of the sum or difference of two angles.[reference:39]
sin(A + B)
sinA cosB + cosA sinB
Sum formula[reference:40]
sin(A − B)
sinA cosB − cosA sinB
Difference formula[reference:41]
cos(A + B)
cosA cosB − sinA sinB
Sum formula[reference:42]
cos(A − B)
cosA cosB + sinA sinB
Difference formula[reference:43]
tan(A + B)
(tanA + tanB) / (1 − tanA tanB)
Sum formula[reference:44]
tan(A − B)
(tanA − tanB) / (1 + tanA tanB)
Difference formula[reference:45]
Additional Compound Angle Results
sin(A+B) sin(A−B)
sin²A − sin²B = cos²B − cos²A
Product identity[reference:46]
cos(A+B) cos(A−B)
cos²A − sin²B = cos²B − sin²A
Product identity[reference:47]
Important: Compound angle formulas are the backbone of trigonometry. They are used to derive all other identities and are essential for solving complex problems.
Multiple and Submultiple Angles
Definition: Multiple angle formulas express trigonometric functions of 2θ, 3θ, etc., in terms of functions of θ.[reference:48]
Double Angle Formulas
sin 2θ
2sinθ cosθ = 2tanθ/(1+tan²θ)
Double angle[reference:49]
cos 2θ
cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
Three forms[reference:50]
tan 2θ
2tanθ/(1 − tan²θ)
Double angle[reference:51]
Triple Angle Formulas
sin 3θ
3sinθ − 4sin³θ
Triple angle[reference:52]
cos 3θ
4cos³θ − 3cosθ
Triple angle[reference:53]
tan 3θ
(3tanθ − tan³θ)/(1 − 3tan²θ)
Triple angle[reference:54]
Key Insight: Double and triple angle formulas are frequently used in integration, differentiation, and solving trigonometric equations.
Transformation Formulas — Sum to Product & Product to Sum
Definition: Transformation formulas convert sums/differences of trigonometric functions into products, and vice versa.[reference:55]
Product to Sum Formulas
sinA sinB
½[cos(A−B) − cos(A+B)]
Product to sum[reference:56]
cosA cosB
½[cos(A−B) + cos(A+B)]
Product to sum[reference:57]
sinA cosB
½[sin(A+B) + sin(A−B)]
Product to sum[reference:58]
cosA sinB
½[sin(A+B) − sin(A−B)]
Product to sum[reference:59]
Sum to Product Formulas
sinC + sinD
2 sin[(C+D)/2] cos[(C−D)/2]
Sum to product[reference:60]
sinC − sinD
2 cos[(C+D)/2] sin[(C−D)/2]
Difference to product[reference:61]
cosC + cosD
2 cos[(C+D)/2] cos[(C−D)/2]
Sum to product[reference:62]
cosC − cosD
−2 sin[(C+D)/2] sin[(C−D)/2]
Difference to product[reference:63]
Important: Transformation formulas are essential for simplifying expressions and solving trigonometric equations quickly.
Trigonometric Equations — Principal and General Solutions
Definition: A trigonometric equation is an equation involving trigonometric functions. The solution for 0 ≤ θ < 2π is the principal solution. The general solution gives all possible solutions using an integer n.[reference:64]
General Solutions of Standard Equations
sin θ = sin α
θ = nπ + (−1)ⁿα
n ∈ Z[reference:65]
cos θ = cos α
θ = 2nπ ± α
n ∈ Z[reference:66]
tan θ = tan α
θ = nπ + α
n ∈ Z[reference:67]
Special Cases
sin θ = 0
θ = nπ
n ∈ Z[reference:68]
cos θ = 0
θ = (2n+1)π/2
n ∈ Z[reference:69]
tan θ = 0
θ = nπ
n ∈ Z
sin²θ = sin²α
θ = nπ ± α
n ∈ Z[reference:70]
cos²θ = cos²α
θ = nπ ± α
n ∈ Z[reference:71]
tan²θ = tan²α
θ = nπ ± α
n ∈ Z[reference:72]
Golden Rule: Always find the principal solution first (0 ≤ θ < 2π), then use the general solution formulas to get all possible values.
Conditional Identities — When A + B + C = π
Definition: When A + B + C = π (as in a triangle), special trigonometric identities arise.[reference:73]
sin(A+B)
= sinC
Since A+B = π−C[reference:74]
cos(A+B)
= −cosC
Since A+B = π−C[reference:75]
sin2A + sin2B + sin2C
= 4 sinA sinB sinC
Key identity[reference:76]
cos2A + cos2B + cos2C
= −1 − 4 cosA cosB cosC
Key identity[reference:77]
cosA + cosB + cosC
= 1 + 4 sin(A/2) sin(B/2) sin(C/2)
Key identity[reference:78]
sinA + sinB + sinC
= 4 cos(A/2) cos(B/2) cos(C/2)
Key identity[reference:79]
tanA + tanB + tanC
= tanA · tanB · tanC
Key identity[reference:80]
cotA cotB + cotB cotC + cotC cotA
= 1
Key identity[reference:81]
Important: Conditional identities are frequently tested in JEE Main and Advanced, especially in triangle-related problems.
