Trigonometry – JEE Main Maths Formula Sheet & Class 11 Notes | All Formulas, Identities & Equations

Angles, Radian Measure, and Conversion — Trigonometry Foundations

Why Angle Measurement Formulas Are Direct JEE Main Trigonometry Questions

Angle in trigonometry: A figure traced by rotating a ray about its endpoint. Anticlockwise rotation = positive angle; clockwise rotation = negative angle. An angle has an initial side (starting ray) and terminal side (final ray).

Radian (circular measure) in trigonometry: 1 radian = angle subtended at the centre of a circle by an arc equal in length to the radius. Notation: 1ᶜ. When no unit is mentioned for an angle, it is always in radians.

Arc-angle relation (fundamental trigonometry formula): θ = l/r where θ is in radians, l = arc length, r = radius. So l = rθ and r = l/θ. Arc AB ≈ Chord AB for very small angles θ.

Degree–radian conversion (trigonometry formula for JEE Main): π radians = 180°. So: 1° = π/180 radians; 1 radian = 180/π degrees ≈ 57.3°. To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.

Key angle conversions in trigonometry (JEE Main values to memorise): 30° = π/6; 45° = π/4; 60° = π/3; 90° = π/2; 120° = 2π/3; 135° = 3π/4; 150° = 5π/6; 180° = π; 270° = 3π/2; 360° = 2π.

Interior angles of polygon (trigonometry application): Sum of interior angles of n-sided polygon = (2n–4) × 90° = (n–2) × π radians. Each interior angle of a regular polygon of n sides = (n–2)·180°/n. Download the Free PDF for Trigonometry for all angle measurement examples and polygon formulas.

Trigonometric Functions — Definitions, Domain, Range, and ASTC Rule

Why Domain-Range and ASTC Sign Rules Are Core Trigonometry JEE Main Questions

Definitions of the six trigonometric functions (trigonometry fundamentals): For angle θ in a right triangle with perpendicular P, base B, hypotenuse H: sin θ = P/H; cos θ = B/H; tan θ = P/B; cot θ = B/P; sec θ = H/B; cosec θ = H/P.

ASTC rule for signs of trigonometric functions (JEE Main trigonometry): Divide the xy-plane into four quadrants: Q1 (0 to 90°): ALL positive. Q2 (90° to 180°): Sin (and cosec) positive. Q3 (180° to 270°): Tan (and cot) positive. Q4 (270° to 360°): Cos (and sec) positive. Memory aid: "All Students Take Calculus" or "Add Sugar To Coffee." Mnemonic from the PDF: Each ratio and its reciprocal share the same sign.

Domain and Range of all 6 trigonometric functions (trigonometry JEE Main table):

sin θ: Domain = ℝ; Range = [–1, 1]

cos θ: Domain = ℝ; Range = [–1, 1]

tan θ: Domain = ℝ – {(2n+1)π/2 : n∈ℤ}; Range = ℝ

cot θ: Domain = ℝ – {nπ : n∈ℤ}; Range = ℝ

sec θ: Domain = ℝ – {(2n+1)π/2 : n∈ℤ}; Range = (–∞, –1] ∪ [1, ∞)

cosec θ: Domain = ℝ – {nπ : n∈ℤ}; Range = (–∞, –1] ∪ [1, ∞)

Trigonometric function ranges (trigonometry inequalities — JEE Main): –1 ≤ sin θ ≤ 1; |sin θ| ≤ 1. –1 ≤ cos θ ≤ 1; |cos θ| ≤ 1. 0 ≤ sin²θ ≤ 1; 0 ≤ cos²θ ≤ 1. cosec θ ≤ –1 or cosec θ ≥ 1. sec θ ≤ –1 or sec θ ≥ 1. If θ is very small: sin θ ≈ θ (in radians) — Trigonometry small angle approximation.

Fundamental trigonometric identities (trigonometry JEE Main — must know): sin²θ + cos²θ = 1 (→ sin²θ = 1–cos²θ; cos²θ = 1–sin²θ). 1 + tan²θ = sec²θ (→ sec²θ – tan²θ = 1). 1 + cot²θ = cosec²θ (→ cosec²θ – cot²θ = 1). Download the Free PDF for Trigonometry for the complete T-ratio domain-range table and fundamental identity applications.

