1. Fundamentals & Definitions
Angles and Measurements
Angle: The measure of rotation of a given ray about its initial point.
Vertex: The point of rotation.Initial Side: The original position of the ray before rotation.Terminal Side: The final position of the ray after rotation.Positive Angle: Angle generated when the rotation is anticlockwise.Negative Angle: Angle generated when the rotation is clockwise.
Systems of Angle Measurement:
Sexagesimal System (Degree Measure):
If a rotation from the initial side to the terminal side is 1 360 th \frac{1}{360}^{\text{th}} 360 1 th 1 ∘ 1^\circ 1 ∘
A degree is divided into 60 60 60 1 ∘ = 60 ′ 1^\circ = 60' 1 ∘ = 6 0 ′
A minute is divided into 60 60 60 1 ′ = 60 ′ ′ 1' = 60'' 1 ′ = 6 0 ′′
Circular System (Radian Measure):
An angle subtended at the center of a unit circle (radius r = 1 r = 1 r = 1 1 1 1 1 1 1 1 rad 1\text{ rad} 1 rad
In a circle of radius r r r l l l θ \theta θ θ = l r ⟹ l = r θ \theta = \frac{l}{r} \implies l = r\theta θ = r l ⟹ l = r θ
Relation between Degrees and Radians:
π radians = 180 ∘ \pi \text{ radians} = 180^\circ π radians = 18 0 ∘ 1 radian = 180 ∘ π ≈ 57 ∘ 16 ′ 22 ′ ′ 1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57^\circ 16' 22'' 1 radian = π 18 0 ∘ ≈ 5 7 ∘ 1 6 ′ 2 2 ′′ 1 ∘ = π 180 radians ≈ 0.01745 radians 1^\circ = \frac{\pi}{180} \text{ radians} \approx 0.01745 \text{ radians} 1 ∘ = 180 π radians ≈ 0.01745 radians Conversion Formulas:
Radian measure = π 180 × Degree measure \text{Radian measure} = \frac{\pi}{180} \times \text{Degree measure} Radian measure = 180 π × Degree measure Degree measure = 180 π × Radian measure \text{Degree measure} = \frac{180}{\pi} \times \text{Radian measure} Degree measure = π 180 × Radian measure
Trigonometric Ratios (Right-Angled Triangle)
For a right-angled triangle A B C ABC A B C B B B ∠ B = 90 ∘ \angle B = 90^\circ ∠ B = 9 0 ∘ ∠ A = θ \angle A = \theta ∠ A = θ
Base (Adjacent Side, b b b The side adjacent to angle θ \theta θ A B AB A B Perpendicular (Opposite Side, p p p The side opposite to angle θ \theta θ B C BC B C Hypotenuse (h h h The side opposite to the right angle (A C AC A C Definitions:
sin θ = Perpendicular Hypotenuse = p h \sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{p}{h} sin θ = Hypotenuse Perpendicular = h p cos θ = Base Hypotenuse = b h \cos\theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{b}{h} cos θ = Hypotenuse Base = h b tan θ = Perpendicular Base = p b = sin θ cos θ \tan\theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{p}{b} = \frac{\sin\theta}{\cos\theta} tan θ = Base Perpendicular = b p = c o s θ s i n θ csc θ = Hypotenuse Perpendicular = h p = 1 sin θ \csc\theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{h}{p} = \frac{1}{\sin\theta} csc θ = Perpendicular Hypotenuse = p h = s i n θ 1 sec θ = Hypotenuse Base = h b = 1 cos θ \sec\theta = \frac{\text{Hypotenuse}}{\text{Base}} = \frac{h}{b} = \frac{1}{\cos\theta} sec θ = Base Hypotenuse = b h = c o s θ 1 cot θ = Base Perpendicular = b p = 1 tan θ \cot\theta = \frac{\text{Base}}{\text{Perpendicular}} = \frac{b}{p} = \frac{1}{\tan\theta} cot θ = Perpendicular Base = p b = t a n θ 1
Unit Circle Trigonometry
Let P ( x , y ) P(x, y) P ( x , y ) O ( 0 , 0 ) O(0, 0) O ( 0 , 0 ) O P OP O P θ \theta θ
cos θ = x \cos\theta = x cos θ = x sin θ = y \sin\theta = y sin θ = y tan θ = y x ( x ≠ 0 ) \tan\theta = \frac{y}{x} \quad (x \neq 0) tan θ = x y ( x = 0 ) Quadrantal Angles: Angles that are integral multiples of π 2 \frac{\pi}{2} 2 π 0 , π 2 , π , 3 π 2 , 2 π 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi 0 , 2 π , π , 2 3 π , 2 π
Height & Distance Terminology
Line of Sight: The straight line path from the observer's eye to the object being viewed.Angle of Elevation: The angle between the line of sight and the horizontal plane when the object is above the horizontal plane.Angle of Depression: The angle between the line of sight and the horizontal plane when the object is below the horizontal plane.
