Trigonometry Tutorials: At a glance
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Trigonometry 1: Basic Trigonometric Ratios and Identities
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Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE, Anyone else who needs this Tutorial as a reference!
Important points to remember:
Angle A measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative. Figure 1(i)  Positive Angle Figure 1(ii) Negative Angle
 Degree measure: If a rotation from the initial side to terminal side is (1360)th of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″.Thus, 1° = 60′,1′= 60″
 Radian measure: There is another unit for measurement of an angle, called the radian measure. Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.
 If in a circle of radius r, an arc of length l subtends an angle θ radian at the centre, we have l=θ/r.
 Relation between degrees and radians : 2π radian = 360°or π radians = 180°
Degree  30o  45o  60o  90o  180o  270o  360o  Radian  π/6  π/4  π/3  π/2  π  3π/2  2π 
A right angled triangle
 sinθ= perpendicular/hypotenuse
 tanθ= perpendicular/base , θ ≠ (π/2)(2n +1) where n is any integer
 cotθ=1/tanθ , θ ≠nπ, where n is any integer
 secθ=1/cosθ , θ ≠ (π/2)(2n + 1) where n is any integer
 cosecθ=1/sinθ , θ ≠ nπ where n is any integer.
 Signs of trigonometric functions:
Quadrant  I  II  III  IV  sin  +  +      cos  +      +  tan  +    +    cot  +    +    sec  +      +  cosec  +  +     
 Domains and ranges of trigonometric functions:
 Domain of y = sin x and y = cos x is the set of all real numbers and range is the interval [–1, 1], i.e., – 1 ≤ y ≤ 1.
 The domain of y = cosec x is the set {x: x ϵ R and x ≠ n π, n ϵ Z} and range is the set {y: y ϵ R, y ≥ 1 or y ≤– 1}.
 The domain of y = sec x is the set {x: x ϵ R and x ≠ π2(2n + 1), n ϵ Z} and range is the set{y: y ϵ R, y≤ 1or y ≥ 1}.
 The domain of y = tan x is the set {x: x ϵ R and x ≠ π2(2n +1), n ϵ Z} and range is the set of all real numbers.
 The domain of y = cot x is the set {x: x ϵ R and x ≠n π , n ϵ Z} and the range is the set of all real numbers.
 Monotonicity of trigonometric functions:
Quadrant  I  II  III  IV  sin  Increases from 0 to 1  Decreases from 1 to 0  Decreases from 0 to 1  Increases from 1 to 0  cos  Decreases from 1 to 0  Decreases from 0 to 1  Increases from 1 to 0  Increases from 0 to 1  tan  Increases from 0 to ∞  Increases from ∞ to 0  Increases from 0 to ∞  Increases from ∞ to 0  cot  Decreases from ∞ to 0  Decreases from 0 to ∞  decreases from ∞ to 0  Decreases from 0 to ∞  sec  Increases from 1 to ∞  Increases from ∞ to 1  Decreases from 1 to ∞  Decreases from ∞ to 1  cosec  Decreases from ∞ to 1  Increases from 1 to ∞  Increases from ∞ to 1  Decreases from 1 to ∞ 
 Plots of trigonometric functions:
 sin2 x + cos2 x = 1
 1 + tan2 x = sec2 x
 1 + cot2 x = cosec2 x
 cos (– x) = cos x
 sin (– x) = – sin x
 Trigonometric Functions of Sum and Difference of Two Angles
 cos (x + y) = cos x cos y – sin x sin y
 cos (x – y) = cos x cos y + sin x sin y
 cos (π2 – x) = sin x & sin (π2 – x) = cos x
 sin (x + y) = sin x cos y + cos x sin y
 sin (x – y) = sin x cos y – cos x sin y
 cos (π2 + x) = – sin x & sin (π2 + x) = cos x
 cos (π – x) = – cos x & sin (π – x) = sin x
 cos (π + x) = – cos x & sin (π + x) = – sin x
 cos (2π – x) = cos x & sin (2π – x) = – sin x
 tan (x + y) = tanx+tany1tanxtany
 tan (x – y) =tanxtany1+tanxtany
 cot ( x + y) =cotxcoty1coty+cotx
 cot (x – y)= cotxcoty+1cotycotx
Double Angle Formulae cos 2x = cos2x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x = 1tan2x1+tan2x
 sin 2x = 2 sinx cos x =2tanx1+tan2x
 tan 2x =2tanx1tan2x
 sin 3x = 3 sin x – 4 sin3 x
 cos 3x= 4 cos3 x – 3 cos x
 tan 3x =3tanxtan3x13tan2x
 cos x + cos y =2cos(x + y2) cos(x y2)
 cos x – cos y = –2sin(x + y2) sin(x y2)
 sin x + sin y = 2sin(x + y2) cos(x y2)
 sin x – sin y =2cos(x + y2) sin(x y2)
 2 cos x cos y = cos (x + y) + cos (x – y)
 –2 sin x sin y = cos (x + y) – cos (x – y)
 2 sin x cos y = sin (x + y) + sin (x – y)
 2 cos x sin y = sin (x + y) – sin (x – y).
 General solutions of trigonometric equations
 For any real numbers x and y, sin x = sin y implies x = nπ + (–1)n y, where n ϵ Z
 For any real numbers x and y, cos x = cos y, implies x = 2nπ ± y, where nϵZ
 if x and y are not odd multiple of π2, then tan x = tan y implies x = nπ + y, where n ϵ Z
Those general solutions which lie between 0 to 2π i.e. 0≤x≤2 π.
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