Allied Angles — Complete Trigonometry Table for JEE Main

Why the Allied Angle Rules Are the Fastest-Answer Trigonometry Section in JEE Main

Allied angles in trigonometry are the angles –θ, 90°±θ, 180°±θ, 270°±θ, and 360°±θ. Every trigonometric function of an allied angle can be expressed in terms of the same (or complementary) function of θ. The rules have two parts: (1) find the sign using ASTC in the appropriate quadrant; (2) find the function (changes or stays) based on whether the allied angle involves 90°/270° or 180°/360°.

Allied angle rules for trigonometry (from the PDF — complete table):

Rule 1 (Sign): Use ASTC to determine the sign of the given function in the quadrant where the allied angle falls (treating θ as acute).

Rule 2 (Function change): (a) If the multiple is 0°, 180°, 360° (i.e., multiples of π): NO change of function — sin stays sin, cos stays cos, tan stays tan, etc. (b) If the multiple is 90°, 270° (i.e., odd multiples of π/2): CHANGE of function — sin ↔ cos; tan ↔ cot; sec ↔ cosec.

Complete Allied Angle Table (Trigonometry — JEE Main direct substitution):

–θ: sin(–θ) = –sin θ; cos(–θ) = cos θ; tan(–θ) = –tan θ

90°–θ (Q1): sin(90°–θ) = cos θ; cos(90°–θ) = sin θ; tan(90°–θ) = cot θ; cot = tan; sec = cosec; cosec = sec

90°+θ (Q2): sin(90°+θ) = cos θ; cos(90°+θ) = –sin θ; tan(90°+θ) = –cot θ

180°–θ (Q2): sin(180°–θ) = sin θ; cos(180°–θ) = –cos θ; tan(180°–θ) = –tan θ

180°+θ (Q3): sin(180°+θ) = –sin θ; cos(180°+θ) = –cos θ; tan(180°+θ) = tan θ

270°–θ (Q3): sin(270°–θ) = –cos θ; cos(270°–θ) = –sin θ; tan(270°–θ) = cot θ

270°+θ (Q4): sin(270°+θ) = –cos θ; cos(270°+θ) = sin θ; tan(270°+θ) = –cot θ

360°–θ (Q4): sin(360°–θ) = –sin θ; cos(360°–θ) = cos θ; tan(360°–θ) = –tan θ

360°+θ: same as θ (periodicity). Download the Free PDF for Trigonometry for the complete allied angle table with all 6 functions.

Compound Angle Formulas — All Trigonometry Addition and Subtraction Formulas

Why Compound Angle Formulas Are the Foundation of All Higher Trigonometry for JEE Main

Compound angle formulas in trigonometry express the trigonometric functions of the sum or difference of two angles in terms of functions of the individual angles. These formulas are the foundation for all subsequent trigonometry — every other formula (double angle, half angle, sum-to-product, product-to-sum) is derived from them.

All compound angle formulas in trigonometry (from the PDF — JEE Main):

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

2. sin(A–B) = sinA cosB – cosA sinB

3. cos(A+B) = cosA cosB – sinA sinB

4. cos(A–B) = cosA cosB + sinA sinB

5. tan(A+B) = (tanA + tanB) / (1 – tanA tanB)

6. tan(A–B) = (tanA – tanB) / (1 + tanA tanB)

7. cot(A+B) = (cotA cotB – 1) / (cotB + cotA)

8. cot(A–B) = (cotA cotB + 1) / (cotB – cotA)

9. sin(A+B)·sin(A–B) = sin²A – sin²B = cos²B – cos²A

10. cos(A+B)·cos(A–B) = cos²A – sin²B = cos²B – sin²A

Three-angle compound formulas (trigonometry — JEE Main):

11. sin(A+B+C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC – sinA sinB sinC

     = cosA cosB cosC (tanA + tanB + tanC – tanA tanB tanC)

12. cos(A+B+C) = cosA cosB cosC – sinA sinB cosC – sinA cosB sinC – cosA sinB sinC