2. Core Concepts & Formulas
Trigonometric Values of Standard Angles
Angle (Degrees) Angle (Radians) sin θ \sin\theta sin θ cos θ \cos\theta cos θ tan θ \tan\theta tan θ csc θ \csc\theta csc θ sec θ \sec\theta sec θ cot θ \cot\theta cot θ 0 ∘ 0^\circ 0 ∘ 0 0 0 0 0 0 1 1 1 0 0 0 Undefined 1 1 1 Undefined 30 ∘ 30^\circ 3 0 ∘ π 6 \frac{\pi}{6} 6 π 1 2 \frac{1}{2} 2 1 3 2 \frac{\sqrt{3}}{2} 2 3 1 3 \frac{1}{\sqrt{3}} 3 1 2 2 2 2 3 \frac{2}{\sqrt{3}} 3 2 3 \sqrt{3} 3 45 ∘ 45^\circ 4 5 ∘ π 4 \frac{\pi}{4} 4 π 1 2 \frac{1}{\sqrt{2}} 2 1 1 2 \frac{1}{\sqrt{2}} 2 1 1 1 1 2 \sqrt{2} 2 2 \sqrt{2} 2 1 1 1 60 ∘ 60^\circ 6 0 ∘ π 3 \frac{\pi}{3} 3 π 3 2 \frac{\sqrt{3}}{2} 2 3 1 2 \frac{1}{2} 2 1 3 \sqrt{3} 3 2 3 \frac{2}{\sqrt{3}} 3 2 2 2 2 1 3 \frac{1}{\sqrt{3}} 3 1 90 ∘ 90^\circ 9 0 ∘ π 2 \frac{\pi}{2} 2 π 1 1 1 0 0 0 Undefined 1 1 1 Undefined 0 0 0 180 ∘ 180^\circ 18 0 ∘ π \pi π 0 0 0 − 1 -1 − 1 0 0 0 Undefined − 1 -1 − 1 Undefined 270 ∘ 270^\circ 27 0 ∘ 3 π 2 \frac{3\pi}{2} 2 3 π − 1 -1 − 1 0 0 0 Undefined − 1 -1 − 1 Undefined 0 0 0 360 ∘ 360^\circ 36 0 ∘ 2 π 2\pi 2 π 0 0 0 1 1 1 0 0 0 Undefined 1 1 1 Undefined
Special Competitive Exam Angles
Angle (Degrees) sin θ \sin\theta sin θ cos θ \cos\theta cos θ tan θ \tan\theta tan θ cot θ \cot\theta cot θ 15 ∘ 15^\circ 1 5 ∘ π 12 \frac{\pi}{12} 12 π 3 − 1 2 2 \frac{\sqrt{3}-1}{2\sqrt{2}} 2 2 3 − 1 3 + 1 2 2 \frac{\sqrt{3}+1}{2\sqrt{2}} 2 2 3 + 1 2 − 3 2-\sqrt{3} 2 − 3 2 + 3 2+\sqrt{3} 2 + 3 75 ∘ 75^\circ 7 5 ∘ 5 π 12 \frac{5\pi}{12} 12 5 π 3 + 1 2 2 \frac{\sqrt{3}+1}{2\sqrt{2}} 2 2 3 + 1 3 − 1 2 2 \frac{\sqrt{3}-1}{2\sqrt{2}} 2 2 3 − 1 2 + 3 2+\sqrt{3} 2 + 3 2 − 3 2-\sqrt{3} 2 − 3 18 ∘ 18^\circ 1 8 ∘ π 10 \frac{\pi}{10} 10 π 5 − 1 4 \frac{\sqrt{5}-1}{4} 4 5 − 1 10 + 2 5 4 \frac{\sqrt{10+2\sqrt{5}}}{4} 4 10 + 2 5 1 − 2 5 \sqrt{1 - \frac{2}{\sqrt{5}}} 1 − 5 2 5 + 2 5 \sqrt{5+2\sqrt{5}} 5 + 2 5 36 ∘ 36^\circ 3 6 ∘ π 5 \frac{\pi}{5} 5 π 10 − 2 5 4 \frac{\sqrt{10-2\sqrt{5}}}{4} 4 10 − 2 5 5 + 1 4 \frac{\sqrt{5}+1}{4} 4 5 + 1 5 − 2 5 \sqrt{5-2\sqrt{5}} 5 − 2 5 1 + 2 5 \sqrt{1 + \frac{2}{\sqrt{5}}} 1 + 5 2 22.5 ∘ 22.5^\circ 22. 5 ∘ π 8 \frac{\pi}{8} 8 π 2 − 2 2 \frac{\sqrt{2-\sqrt{2}}}{2} 2 2 − 2 2 + 2 2 \frac{\sqrt{2+\sqrt{2}}}{2} 2 2 + 2 2 − 1 \sqrt{2}-1 2 − 1 2 + 1 \sqrt{2}+1 2 + 1
Behavior & Signs of Trigonometric Functions
ASTC Quadrant Rule:
Quadrant I (0 < θ < 90 ∘ 0 < \theta < 90^\circ 0 < θ < 9 0 ∘ A ll functions are positive.Quadrant II (90 ∘ < θ < 180 ∘ 90^\circ < \theta < 180^\circ 9 0 ∘ < θ < 18 0 ∘ Only S ine (and Cosecant) are positive.Quadrant III (180 ∘ < θ < 270 ∘ 180^\circ < \theta < 270^\circ 18 0 ∘ < θ < 27 0 ∘ Only T angent (and Cotangent) are positive.Quadrant IV (270 ∘ < θ < 360 ∘ 270^\circ < \theta < 360^\circ 27 0 ∘ < θ < 36 0 ∘ Only C osine (and Secant) are positive.
Quadrant sin x \sin x sin x csc x \csc x csc x cos x \cos x cos x sec x \sec x sec x tan x \tan x tan x cot x \cot x cot x I + + + + + + + + + II + + + − - − − - − III − - − − - − + + + IV − - − + + + − - −
Ranges of Trigonometric Functions:
− 1 ≤ sin θ ≤ 1 -1 \le \sin\theta \le 1 − 1 ≤ sin θ ≤ 1 − 1 ≤ cos θ ≤ 1 -1 \le \cos\theta \le 1 − 1 ≤ cos θ ≤ 1 sec θ ≥ 1 \sec\theta \ge 1 sec θ ≥ 1 sec θ ≤ − 1 \sec\theta \le -1 sec θ ≤ − 1 csc θ ≥ 1 \csc\theta \ge 1 csc θ ≥ 1 csc θ ≤ − 1 \csc\theta \le -1 csc θ ≤ − 1 − ∞ < tan θ < ∞ -\infty < \tan\theta < \infty − ∞ < tan θ < ∞ − ∞ < cot θ < ∞ -\infty < \cot\theta < \infty − ∞ < cot θ < ∞
Fundamental Trigonometric Identities
Pythagorean Identities:
sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ ⟹ sec 2 θ − tan 2 θ = 1 ⟹ ( sec θ − tan θ ) = 1 sec θ + tan θ 1 + \tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1 \implies (\sec\theta - \tan\theta) = \frac{1}{\sec\theta + \tan\theta} 1 + tan 2 θ = sec 2 θ ⟹ sec 2 θ − tan 2 θ = 1 ⟹ ( sec θ − tan θ ) = s e c θ + t a n θ 1 1 + cot 2 θ = csc 2 θ ⟹ csc 2 θ − cot 2 θ = 1 ⟹ ( csc θ − cot θ ) = 1 csc θ + cot θ 1 + \cot^2\theta = \csc^2\theta \implies \csc^2\theta - \cot^2\theta = 1 \implies (\csc\theta - \cot\theta) = \frac{1}{\csc\theta + \cot\theta} 1 + cot 2 θ = csc 2 θ ⟹ csc 2 θ − cot 2 θ = 1 ⟹ ( csc θ − cot θ ) = c s c θ + c o t θ 1
Even-Odd (Negative Angle) Identities
sin ( − θ ) = − sin θ \sin(-\theta) = -\sin\theta sin ( − θ ) = − sin θ cos ( − θ ) = cos θ \cos(-\theta) = \cos\theta cos ( − θ ) = cos θ tan ( − θ ) = − tan θ \tan(-\theta) = -\tan\theta tan ( − θ ) = − tan θ csc ( − θ ) = − csc θ \csc(-\theta) = -\csc\theta csc ( − θ ) = − csc θ sec ( − θ ) = sec θ \sec(-\theta) = \sec\theta sec ( − θ ) = sec θ cot ( − θ ) = − cot θ \cot(-\theta) = -\cot\theta cot ( − θ ) = − cot θ
Allied and Complementary Angle Formulas