Maximum and Minimum Values of Trigonometric Expressions
Definition: The expression a cosθ + b sinθ can be written in the form r cos(θ−φ), where r = √(a² + b²).[reference:82]
a cosθ + b sinθ
Max = √(a²+b²)
Min = −√(a²+b²)[reference:83]
a sinθ + b cosθ
Max = √(a²+b²)
Min = −√(a²+b²)[reference:84]
(a cosθ + b sinθ)²
Max = a²+b²
Min = 0[reference:85]
a sin²θ + b sinθcosθ + c cos²θ
Max = (a+c)/2 + ½√((a−c)²+b²)
Min = (a+c)/2 − ½√((a−c)²+b²)[reference:86]
Key Insight: The method of writing a cosθ + b sinθ as r cos(θ−φ) is extremely useful for finding maximum and minimum values in JEE problems.[reference:87]
Practice Questions — From JEE and Boards
| Question | Answer |
|---|---|
| Q1: Find the value of sin 75° using compound angle formulas.[reference:88] | sin75° = sin(45°+30°) = sin45°cos30° + cos45°sin30° = (√6+√2)/4. |
| Q2: Simplify: (sin75° + sin15°)/(cos75° + cos15°).[reference:89] | Using sum-to-product formulas, the expression simplifies to tan45° = 1. |
| Q3: Find the general solution of sin x = 1/2. | sin x = sin(π/6). General solution: x = nπ + (−1)ⁿ(π/6), n ∈ Z.[reference:90] |
| Q4: Find the maximum value of sin x + cos x. | Max = √(1²+1²) = √2.[reference:91] |
| Q5: Prove that: (sinA + sinB)/(cosA + cosB) = tan((A+B)/2).[reference:92] | Using sum-to-product formulas: LHS = [2sin((A+B)/2)cos((A−B)/2)]/[2cos((A+B)/2)cos((A−B)/2)] = tan((A+B)/2). |
| Q6: If A + B + C = π, find sin2A + sin2B + sin2C. | = 4 sinA sinB sinC (conditional identity).[reference:93] |
| Q7: Find the general solution of cos x = cos(π/3). | x = 2nπ ± π/3, n ∈ Z.[reference:94] |
| Q8: Simplify: sin7x + sin3x into product form.[reference:95] | 2 sin5x cos2x. |
| Practise these types of questions to become comfortable with applying Trigonometry concepts in exam scenarios. | |
All Trigonometry Formulas at a Glance
| Category | Formula |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 · 1 + tan²θ = sec²θ · 1 + cot²θ = cosec²θ[reference:96] |
| Reciprocal | cosecθ = 1/sinθ · secθ = 1/cosθ · cotθ = 1/tanθ[reference:97] |
| Compound Angles | sin(A±B) = sinA cosB ± cosA sinB · cos(A±B) = cosA cosB ∓ sinA sinB[reference:98] |
| Double Angle | sin2θ = 2sinθcosθ · cos2θ = cos²θ−sin²θ = 2cos²θ−1 = 1−2sin²θ[reference:99] |
| Triple Angle | sin3θ = 3sinθ−4sin³θ · cos3θ = 4cos³θ−3cosθ[reference:100] |
| Product to Sum | sinA sinB = ½[cos(A−B)−cos(A+B)] · cosA cosB = ½[cos(A−B)+cos(A+B)][reference:101] |
| Sum to Product | sinC+sinD = 2sin[(C+D)/2]cos[(C−D)/2] · cosC+cosD = 2cos[(C+D)/2]cos[(C−D)/2][reference:102] |
| General Solutions | sinθ=sinα → θ=nπ+(−1)ⁿα · cosθ=cosα → θ=2nπ±α · tanθ=tanα → θ=nπ+α[reference:103] |
| Max-Min | −√(a²+b²) ≤ a cosθ + b sinθ ≤ √(a²+b²)[reference:104] |
| Memorise these formulas for Trigonometry. They are the key to scoring full marks in this chapter. | |
Common Mistakes in Trigonometry
- Confusing degrees and radians: Always check whether the angle is in degrees or radians before applying formulas. In calculus, radians are always used.