     = cosA cosB cosC (1 – tanA tanB – tanB tanC – tanC tanA)

13. tan(A+B+C) = (tanA + tanB + tanC – tanA tanB tanC) / (1 – tanA tanB – tanB tanC – tanC tanA)

Special compound angle trigonometry results (JEE Main):

14. tan(π/4 + A) = (1 + tanA) / (1 – tanA)

15. tan(π/4 – A) = (1 – tanA) / (1 + tanA)

Trigonometry series sum formulas (from PDF — JEE Main):

16. sinα + sin(α+β) + sin(α+2β) + … + sin(α+(n–1)β) = sin(α + (n–1)β/2) · sin(nβ/2) / sin(β/2)

17. cosα + cos(α+β) + cos(α+2β) + … + cos(α+(n–1)β) = cos(α + (n–1)β/2) · sin(nβ/2) / sin(β/2)

Note: These are sums of n terms in AP with common difference β. Download the Free PDF for Trigonometry for all compound angle derivations and worked examples for JEE Main.

Transformation Formulas — Product-to-Sum and Sum-to-Product Trigonometry

Why Transformation Formulas Are the Most Used Trigonometry Tools in JEE Main Problems

Transformation formulas in trigonometry convert between products of T-functions and sums/differences, and vice versa. They are derived directly from the compound angle formulas.

Product-to-Sum formulas in trigonometry (JEE Main):

1. 2 sinA cosB = sin(A+B) + sin(A–B) [A > B]

2. 2 cosA sinB = sin(A+B) – sin(A–B) [A > B]

3. 2 cosA cosB = cos(A+B) + cos(A–B)

4. 2 sinA sinB = cos(A–B) – cos(A+B)

Sum-to-Product formulas in trigonometry (JEE Main — from PDF):

5. sinC + sinD = 2 sin((C+D)/2) cos((C–D)/2)

6. sinC – sinD = 2 cos((C+D)/2) sin((C–D)/2)

7. cosC + cosD = 2 cos((C+D)/2) cos((C–D)/2)

8. cosC – cosD = –2 sin((C+D)/2) sin((C–D)/2)

Tan and Cot transformation formulas in trigonometry (PDF — JEE Main):

9. tanC + tanD = sin(C+D) / (cosC cosD)

10. tanC – tanD = sin(C–D) / (cosC cosD)

11. cotC + cotD = sin(C+D) / (sinC sinD)

12. cotC – cotD = –sin(C–D) / (sinC sinD)

How to memorise transformation formulas in trigonometry (JEE Main mnemonic): Product-to-sum: 2sincos = Σsin (sin(A+B)+sin(A–B)); 2coscos = Σcos; 2sinsin = Δcos (cos(A–B)–cos(A+B), note minus). Sum-to-product: sinC+sinD → 2sin(sum/2)cos(diff/2); cosC+cosD → 2cos(sum/2)cos(diff/2); cosC–cosD → –2sin(sum/2)sin(diff/2). The minus sign in cosC–cosD sum-to-product is the most commonly missed sign in JEE Main trigonometry. Download the Free PDF for Trigonometry for all transformation formula worked examples.

Multiple and Submultiple Angle Formulas — Double, Triple, Half Angle Trigonometry

Why Double and Triple Angle Formulas Are the Most Calculation-Dense Trigonometry Section for JEE Main

Double angle formulas in trigonometry (from PDF — JEE Main):

sin 2A = 2 sinA cosA = 2tanA / (1+tan²A)

cos 2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A = (1–tan²A)/(1+tan²A)

tan 2A = 2tanA / (1–tan²A); cot 2A = (cot²A – 1) / (2cotA)

Key double angle derived results in trigonometry (JEE Main):

1 + cos 2A = 2cos²A → cos²A = (1+cos2A)/2

1 – cos 2A = 2sin²A → sin²A = (1–cos2A)/2

1 + cos A = 2cos²(A/2); 1 – cos A = 2sin²(A/2) [crucial half-angle results]

