Cheat Sheet: Trigonometric Ratios, Functions & Equations | Mathematics (Maths) for JEE Main & Advanced PDF Download

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Trigonometric Ratios, Functions, and 
Equations Cheat Sheet 
Trigonometric Ratios 
Definitions 
? sin?: Opposite/Hypotenuse. 
? cos?: Adjacent/Hypotenuse. 
? tan?: Opposite/Adjacent = sin?/cos?. 
? cot?: 1/tan? = cos?/sin?. 
? sec?: 1/cos?. 
? csc?: 1/sin?. 
Standard Angle Values 
? 0° 30° 45° 60° 90° 
sin? 0 1/2 v2/2 v3/2 1 
cos? 1 v3/2 v2/2 1/2 0 
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Trigonometric Ratios, Functions, and 
Equations Cheat Sheet 
Trigonometric Ratios 
Definitions 
? sin?: Opposite/Hypotenuse. 
? cos?: Adjacent/Hypotenuse. 
? tan?: Opposite/Adjacent = sin?/cos?. 
? cot?: 1/tan? = cos?/sin?. 
? sec?: 1/cos?. 
? csc?: 1/sin?. 
Standard Angle Values 
? 0° 30° 45° 60° 90° 
sin? 0 1/2 v2/2 v3/2 1 
cos? 1 v3/2 v2/2 1/2 0 
? 0° 30° 45° 60° 90° 
tan? 0 v3/3 1 v3 undefine
d 
Signs in Quadrants 
? I: All positive. 
? II: sin, csc positive; others negative. 
? III: tan, cot positive; others negative. 
? IV: cos, sec positive; others negative. 
Relationships 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Trigonometric Identities 
Pythagorean Identities 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Sum and Difference Identities 
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Trigonometric Ratios, Functions, and 
Equations Cheat Sheet 
Trigonometric Ratios 
Definitions 
? sin?: Opposite/Hypotenuse. 
? cos?: Adjacent/Hypotenuse. 
? tan?: Opposite/Adjacent = sin?/cos?. 
? cot?: 1/tan? = cos?/sin?. 
? sec?: 1/cos?. 
? csc?: 1/sin?. 
Standard Angle Values 
? 0° 30° 45° 60° 90° 
sin? 0 1/2 v2/2 v3/2 1 
cos? 1 v3/2 v2/2 1/2 0 
? 0° 30° 45° 60° 90° 
tan? 0 v3/3 1 v3 undefine
d 
Signs in Quadrants 
? I: All positive. 
? II: sin, csc positive; others negative. 
? III: tan, cot positive; others negative. 
? IV: cos, sec positive; others negative. 
Relationships 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Trigonometric Identities 
Pythagorean Identities 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Sum and Difference Identities 
? sin(A ± B) = sinA cosB ± cosA sinB. 
? cos(A ± B) = cosA cosB ± sinA sinB. 
? tan(A ± B) = (tanA ± tanB)/(1 ± tanA tanB). 
Double Angle Identities 
? sin2? = 2sin? cos?. 
? cos2? = cos²? - sin²? = 2cos²? - 1 = 1 - 2sin²?. 
? tan2? = 2tan?/(1 - tan²?). 
Half Angle Identities 
? sin(?/2) = ±v[(1 - cos?)/2]. 
? cos(?/2) = ±v[(1 + cos?)/2]. 
? tan(?/2) = (1 - cos?)/sin? = sin?/(1 + cos?). 
Product-to-Sum Identities 
? sinA sinB = (1/2)[cos(A-B) - cos(A+B)]. 
? cosA cosB = (1/2)[cos(A-B) + cos(A+B)]. 
? sinA cosB = (1/2)[sin(A+B) + sin(A-B)]. 
Sum-to-Product Identities 
? 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). 
Trigonometric Functions 
Properties 
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Trigonometric Ratios, Functions, and 
Equations Cheat Sheet 
Trigonometric Ratios 
Definitions 
? sin?: Opposite/Hypotenuse. 
? cos?: Adjacent/Hypotenuse. 