- Forgetting the signs in different quadrants: Use the ASTC rule to determine the sign of trigonometric functions.[reference:105]
- Misapplying compound angle formulas: Pay attention to the signs in sin(A−B) and cos(A−B).[reference:106]
- Forgetting the general solution formulas: sinθ = sinα gives θ = nπ + (−1)ⁿα, not nπ ± α.[reference:107]
Not checking the domain of inverse trigonometric functions: sin⁻¹x is defined only for x ∈ [-1, 1]. Understanding the domain and range is essential for solving ITF problems correctly.
Key Insight: The domain and range of inverse trigonometric functions are the single most important thing to memorise. Every problem in this chapter starts with checking them.
Important Properties of Inverse Trigonometric Functions
1. Negative Arguments
sin⁻¹(−x)
= −sin⁻¹x
Odd function
tan⁻¹(−x)
= −tan⁻¹x
Odd function
cos⁻¹(−x)
= π − cos⁻¹x
Even function (shifted)
cot⁻¹(−x)
= π − cot⁻¹x
Even function (shifted)
2. Complementary Pairs (Sum = π/2)
sin⁻¹x + cos⁻¹x
= π/2
Valid for x ∈ [-1, 1]
tan⁻¹x + cot⁻¹x
= π/2
Valid for x ∈ R
sec⁻¹x + cosec⁻¹x
= π/2
Valid for |x| ≥ 1
3. Reciprocal Arguments
sin⁻¹(1/x)
= cosec⁻¹x
x ∈ [-1, 0) ∪ (0, 1]
cos⁻¹(1/x)
= sec⁻¹x
x ∈ [-1, 0) ∪ (0, 1]
tan⁻¹(1/x)
= cot⁻¹x
x > 0
4. Conversion Formulas
sin⁻¹x
= tan⁻¹(x/√(1−x²))
|x| < 1
cos⁻¹x
= tan⁻¹(√(1−x²)/x)
x > 0
tan⁻¹x
= sin⁻¹(x/√(1+x²))
x ∈ R
tan⁻¹x
= cos⁻¹(1/√(1+x²))
x ≥ 0
Key Insight: These properties are essential for simplifying expressions and solving equations involving inverse trigonometric functions. Memorise them thoroughly.
Addition and Subtraction Formulas for ITF
tan⁻¹x + tan⁻¹y
tan⁻¹((x+y)/(1−xy))
xy < 1
tan⁻¹x + tan⁻¹y
π + tan⁻¹((x+y)/(1−xy))
xy > 1, x > 0, y > 0
tan⁻¹x + tan⁻¹y
−π + tan⁻¹((x+y)/(1−xy))
xy > 1, x < 0, y < 0
tan⁻¹x − tan⁻¹y
tan⁻¹((x−y)/(1+xy))
xy > −1
2tan⁻¹x
sin⁻¹(2x/(1+x²))
|x| ≤ 1
2tan⁻¹x
cos⁻¹((1−x²)/(1+x²))
x ≥ 0
2tan⁻¹x
tan⁻¹(2x/(1−x²))
|x| < 1
sin⁻¹x + sin⁻¹y
sin⁻¹(x√(1−y²) + y√(1−x²))
Valid under certain conditions
cos⁻¹x + cos⁻¹y
cos⁻¹(xy − √(1−x²)√(1−y²))
Valid under certain conditions
Important: The addition formulas for tan⁻¹ require careful attention to the sign and value of xy. These conditions are frequently tested in JEE.
Graphs of Inverse Trigonometric Functions
Definition: The graph of an inverse function is the reflection of the original function's graph across the line y = x, after restricting the domain appropriately.
Graph of sin⁻¹x
The graph of sin⁻¹x is the reflection of sin x (restricted to [-π/2, π/2]) across the line y = x.
Graph of tan⁻¹x
tan⁻¹x has horizontal asymptotes at y = π/2 and y = −π/2.
Key Insight: The graphs of inverse trigonometric functions are reflections of the restricted trigonometric functions across the line y = x. Understanding these graphs helps in visualising domain and range.