Half angle (submultiple) formulas in trigonometry (from PDF — JEE Main):

sin θ = 2sin(θ/2)cos(θ/2) = 2tan(θ/2) / (1+tan²(θ/2))

cos θ = cos²(θ/2) – sin²(θ/2) = 2cos²(θ/2) – 1 = 1 – 2sin²(θ/2) = (1–tan²(θ/2))/(1+tan²(θ/2))

tan θ = 2tan(θ/2) / (1–tan²(θ/2))

Half-angle values from cos θ: sin²(θ/2) = (1–cosθ)/2 → sin(θ/2) = ±√((1–cosθ)/2)

cos²(θ/2) = (1+cosθ)/2 → cos(θ/2) = ±√((1+cosθ)/2)

tan²(θ/2) = (1–cosθ)/(1+cosθ) → tan(θ/2) = sinθ/(1+cosθ) = (1–cosθ)/sinθ

Triple angle formulas in trigonometry (from PDF — JEE Main):

sin 3A = 3sinA – 4sin³A

cos 3A = 4cos³A – 3cosA

tan 3A = (3tanA – tan³A) / (1 – 3tan²A)

cot 3A = (3cotA – cot³A) / (1 – 3cot²A) [from PDF]

Product formula for repeated cosines in trigonometry (from PDF — JEE Main):

cosA · cos2A · cos2²A · cos2³A · … · cos(2^(n–1))A = sin(2ⁿA) / (2ⁿ sinA)

Example: cos(π/7)·cos(2π/7)·cos(4π/7) = sin(8π/7)/(8sin(π/7)) = 1/8.

Important results from double angle (trigonometry — JEE Main shortcuts):

sin A + sin(60°–A) + sin(60°+A) = (3/2)sinA ... wait PDF gives: sin 3A = 4sinA·sin(60°+A)·sin(60°–A) → sinA = (1/4sin3A) × ...; similarly from PDF: cos3A = 4cosA·cos(60°–A)·cos(60°+A).

tan 3A = tanA·tan(60°–A)·tan(60°+A)... these are factorization results in trigonometry. Download the Free PDF for Trigonometry for all multiple angle formula derivations and applications for JEE Main.

Special Angle Values in Trigonometry — 15°, 18°, 22½°, 36°

Why Special Angle Values Are Direct JEE Main One-Mark Trigonometry Questions

Beyond the standard 0°, 30°, 45°, 60°, 90° values, JEE Main tests the following special angle values in trigonometry (all from the PDF):

Trigonometric values at 15° and 75° (from PDF):

sin 15° = cos 75° = (√6–√2)/4

cos 15° = sin 75° = (√6+√2)/4

tan 15° = cot 75° = 2–√3

cot 15° = tan 75° = 2+√3

Trigonometric values at 18°, 36°, 54°, 72° (from PDF):

sin 18° = cos 72° = (√5–1)/4

cos 18° = sin 72° = √(10+2√5)/4

sin 36° = cos 54° = √(10–2√5)/4

cos 36° = sin 54° = (√5+1)/4

Trigonometric values at 22½° (from PDF — JEE Main):

tan 22½° = √2 – 1

cot 22½° = √2 + 1

sin 22½° = √((√2–1)/(2√2)); cos 22½° = √((√2+1)/(2√2))

Standard table values in trigonometry (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, undefined. Pattern for sin: √0/2, √1/2, √2/2, √3/2, √4/2.

Sign/square root results for sin(A/2) ± cos(A/2) in trigonometry (from PDF):

sin A + cos A = ±√(1 + sin A) [sign depends on range of A/2]

Specifically: sin(A/2) + cos(A/2) = +√(1+sinA) if A/2 ∈ [–π/4, 3π/4]; = –√(1+sinA) otherwise.

sin(A/2) – cos(A/2) = +√(1–sinA) if A/2 ∈ [π/4, 5π/4]; = –√(1–sinA) otherwise. Download the Free PDF for Trigonometry for all special angle values and their derivations.

Maximum and Minimum of a cosθ + b sinθ — Trigonometry Optimization Formula

Why the √(a²+b²) Formula Is a Direct JEE Main Trigonometry Result

The expression S = a cosθ + b sinθ appears frequently in JEE Main trigonometry problems. Its maximum and minimum values can be found without calculus using a single formula.

