We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions.
Consider a unit circle with centre at origin of the coordinate axes.
Let `P (a, b)` be any point on the circle with angle `AOP = x` radian, i.e., length of arc `AP = x` (Fig 3.6).
We define cos x = a and sin x = b Since ΔOMP is a right triangle, we have
` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(blue)(OM^2 + MP^2 = OP^2) ` or `color(green)(a^2 + b^2 = 1)`
Thus, for every point on the unit circle, we have
` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(green)(a^2 + b^2 = 1)` or `color(red)(cos^2 x + sin^2 x = 1)`
`=>` Since one complete revolution subtends an angle of `2π` radian at the centre of the circle, `color(purple)(∠AOB =π/2),` `color(purple)(∠AOC = π)` and `color(purple)(∠AOD = 3π /2) .`
`color(blue)(=>"All angles which are integral multiples of"\ \ π /2 "are called quadrantal angles.")`
The coordinates of the points `A, B, C` and `D` are, respectively, `(1, 0), (0, 1), (–1, 0)` and `(0, –1).` Therefore, for quadrantal angles, we have
` \ \ \ \ \ \ \ \ \ color(green)(cos 0° = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin 0° = 0,)`
` \ \ \ \ \ \ \ \ \color(green)( cos (π/ 2) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin (π /2) = 1)`
` \ \ \ \ \ \ \ \ \color(green)(cosπ = − 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sinπ = 0)`
` \ \ \ \ \ \ \ \ \color(green)(cos ((3π)/2 )= 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin((3π)/2) = –1)`
` \ \ \ \ \ \ \ \ \color(green)( cos 2π = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin 2π = 0)`
Now, if we take one complete revolution from the point P, we again come back to same point `P.` Thus, we also observe that if `x` increases (or decreases) by `color(blue)("any integral multiple of 2π")` `color(red)("the values of sine and cosine functions do not change.")` Thus,
` \ \ \ \ \ \ \ \ \ \ \ \ \ color(red)(sin (2nπ + x) = sin x, n ∈ Z , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cos (2nπ + x) = cos x , n ∈ Z)`
Further, `color(green)(sin x = 0),` if `color(green)(x = 0, ± π, ± 2π , ± 3π, ........,)` i.e., when `x` is an integral multiple of `π`
and `color(green)(cos x = 0)` , if `color(green)(x = pm pi/2 , pm (3 pi)/2 , pm (5 pi)/2)`,.... i.e., `cos x` vanishes when `x` is an odd multiple of `π/2` . Thus
` color(red)(sin x = 0)` implies `color(red)(x = nπ)`, where `n` is any integer
` color(red)(cos x = 0)` implies ` color(red)(x = (2n + 1)π/2)` , where `n` is any integer.
We now define other trigonometric functions in terms of sine and cosine functions:
` color(red)(cosec x = 1/(sin x)) , \ \ color(blue)(x ≠ nπ)` , where `n` is any integer.
` color(red)(sec x = 1/(cos x)) , \ \ color(blue)(x ≠ (2n +1 ) pi/2)` , where `n` is any integer
` color(red)(tan x = (sin x)/(cos x) ), \ \ color(blue)(x ≠ (2n +1) pi/2 )`, where `n` is any integer.
` color(red)(cot x = ( cos x)/(sin x)), \ \ color(blue)(x ≠ n π)`, where `n` is any integer
We have shown that for all real `x`
`color(purple)( sin^2 x + cos^2 x = 1)`
`color(purple)(1 + tan^2 x = sec^2 x)`
`color(purple)(1 + cot^2 x = cosec^2 x)`
We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions.
Consider a unit circle with centre at origin of the coordinate axes.
Let `P (a, b)` be any point on the circle with angle `AOP = x` radian, i.e., length of arc `AP = x` (Fig 3.6).
We define cos x = a and sin x = b Since ΔOMP is a right triangle, we have
` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(blue)(OM^2 + MP^2 = OP^2) ` or `color(green)(a^2 + b^2 = 1)`
Thus, for every point on the unit circle, we have
` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(green)(a^2 + b^2 = 1)` or `color(red)(cos^2 x + sin^2 x = 1)`
`=>` Since one complete revolution subtends an angle of `2π` radian at the centre of the circle, `color(purple)(∠AOB =π/2),` `color(purple)(∠AOC = π)` and `color(purple)(∠AOD = 3π /2) .`
`color(blue)(=>"All angles which are integral multiples of"\ \ π /2 "are called quadrantal angles.")`
The coordinates of the points `A, B, C` and `D` are, respectively, `(1, 0), (0, 1), (–1, 0)` and `(0, –1).` Therefore, for quadrantal angles, we have
` \ \ \ \ \ \ \ \ \ color(green)(cos 0° = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin 0° = 0,)`
` \ \ \ \ \ \ \ \ \color(green)( cos (π/ 2) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin (π /2) = 1)`
` \ \ \ \ \ \ \ \ \color(green)(cosπ = − 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sinπ = 0)`
` \ \ \ \ \ \ \ \ \color(green)(cos ((3π)/2 )= 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin((3π)/2) = –1)`
` \ \ \ \ \ \ \ \ \color(green)( cos 2π = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin 2π = 0)`
Now, if we take one complete revolution from the point P, we again come back to same point `P.` Thus, we also observe that if `x` increases (or decreases) by `color(blue)("any integral multiple of 2π")` `color(red)("the values of sine and cosine functions do not change.")` Thus,
` \ \ \ \ \ \ \ \ \ \ \ \ \ color(red)(sin (2nπ + x) = sin x, n ∈ Z , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cos (2nπ + x) = cos x , n ∈ Z)`
Further, `color(green)(sin x = 0),` if `color(green)(x = 0, ± π, ± 2π , ± 3π, ........,)` i.e., when `x` is an integral multiple of `π`
and `color(green)(cos x = 0)` , if `color(green)(x = pm pi/2 , pm (3 pi)/2 , pm (5 pi)/2)`,.... i.e., `cos x` vanishes when `x` is an odd multiple of `π/2` . Thus
` color(red)(sin x = 0)` implies `color(red)(x = nπ)`, where `n` is any integer
` color(red)(cos x = 0)` implies ` color(red)(x = (2n + 1)π/2)` , where `n` is any integer.
We now define other trigonometric functions in terms of sine and cosine functions:
` color(red)(cosec x = 1/(sin x)) , \ \ color(blue)(x ≠ nπ)` , where `n` is any integer.
` color(red)(sec x = 1/(cos x)) , \ \ color(blue)(x ≠ (2n +1 ) pi/2)` , where `n` is any integer
` color(red)(tan x = (sin x)/(cos x) ), \ \ color(blue)(x ≠ (2n +1) pi/2 )`, where `n` is any integer.
` color(red)(cot x = ( cos x)/(sin x)), \ \ color(blue)(x ≠ n π)`, where `n` is any integer
We have shown that for all real `x`
`color(purple)( sin^2 x + cos^2 x = 1)`
`color(purple)(1 + tan^2 x = sec^2 x)`
`color(purple)(1 + cot^2 x = cosec^2 x)`