Function − θ -\theta − θ 90 ∘ − θ 90^\circ - \theta 9 0 ∘ − θ 90 ∘ + θ 90^\circ + \theta 9 0 ∘ + θ 180 ∘ − θ 180^\circ - \theta 18 0 ∘ − θ 180 ∘ + θ 180^\circ + \theta 18 0 ∘ + θ 270 ∘ − θ 270^\circ - \theta 27 0 ∘ − θ 270 ∘ + θ 270^\circ + \theta 27 0 ∘ + θ 360 ∘ − θ 360^\circ - \theta 36 0 ∘ − θ sin \sin sin − sin θ -\sin\theta − sin θ cos θ \cos\theta cos θ cos θ \cos\theta cos θ sin θ \sin\theta sin θ − sin θ -\sin\theta − sin θ − cos θ -\cos\theta − cos θ − cos θ -\cos\theta − cos θ − sin θ -\sin\theta − sin θ cos \cos cos cos θ \cos\theta cos θ sin θ \sin\theta sin θ − sin θ -\sin\theta − sin θ − cos θ -\cos\theta − cos θ − cos θ -\cos\theta − cos θ − sin θ -\sin\theta − sin θ sin θ \sin\theta sin θ cos θ \cos\theta cos θ tan \tan tan − tan θ -\tan\theta − tan θ cot θ \cot\theta cot θ − cot θ -\cot\theta − cot θ − tan θ -\tan\theta − tan θ tan θ \tan\theta tan θ cot θ \cot\theta cot θ − cot θ -\cot\theta − cot θ − tan θ -\tan\theta − tan θ csc \csc csc − csc θ -\csc\theta − csc θ sec θ \sec\theta sec θ sec θ \sec\theta sec θ csc θ \csc\theta csc θ − csc θ -\csc\theta − csc θ − sec θ -\sec\theta − sec θ − sec θ -\sec\theta − sec θ − csc θ -\csc\theta − csc θ sec \sec sec sec θ \sec\theta sec θ csc θ \csc\theta csc θ − csc θ -\csc\theta − csc θ − sec θ -\sec\theta − sec θ − sec θ -\sec\theta − sec θ − csc θ -\csc\theta − csc θ csc θ \csc\theta csc θ sec θ \sec\theta sec θ cot \cot cot − cot θ -\cot\theta − cot θ tan θ \tan\theta tan θ − tan θ -\tan\theta − tan θ − cot θ -\cot\theta − cot θ cot θ \cot\theta cot θ tan θ \tan\theta tan θ − tan θ -\tan\theta − tan θ − cot θ -\cot\theta − cot θ
Compound Angle Formulas
sin ( A ± B ) = sin A cos B ± cos A sin B \sin(A \pm B) = \sin A \cos B \pm \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 \pm B) = \cos A \cos B \mp \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 \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} tan ( A ± B ) = 1 ∓ t a n A t a n B t a n A ± t a n B cot ( A ± B ) = cot A cot B ∓ 1 cot B ± cot A \cot(A \pm B) = \frac{\cot A \cot B \mp 1}{\cot B \pm \cot A} cot ( A ± B ) = c o t B ± c o t A c o t A c o t B ∓ 1
Double Angle Formulas
sin 2 θ = 2 sin θ cos θ = 2 tan θ 1 + tan 2 θ \sin 2\theta = 2\sin\theta\cos\theta = \frac{2\tan\theta}{1+\tan^2\theta} sin 2 θ = 2 sin θ cos θ = 1 + t a n 2 θ 2 t a n θ cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ = 1 − tan 2 θ 1 + tan 2 θ \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta = \frac{1-\tan^2\theta}{1+\tan^2\theta} cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ = 1 + t a n 2 θ 1 − t a n 2 θ