Trigonometry max-min formula (from PDF — JEE Main):

Let r = √(a²+b²). Write: S = a cosθ + b sinθ = r[(a/r)cosθ + (b/r)sinθ]

Let cosφ = a/r and sinφ = b/r (which is consistent since (a/r)² + (b/r)² = 1).

Then S = r[cosφ cosθ + sinφ sinθ] = r cos(θ–φ).

Since –1 ≤ cos(θ–φ) ≤ 1: –√(a²+b²) ≤ a cosθ + b sinθ ≤ √(a²+b²).

Maximum value = √(a²+b²), achieved when θ–φ = 0 → θ = φ = arctan(b/a).

Minimum value = –√(a²+b²), achieved when θ–φ = π.

Extension to trigonometry combinations (JEE Main):

Maximum of a sinθ + b cosθ = √(a²+b²); Minimum = –√(a²+b²).

Maximum of (a sinθ + b cosθ)² = a²+b² (since max of the linear expression is √(a²+b²)).

If p ≤ a cosθ + b sinθ + c ≤ q, then: c – √(a²+b²) ≤ a cosθ + b sinθ + c ≤ c + √(a²+b²).

Maximum of a sin²θ + b sinθ cosθ + c cos²θ: write using double angles: a sin²θ = a(1–cos2θ)/2; b sinθ cosθ = (b/2)sin2θ; c cos²θ = c(1+cos2θ)/2. Then expression = (a+c)/2 + ((c–a)/2)cos2θ + (b/2)sin2θ. Max = (a+c)/2 + √(((c–a)/2)² + (b/2)²) = (a+c)/2 + (1/2)√((c–a)² + b²). Download the Free PDF for Trigonometry max-min formula and all quadratic trigonometry optimization examples.

Conditional Identities — When A+B+C = π (Triangle Trigonometry)

Why Conditional Identities Are Multi-Mark JEE Main Trigonometry Topics

When A+B+C = π (as in a triangle), trigonometric identities take special forms because the constraint creates additional relationships. These are called conditional identities in trigonometry and are directly from the PDF.

Key conditional identities when A+B+C = π (trigonometry — from PDF):

Since A+B+C = π: sin(A+B) = sin(π–C) = sinC. cos(A+B) = cos(π–C) = –cosC.

Also: A/2 + B/2 + C/2 = π/2, so sin((A+B)/2) = cos(C/2); cos((A+B)/2) = sin(C/2).

Standard conditional identities in trigonometry (from PDF — JEE Main):

(i) sin2A + sin2B + sin2C = 4 sinA sinB sinC

(ii) cos2A + cos2B + cos2C = –1 – 4 cosA cosB cosC

(iii) cosA + cosB + cosC = 1 + 4 sin(A/2) sin(B/2) sin(C/2)

(iv) sinA + sinB + sinC = 4 cos(A/2) cos(B/2) cos(C/2)

(v) tanA + tanB + tanC = tanA·tanB·tanC

(vi) cotA cotB + cotB cotC + cotC cotA = 1

(vii) cot(A/2) + cot(B/2) + cot(C/2) = cot(A/2)·cot(B/2)·cot(C/2)

(viii) tan(A/2)tan(B/2) + tan(B/2)tan(C/2) + tan(C/2)tan(A/2) = 1

Proof of key identity (i) using trigonometry conditional: sin2A + sin2B + sin2C = 2sin(A+B)cos(A–B) + 2sinC cosC. Since sin(A+B) = sinC: = 2sinC[cos(A–B) + cosC] = 2sinC[cos(A–B) + cos(π–A–B)] = 2sinC[cos(A–B) – cos(A+B)] = 2sinC·2sinA sinB = 4sinA sinB sinC. Download the Free PDF for Trigonometry for all 8 conditional identities with proofs.

Trigonometric Equations — General Solutions for JEE Main

Why All 8 General Solution Cases Are Direct JEE Main Trigonometric Equation Formulas

Trigonometric equations are equations involving trigonometric functions of an unknown angle. Since all T-functions are periodic, trigonometric equations have infinitely many solutions — the general solution captures all of them using integer n.