? tan?: Opposite/Adjacent = sin?/cos?. 
? cot?: 1/tan? = cos?/sin?. 
? sec?: 1/cos?. 
? csc?: 1/sin?. 
Standard Angle Values 
? 0° 30° 45° 60° 90° 
sin? 0 1/2 v2/2 v3/2 1 
cos? 1 v3/2 v2/2 1/2 0 
? 0° 30° 45° 60° 90° 
tan? 0 v3/3 1 v3 undefine
d 
Signs in Quadrants 
? I: All positive. 
? II: sin, csc positive; others negative. 
? III: tan, cot positive; others negative. 
? IV: cos, sec positive; others negative. 
Relationships 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Trigonometric Identities 
Pythagorean Identities 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Sum and Difference Identities 
? sin(A ± B) = sinA cosB ± cosA sinB. 
? cos(A ± B) = cosA cosB ± sinA sinB. 
? tan(A ± B) = (tanA ± tanB)/(1 ± tanA tanB). 
Double Angle Identities 
? sin2? = 2sin? cos?. 
? cos2? = cos²? - sin²? = 2cos²? - 1 = 1 - 2sin²?. 
? tan2? = 2tan?/(1 - tan²?). 
Half Angle Identities 
? sin(?/2) = ±v[(1 - cos?)/2]. 
? cos(?/2) = ±v[(1 + cos?)/2]. 
? tan(?/2) = (1 - cos?)/sin? = sin?/(1 + cos?). 
Product-to-Sum Identities 
? sinA sinB = (1/2)[cos(A-B) - cos(A+B)]. 
? cosA cosB = (1/2)[cos(A-B) + cos(A+B)]. 
? sinA cosB = (1/2)[sin(A+B) + sin(A-B)]. 
Sum-to-Product Identities 
? 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). 
Trigonometric Functions 
Properties 
? sin?, cos?: Period = 2p, Domain = R, Range = [-1, 1]. 
? tan?, cot?: Period = p, Domain = {? | ? ? (2k+1)p/2}, Range = R. 
? sec?, csc?: Period = 2p, Domain = {? | ? ? kp (csc), ? ? (2k+1)p/2 (sec)}, Range = 
(-8,-1] ? [1,8). 
? Even/Odd: cos, sec are even; sin, tan, csc, cot are odd. 
Graphs 
? sin?: Wave, amplitude 1, crosses x-axis at kp. 
? cos?: Similar to sin?, shifted by p/2. 
? tan?: Asymptotes at (2k+1)p/2, period p. 
Trigonometric Equations 
General Solutions 
? sin? = k (|k| = 1): ? = sin?¹k + 2np or ? = p - sin?¹k + 2np, n ? Z. 
? cos? = k (|k| = 1): ? = ±cos?¹k + 2np, n ? Z. 
? tan? = k: ? = tan?¹k + np, n ? Z. 
? sin(m?) = k: ? = (1/m)[sin?¹k + 2np] or ? = (1/m)[p - sin?¹k + 2np]. 
Solving Techniques 
? Use identities to simplify (e.g., sin²? = 1 - cos²?). 
? Factorize or use quadratic formulas for complex equations. 
? Apply Weierstrass substitution (t = tan(?/2)) for rationalizing. 
Inverse Trigonometric Functions 
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Trigonometric Ratios, Functions, and 
Equations Cheat Sheet 
Trigonometric Ratios 
Definitions 
? sin?: Opposite/Hypotenuse. 
? cos?: Adjacent/Hypotenuse. 
? tan?: Opposite/Adjacent = sin?/cos?. 
? cot?: 1/tan? = cos?/sin?. 
? sec?: 1/cos?. 
? csc?: 1/sin?. 
Standard Angle Values 
? 0° 30° 45° 60° 90° 
sin? 0 1/2 v2/2 v3/2 1 
cos? 1 v3/2 v2/2 1/2 0 
? 0° 30° 45° 60° 90° 
tan? 0 v3/3 1 v3 undefine
d 
Signs in Quadrants 
? I: All positive. 
? II: sin, csc positive; others negative. 