Practice Questions — From JEE and Boards
| Question | Answer |
|---|---|
| Q1: Find the principal value of sin⁻¹(1/2). | sin⁻¹(1/2) = π/6 (since sin π/6 = 1/2 and π/6 ∈ [-π/2, π/2]). |
| Q2: Find the principal value of cos⁻¹(−1/2). | cos⁻¹(−1/2) = 2π/3 (since cos 2π/3 = −1/2 and 2π/3 ∈ [0, π]). |
| Q3: Evaluate sin⁻¹(1/2) + cos⁻¹(1/2). | sin⁻¹(1/2) + cos⁻¹(1/2) = π/2 (complementary pair property). |
| Q4: Find the value of tan⁻¹(1) + cot⁻¹(1). | tan⁻¹(1) + cot⁻¹(1) = π/2. |
| Q5: Evaluate sin⁻¹(−1/2). | sin⁻¹(−1/2) = −sin⁻¹(1/2) = −π/6. |
| Q6: Find the value of cos⁻¹(−1/2) using the property. | cos⁻¹(−1/2) = π − cos⁻¹(1/2) = π − π/3 = 2π/3. |
| Q7: Evaluate tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3). | tan⁻¹(1) = π/4. tan⁻¹(2) + tan⁻¹(3) = tan⁻¹((2+3)/(1−6)) = tan⁻¹(−1) = −π/4. Sum = 0. |
| Q8: Find the domain of f(x) = sin⁻¹(2x). | Domain: −1 ≤ 2x ≤ 1 → x ∈ [-1/2, 1/2]. |
| Practise these types of questions to become comfortable with applying Inverse Trigonometric Functions concepts in exam scenarios. | |
All ITF Formulas at a Glance
| Formula | What It Means |
|---|---|
| sin⁻¹(−x) = −sin⁻¹x | Odd function |
| cos⁻¹(−x) = π − cos⁻¹x | Even function (shifted) |
| tan⁻¹(−x) = −tan⁻¹x | Odd function |
| sin⁻¹x + cos⁻¹x = π/2 | Complementary pairs |
| tan⁻¹x + cot⁻¹x = π/2 | Complementary pairs |
| sec⁻¹x + cosec⁻¹x = π/2 | Complementary pairs |
| sin⁻¹(1/x) = cosec⁻¹x | Reciprocal arguments |
| cos⁻¹(1/x) = sec⁻¹x | Reciprocal arguments |
| tan⁻¹(1/x) = cot⁻¹x | Reciprocal arguments |
| tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) | Addition formula |
| 2tan⁻¹x = sin⁻¹(2x/(1+x²)) | Double angle (sin) |
| 2tan⁻¹x = cos⁻¹((1−x²)/(1+x²)) | Double angle (cos) |
| Memorise these formulas for Inverse Trigonometric Functions. They are the key to scoring full marks in this chapter. | |
Common Mistakes in ITF
- Confusing domain and range: Remember that the domain of sin⁻¹x is [-1, 1] and the range is [-π/2, π/2]. The domain of tan⁻¹x is R and the range is (-π/2, π/2).
- Forgetting the principal value branch: Always use the principal value branch when evaluating inverse trigonometric functions.
- Misapplying the addition formulas: The formula tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) has conditions on xy. Check the conditions before applying.
- Confusing sin⁻¹x with (sin x)⁻¹: sin⁻¹x is the inverse sine function, not the reciprocal of sine (which is cosec x).
- Not checking the domain of composite functions: When dealing with composite functions like sin⁻¹(2x), always check the domain of the inner function.
- Forgetting the complementary pair properties: sin⁻¹x + cos⁻¹x = π/2 is a very useful property. Memorise it.
Golden Rule: In Inverse Trigonometric Functions, always check the domain and range before applying any formula. Use the principal value branch for single-valued answers.
Why ITF Matters for JEE and Boards
- Moderate weightage: Inverse Trigonometric Functions appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
- Foundation for calculus: Inverse trigonometric functions are used in integration and differentiation, making them essential for advanced mathematics.
- Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
- Practical relevance: Inverse trigonometric functions are used in physics, engineering, and computer graphics.
Why this guide helps: A comprehensive Inverse Trigonometric Functions guide with all concepts, definitions, formulas, and practice questions saves you time during revision and helps you quickly recall everything during the exam. You won't need to look anywhere else.
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Frequently Asked Questions — Trigonometry
What are the basic trigonometric ratios?
The six basic trigonometric ratios are: sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent, cosec θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.[reference:108]
What are the Pythagorean identities in trigonometry?
The three Pythagorean identities are: (1) sin²θ + cos²θ = 1, (2) 1 + tan²θ = sec²θ, and (3) 1 + cot²θ = cosec²θ.[reference:109]
What are the compound angle formulas?
Compound angle formulas are: sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB, and tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).[reference:110]
What is the general solution of a trigonometric equation?
The general solution of sin θ = sin α is θ = nπ + (-1)ⁿα. For cos θ = cos α, it is θ = 2nπ ± α. For tan θ = tan α, it is θ = nπ + α.[reference:111]
Can I download the Trigonometry formula sheet PDF for free?
Yes. You can download the complete Trigonometry formula sheet PDF for free using the download button on this page. It covers all formulas, identities, compound angles, and trigonometric equations in one comprehensive place for quick revision before JEE and board exams.
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