General solutions of trigonometric equations (from PDF — all 8 cases for JEE Main):

Case 1: sin θ = 0 → θ = nπ, n ∈ ℤ

Case 2: cos θ = 0 → θ = (2n+1)π/2, n ∈ ℤ

Case 3: tan θ = 0 → θ = nπ, n ∈ ℤ

Case 4: sin θ = sin α → θ = nπ + (–1)ⁿα, n ∈ ℤ; α ∈ [–π/2, π/2]

Case 5: cos θ = cos α → θ = 2nπ ± α, n ∈ ℤ; α ∈ [0, π]

Case 6: tan θ = tan α → θ = nπ + α, n ∈ ℤ; α ∈ (–π/2, π/2)

Case 7: sin θ = 1 → θ = (4n+1)π/2; sin θ = –1 → θ = (4n–1)π/2 = (4n+3)π/2.

       cos θ = 1 → θ = 2nπ; cos θ = –1 → θ = (2n+1)π. n ∈ ℤ.

Case 8: sin²θ = sin²α → θ = nπ ± α, n ∈ ℤ (also valid for cos²θ = cos²α and tan²θ = tan²α)

Solution of a cosθ + b sinθ = c (trigonometric equation from PDF — JEE Main):

Step 1: Divide by √(a²+b²) to get cos(θ–φ) = c/√(a²+b²) where tan φ = b/a.

Step 2: If |c| > √(a²+b²) → no real solution. If |c| ≤ √(a²+b²): θ–φ = ±cos⁻¹(c/√(a²+b²)) + 2nπ → θ = 2nπ ± α + φ where α = cos⁻¹(c/√(a²+b²)).

Important precautions for trigonometric equations in JEE Main (from PDF):

1. Avoid squaring both sides if possible (introduces extraneous solutions). If squared, verify.

2. Never cancel a term containing the unknown (may lose genuine solutions).

3. If tan θ or sec θ involved: θ ≠ (2n+1)π/2. If cot θ or cosec θ involved: θ ≠ nπ.

4. √(f(θ)) is always positive — do not write ±. Example: √(cos²θ) = |cosθ|, not ±cosθ. Download the Free PDF for Trigonometry for all general solution examples and simultaneous equation methods for JEE Main.

Inverse Trigonometric Functions — Domain, Range, and All 8 Properties

Why Inverse Trigonometry Is the Most Property-Rich JEE Main Maths Section

Inverse trigonometric functions assign the principal angle to each value. Because trigonometric functions are periodic and not 1-1 over ℝ, we restrict their domains to make them invertible.

Domain and range of inverse trigonometric functions (from PDF — trigonometry JEE Main):

sin⁻¹x: Domain = [–1, 1]; Range = [–π/2, π/2]

cos⁻¹x: Domain = [–1, 1]; Range = [0, π]

tan⁻¹x: Domain = ℝ; Range = (–π/2, π/2)

cot⁻¹x: Domain = ℝ; Range = (0, π)

sec⁻¹x: Domain = ℝ – (–1, 1); Range = [0, π] – {π/2}

cosec⁻¹x: Domain = ℝ – (–1, 1); Range = [–π/2, π/2] – {0}

Property I — Composition (from PDF — trigonometry inverse):

sin(sin⁻¹x) = x for x ∈ [–1,1]; sin⁻¹(sinθ) = θ only if θ ∈ [–π/2, π/2].

cos(cos⁻¹x) = x for x ∈ [–1,1]; cos⁻¹(cosθ) = θ only if θ ∈ [0, π].

tan(tan⁻¹x) = x for x ∈ ℝ; tan⁻¹(tanθ) = θ only if θ ∈ (–π/2, π/2).

Property II — sin⁻¹(sinθ) etc. (trigonometry — must know for JEE Main):

sin⁻¹(sinx) = x if x ∈ [–π/2, π/2]; = π–x if x ∈ [π/2, 3π/2]; etc. (cyclic reduction to principal range).

Property III — Negative argument (from PDF — trigonometry inverse):

sin⁻¹(–x) = –sin⁻¹x for all x ∈ [–1, 1] (sin⁻¹ is ODD function)

cos⁻¹(–x) = π – cos⁻¹x for all x ∈ [–1, 1]

tan⁻¹(–x) = –tan⁻¹x for all x ∈ ℝ (tan⁻¹ is ODD function)

cosec⁻¹(–x) = –cosec⁻¹x; sec⁻¹(–x) = π – sec⁻¹x; cot⁻¹(–x) = π – cot⁻¹x.