? III: tan, cot positive; others negative. 
? IV: cos, sec positive; others negative. 
Relationships 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Trigonometric Identities 
Pythagorean Identities 
? sin²? + cos²? = 1. 
? 1 + tan²? = sec²?. 
? 1 + cot²? = csc²?. 
Sum and Difference Identities 
? sin(A ± B) = sinA cosB ± cosA sinB. 
? cos(A ± B) = cosA cosB ± sinA sinB. 
? tan(A ± B) = (tanA ± tanB)/(1 ± tanA tanB). 
Double Angle Identities 
? sin2? = 2sin? cos?. 
? cos2? = cos²? - sin²? = 2cos²? - 1 = 1 - 2sin²?. 
? tan2? = 2tan?/(1 - tan²?). 
Half Angle Identities 
? sin(?/2) = ±v[(1 - cos?)/2]. 
? cos(?/2) = ±v[(1 + cos?)/2]. 
? tan(?/2) = (1 - cos?)/sin? = sin?/(1 + cos?). 
Product-to-Sum Identities 
? sinA sinB = (1/2)[cos(A-B) - cos(A+B)]. 
? cosA cosB = (1/2)[cos(A-B) + cos(A+B)]. 
? sinA cosB = (1/2)[sin(A+B) + sin(A-B)]. 
Sum-to-Product Identities 
? 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). 
Trigonometric Functions 
Properties 
? sin?, cos?: Period = 2p, Domain = R, Range = [-1, 1]. 
? tan?, cot?: Period = p, Domain = {? | ? ? (2k+1)p/2}, Range = R. 
? sec?, csc?: Period = 2p, Domain = {? | ? ? kp (csc), ? ? (2k+1)p/2 (sec)}, Range = 
(-8,-1] ? [1,8). 
? Even/Odd: cos, sec are even; sin, tan, csc, cot are odd. 
Graphs 
? sin?: Wave, amplitude 1, crosses x-axis at kp. 
? cos?: Similar to sin?, shifted by p/2. 
? tan?: Asymptotes at (2k+1)p/2, period p. 
Trigonometric Equations 
General Solutions 
? sin? = k (|k| = 1): ? = sin?¹k + 2np or ? = p - sin?¹k + 2np, n ? Z. 
? cos? = k (|k| = 1): ? = ±cos?¹k + 2np, n ? Z. 
? tan? = k: ? = tan?¹k + np, n ? Z. 
? sin(m?) = k: ? = (1/m)[sin?¹k + 2np] or ? = (1/m)[p - sin?¹k + 2np]. 
Solving Techniques 
? Use identities to simplify (e.g., sin²? = 1 - cos²?). 
? Factorize or use quadratic formulas for complex equations. 
? Apply Weierstrass substitution (t = tan(?/2)) for rationalizing. 
Inverse Trigonometric Functions 
Definitions and Ranges 
? sin?¹x: Domain = [-1, 1], Range = [-p/2, p/2]. 
? cos?¹x: Domain = [-1, 1], Range = [0, p]. 
? tan?¹x: Domain = R, Range = (-p/2, p/2). 
Properties 
? sin?¹(-x) = -sin?¹x, cos?¹(-x) = p - cos?¹x, tan?¹(-x) = -tan?¹x. 
? sin?¹x + cos?¹x = p/2, tan?¹x + tan?¹(1/x) = ±p/2 (x > 0: p/2, x < 0: -p/2). 
Applications 
Heights and Distances 
? Angle of Elevation: Angle from horizontal upward to object. 
? Angle of Depression: Angle from horizontal downward to object. 
? Use tan? = height/distance in right triangles. 
Problem-Solving Tactics 
? Simplify expressions using Pythagorean or angle identities. 
? Convert products to sums using product-to-sum identities. 
? Use t = tan(?/2) for equations: sin? = 2t/(1 + t²), cos? = (1 - t²)/(1 + t²). 
? Check quadrants for signs and principal values in inverse functions. 
? Solve multiple-angle equations by reducing to single-angle forms. 
Formula Table