1 − cos 2 θ = 2 sin 2 θ ⟹ sin 2 θ = 1 − cos 2 θ 2 1 - \cos 2\theta = 2\sin^2\theta \implies \sin^2\theta = \frac{1-\cos 2\theta}{2} 1 − cos 2 θ = 2 sin 2 θ ⟹ sin 2 θ = 2 1 − c o s 2 θ 1 + cos 2 θ = 2 cos 2 θ ⟹ cos 2 θ = 1 + cos 2 θ 2 1 + \cos 2\theta = 2\cos^2\theta \implies \cos^2\theta = \frac{1+\cos 2\theta}{2} 1 + cos 2 θ = 2 cos 2 θ ⟹ cos 2 θ = 2 1 + c o s 2 θ
tan 2 θ = 2 tan θ 1 − tan 2 θ \tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta} tan 2 θ = 1 − t a n 2 θ 2 t a n θ
Triple Angle Formulas
sin 3 θ = 3 sin θ − 4 sin 3 θ \sin 3\theta = 3\sin\theta - 4\sin^3\theta sin 3 θ = 3 sin θ − 4 sin 3 θ cos 3 θ = 4 cos 3 θ − 3 cos θ \cos 3\theta = 4\cos^3\theta - 3\cos\theta cos 3 θ = 4 cos 3 θ − 3 cos θ tan 3 θ = 3 tan θ − tan 3 θ 1 − 3 tan 2 θ \tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1-3\tan^2\theta} tan 3 θ = 1 − 3 t a n 2 θ 3 t a n θ − t a n 3 θ
Half-Angle Formulas
sin θ 2 = ± 1 − cos θ 2 \sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}} sin 2 θ = ± 2 1 − c o s θ cos θ 2 = ± 1 + cos θ 2 \cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}} cos 2 θ = ± 2 1 + c o s θ tan θ 2 = ± 1 − cos θ 1 + cos θ = 1 − cos θ sin θ = sin θ 1 + cos θ \tan\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} = \frac{1-\cos\theta}{\sin\theta} = \frac{\sin\theta}{1+\cos\theta} tan 2 θ = ± 1 + c o s θ 1 − c o s θ = s i n θ 1 − c o s θ = 1 + c o s θ s i n θ
Sum, Difference & Product Conversion Formulas
Product to Sum/Difference:
2 sin A cos B = sin ( A + B ) + sin ( A − B ) 2\sin A \cos B = \sin(A + B) + \sin(A - B) 2 sin A cos B = sin ( A + B ) + sin ( A − B ) 2 cos A sin B = sin ( A + B ) − sin ( A − B ) 2\cos A \sin B = \sin(A + B) - \sin(A - B) 2 cos A sin B = sin ( A + B ) − sin ( A − B ) 2 cos A cos B = cos ( A + B ) + cos ( A − B ) 2\cos A \cos B = \cos(A + B) + \cos(A - B) 2 cos A cos B = cos ( A + B ) + cos ( A − B ) 2 sin A sin B = cos ( A − B ) − cos ( A + B ) 2\sin A \sin B = \cos(A - B) - \cos(A + B) 2 sin A sin B = cos ( A − B ) − cos ( A + B )
Sum/Difference to Product:
sin x + sin y = 2 sin ( x + y 2 ) cos ( x − y 2 ) \sin x + \sin y = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) sin x + sin y = 2 sin ( 2 x + y ) cos ( 2 x − y ) sin x − sin y = 2 cos ( x + y 2 ) sin ( x − y 2 ) \sin x - \sin y = 2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right) sin x − sin y = 2 cos ( 2 x + y ) sin ( 2 x − y ) cos x + cos y = 2 cos ( x + y 2 ) cos ( x − y 2 ) \cos x + \cos y = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) cos x + cos y = 2 cos ( 2 x + y ) cos ( 2 x − y ) cos x − cos y = − 2 sin ( x + y 2 ) sin ( x − y 2 ) \cos x - \cos y = -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right) cos x − cos y = − 2 sin ( 2 x + y ) sin ( 2 x − y )
Maximum and Minimum of a cos θ + b sin θ a\cos\theta + b\sin\theta a cos θ + b sin θ
For the expression f ( θ ) = a cos θ + b sin θ f(\theta) = a\cos\theta + b\sin\theta f ( θ ) = a cos θ + b sin θ
Maximum Value: a 2 + b 2 \sqrt{a^2 + b^2} a 2 + b 2 Minimum Value: − a 2 + b 2 -\sqrt{a^2 + b^2} − a 2 + b 2 Range: [ − a 2 + b 2 , a 2 + b 2 ] [-\sqrt{a^2 + b^2}, \sqrt{a^2 + b^2}] [ − a 2 + b 2 , a 2 + b 2 ]
Trigonometric Equations & General Solutions
For n ∈ Z n \in \mathbb{Z} n ∈ Z
sin θ = 0 ⟹ θ = n π \sin\theta = 0 \implies \theta = n\pi sin θ = 0 ⟹ θ = nπ cos θ = 0 ⟹ θ = ( 2 n + 1 ) π 2 \cos\theta = 0 \implies \theta = (2n+1)\frac{\pi}{2} cos θ = 0 ⟹ θ = ( 2 n + 1 ) 2 π tan θ = 0 ⟹ θ = n π \tan\theta = 0 \implies \theta = n\pi tan θ = 0 ⟹ θ = nπ sin θ = sin α ⟹ θ = n π + ( − 1 ) n α \sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n\alpha sin θ = sin α ⟹ θ = nπ + ( − 1 ) n α cos θ = cos α ⟹ θ = 2 n π ± α \cos\theta = \cos\alpha \implies \theta = 2n\pi \pm \alpha cos θ = cos α ⟹ θ = 2 nπ ± α tan θ = tan α ⟹ θ = n π + α \tan\theta = \tan\alpha \implies \theta = n\pi + \alpha tan θ = tan α ⟹ θ = nπ + α sin 2 θ = sin 2 α ⟹ θ = n π ± α \sin^2\theta = \sin^2\alpha \implies \theta = n\pi \pm \alpha sin 2 θ = sin 2 α ⟹ θ = nπ ± α cos 2 θ = cos 2 α ⟹ θ = n π ± α \cos^2\theta = \cos^2\alpha \implies \theta = n\pi \pm \alpha cos 2 θ = cos 2 α ⟹ θ = nπ ± α tan 2 θ = tan 2 α ⟹ θ = n π ± α \tan^2\theta = \tan^2\alpha \implies \theta = n\pi \pm \alpha tan 2 θ = tan 2 α ⟹ θ = nπ ± α
Properties of Triangles (For NDA/CDS/JEE)
In any triangle A B C ABC A B C a , b , c a, b, c a , b , c A , B , C A, B, C A , B , C
Sine Rule:
a sin A = b sin B = c sin C = 2 R ( R = circumradius ) \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \quad (R = \text{circumradius}) sin A a = sin B b = sin C c = 2 R ( R = circumradius ) Cosine Rule:
cos A = b 2 + c 2 − a 2 2 b c \cos A = \frac{b^2 + c^2 - a^2}{2bc} cos A = 2 b c b 2 + c 2 − a 2 cos B = a 2 + c 2 − b 2 2 a c \cos B = \frac{a^2 + c^2 - b^2}{2ac} cos B = 2 a c a 2 + c 2 − b 2 cos C = a 2 + b 2 − c 2 2 a b \cos C = \frac{a^2 + b^2 - c^2}{2ab} cos C = 2 ab a 2 + b 2 − c 2
Projection Formula:
a = b cos C + c cos B a = b\cos C + c\cos B a = b cos C + c cos B b = c cos A + a cos C b = c\cos A + a\cos C b = c cos A + a cos C c = a cos B + b cos A c = a\cos B + b\cos A c = a cos B + b cos A
Napier's Analogy (Law of Tangents):
tan ( A − B 2 ) = a − b a + b cot ( C 2 ) \tan\left(\frac{A-B}{2}\right) = \frac{a-b}{a+b}\cot\left(\frac{C}{2}\right) tan ( 2 A − B ) = a + b a − b cot ( 2 C ) tan ( B − C 2 ) = b − c b + c cot ( A 2 ) \tan\left(\frac{B-C}{2}\right) = \frac{b-c}{b+c}\cot\left(\frac{A}{2}\right) tan ( 2 B − C ) = b + c b − c cot ( 2 A ) tan ( C − A 2 ) = c − a c + a cot ( B 2 ) \tan\left(\frac{C-A}{2}\right) = \frac{c-a}{c+a}\cot\left(\frac{B}{2}\right) tan ( 2 C − A ) = c + a c − a cot ( 2 B )
Mollweide's Equation:
a − b c = sin ( A − B 2 ) cos ( C 2 ) \frac{a-b}{c} = \frac{\sin\left(\frac{A-B}{2}\right)}{\cos\left(\frac{C}{2}\right)} c a − b = cos ( 2 C ) sin ( 2 A − B )
Inverse Trigonometric Functions
Function Domain Principal Value Branch (Range) sin − 1 x \sin^{-1}x sin − 1 x [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] [ − π 2 , π 2 ] [-\frac{\pi}{2}, \frac{\pi}{2}] [ − 2 π , 2 π ] cos − 1 x \cos^{-1}x cos − 1 x [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] [ 0 , π ] [0, \pi] [ 0 , π ] tan − 1 x \tan^{-1}x tan − 1 x R \mathbb{R} R ( − π 2 , π 2 ) (-\frac{\pi}{2}, \frac{\pi}{2}) ( − 2 π , 2 π ) csc − 1 x \csc^{-1}x csc − 1 x R ∖ ( − 1 , 1 ) \mathbb{R} \setminus (-1, 1) R ∖ ( − 1 , 1 ) [ − π 2 , π 2 ] ∖ 0 [-\frac{\pi}{2}, \frac{\pi}{2}] \setminus {0} [ − 2 π , 2 π ] ∖ 0 sec − 1 x \sec^{-1}x sec − 1 x R ∖ ( − 1 , 1 ) \mathbb{R} \setminus (-1, 1) R ∖ ( − 1 , 1 ) [ 0 , π ] ∖ π 2 [0, \pi] \setminus {\frac{\pi}{2}} [ 0 , π ] ∖ 2 π cot − 1 x \cot^{-1}x cot − 1 x R \mathbb{R} R ( 0 , π ) (0, \pi) ( 0 , π )
Typical Exam Weightage
Exam Typical Questions SSC (CGL / CHSL / MTS) 3–4 questions Defense (NDA / CDS) 2–3 questions
Heavily formula-dependent — a small set of standard-angle values and identities covers the large majority of questions asked.
Figures are typical ranges based on recent-year patterns, not a guarantee for any specific upcoming paper — always cross-check against the latest official syllabus and previous-year papers for Trigonometry.