Property IV — Reciprocal arguments (from PDF — trigonometry inverse):

sin⁻¹(1/x) = cosec⁻¹x for |x| ≥ 1

cos⁻¹(1/x) = sec⁻¹x for |x| ≥ 1

tan⁻¹(1/x) = cot⁻¹x for x > 0; = –π + cot⁻¹x for x < 0

cosec⁻¹x = sin⁻¹(1/x) for |x| ≥ 1; sec⁻¹x = cos⁻¹(1/x) for |x| ≥ 1

Property V — Complementary angle sums (from PDF — trigonometry inverse JEE Main):

sin⁻¹x + cos⁻¹x = π/2 for x ∈ [–1, 1]

tan⁻¹x + cot⁻¹x = π/2 for x ∈ ℝ

sec⁻¹x + cosec⁻¹x = π/2 for |x| ≥ 1

Property VI — Sum formulas for inverse trig (from PDF — trigonometry JEE Main):

sin⁻¹x + sin⁻¹y = sin⁻¹(x√(1–y²) + y√(1–x²)) if x²+y² ≤ 1 (or xy < 0 and x²+y² > 1)

sin⁻¹x + sin⁻¹y = π – sin⁻¹(…) if x > 0, y > 0, x²+y² > 1

sin⁻¹x – sin⁻¹y = sin⁻¹(x√(1–y²) – y√(1–x²))

tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1–xy)) if xy < 1

tan⁻¹x + tan⁻¹y = π + tan⁻¹((x+y)/(1–xy)) if x > 0, y > 0, xy > 1

tan⁻¹x + tan⁻¹y = –π + tan⁻¹((x+y)/(1–xy)) if x < 0, y < 0, xy > 1

tan⁻¹x – tan⁻¹y = tan⁻¹((x–y)/(1+xy)) if xy > –1

Property VII — Interconversions (from PDF — trigonometry inverse JEE Main):

sin⁻¹x = cos⁻¹√(1–x²) = tan⁻¹(x/√(1–x²)) = cot⁻¹(√(1–x²)/x) = cosec⁻¹(1/x)

cos⁻¹x = sin⁻¹√(1–x²) = tan⁻¹(√(1–x²)/x) for x > 0

tan⁻¹x = sin⁻¹(x/√(1+x²)) = cos⁻¹(1/√(1+x²)) = cot⁻¹(1/x) for x > 0

Property VIII — Double inverse trig formulas (from PDF — trigonometry JEE Main):

2sin⁻¹x = sin⁻¹(2x√(1–x²)) if |x| ≤ 1/√2; = π – sin⁻¹(2x√(1–x²)) if x > 1/√2; = –π – sin⁻¹(…) if x < –1/√2

2cos⁻¹x = cos⁻¹(2x²–1) if 0 ≤ x ≤ 1; = 2π – cos⁻¹(2x²–1) if –1 ≤ x < 0

2tan⁻¹x = sin⁻¹(2x/(1+x²)) if |x| ≤ 1; = π – sin⁻¹(…) if x > 1; = –π – sin⁻¹(…) if x < –1

2tan⁻¹x = cos⁻¹((1–x²)/(1+x²)) if x ≥ 0; = –cos⁻¹(…) if x < 0

2tan⁻¹x = tan⁻¹(2x/(1–x²)) if |x| < 1; = π + tan⁻¹(…) if x > 1; = –π + tan⁻¹(…) if x < –1

3sin⁻¹x = sin⁻¹(3x–4x³) if |x| ≤ 1/2; 3cos⁻¹x = cos⁻¹(4x³–3x) if 1/2 ≤ x ≤ 1

3tan⁻¹x = tan⁻¹((3x–x³)/(1–3x²)) if |x| < 1/√3; = π + tan⁻¹(…) if x > 1/√3. Download the Free PDF for Trigonometry for all inverse function property applications for JEE Main.