We will cover:
Angles in Degrees and Radians, Introduction to Trigonometric Ratios, Domains and ranges, Basic Identities, Compound Angle Formulas, Double and Triple Angle Formulas
Trigonometry Tutorials: At a glance
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Trigonometry -1: Basic Trigonometric Ratios and Identities
You can also view or download the entire tutorial in PDF format at the end of the page
Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE, 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+tany1-tanxtany
- tan (x – y) =tanx-tany1+tanxtany
- cot ( x + y) =cotxcoty-1coty+cotx
- cot (x – y)= cotxcoty+1coty-cotx
Double Angle Formulae- cos 2x = cos2x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x = 1-tan2x1+tan2x
- sin 2x = 2 sinx cos x =2tanx1+tan2x
- tan 2x =2tanx1-tan2x
- sin 3x = 3 sin x – 4 sin3 x
- cos 3x= 4 cos3 x – 3 cos x
- tan 3x =3tanx-tan3x1-3tan2x
- 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.
In case you'd like to take a look at other Trigonometry tutorials :
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