`star` Trigonometric Functions

`star` Sign of trigonometric functions

`star` Domain and range of trigonometric functions

`star` Sign of trigonometric functions

`star` Domain and range of trigonometric functions

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)`

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)`

Q 3126501471

Find the value of `sin( (31pi)/3)`

We know that values of sin `x` repeats after an interval of `2π.` Therefore

`sin( 31pi/3 )= sin (10pi+pi/3) = sin = sqrt3/2`

Q 3166501475

Find the value of `cos (–1710°).`

We know that values of cos x repeats after an interval of `2π` or `360°.`

Therefore, `cos (–1710°) = cos (–1710° + 5 × 360°)`

`cos (–1710° + 1800°) = cos 90° = 0.`

Let `P (a, b)` be a point on the unit circle with centre at the origin such that `∠AOP = x.` If `∠AOQ = – x,` then the coordinates of the point `Q` will be `(a, –b)` (Fig 3.7). Therefore

`color(blue)(cos (– x) = cos x)`

and `color(blue)(sin (– x) = – sin x)`

Since for every point `P (a, b)` on the unit circle, `– 1 ≤ a ≤ 1` and `– 1 ≤ b ≤ 1,`

we have `– 1 ≤ cos x ≤ 1` and `–1 ≤ sin x ≤ 1` for all `x.`

`=>` In the `color(green)(ul"first quadrant"\ \ ( 0 < x < pi/2)) , ` `color(red)(a)` and `color(red)(b)` are both positive,

`=>` In the `color(green)(ul"second quadrant" \ \ ( pi/2 < x < pi )) , ` `color(red)(a)` is negative and `color(red)(b)` is positive,

`=>` In the `color(green)(ul"third quadrant" \ \ (pi < x < (3pi)/2 )), ` `color(red)(a)` and `color(red)(b)` are both negative and

`=>` In the `color(green)(ul"fourth quadrant" \ \ ( (3 pi)/2 < x < 2 pi )), ` `color(red)(a)` is positive and `color(red)(b)` is negative.

Therefore, `sin x` is positive for `0 < x < π,` and negative for `π < x < 2π.`

Similarly, `cos x` is positive for `0 < x < pi/2` , negative for ` pi/2< x < (3 pi)/2` and also positive for `(3 pi)/2 < x < 2 pi` .

Likewise, we can find the signs of other trigonometric

functions in different quadrants. In fact, we have the following table.

`color(blue)(cos (– x) = cos x)`

and `color(blue)(sin (– x) = – sin x)`

Since for every point `P (a, b)` on the unit circle, `– 1 ≤ a ≤ 1` and `– 1 ≤ b ≤ 1,`

we have `– 1 ≤ cos x ≤ 1` and `–1 ≤ sin x ≤ 1` for all `x.`

`=>` In the `color(green)(ul"first quadrant"\ \ ( 0 < x < pi/2)) , ` `color(red)(a)` and `color(red)(b)` are both positive,

`=>` In the `color(green)(ul"second quadrant" \ \ ( pi/2 < x < pi )) , ` `color(red)(a)` is negative and `color(red)(b)` is positive,

`=>` In the `color(green)(ul"third quadrant" \ \ (pi < x < (3pi)/2 )), ` `color(red)(a)` and `color(red)(b)` are both negative and

`=>` In the `color(green)(ul"fourth quadrant" \ \ ( (3 pi)/2 < x < 2 pi )), ` `color(red)(a)` is positive and `color(red)(b)` is negative.

Therefore, `sin x` is positive for `0 < x < π,` and negative for `π < x < 2π.`

Similarly, `cos x` is positive for `0 < x < pi/2` , negative for ` pi/2< x < (3 pi)/2` and also positive for `(3 pi)/2 < x < 2 pi` .

Likewise, we can find the signs of other trigonometric

functions in different quadrants. In fact, we have the following table.

Q 3126401371

If `cos x = – 3/5 x` lies in the third quadrant, find the values of other five trigonometric functions.

Since cos `x = - 3/5` , we have sec `x = -5/3`

Now `sin^2 x + cos^2 x = 1, i.e., sin^2 x = 1 – cos^2 x`

or `sin_2 x = 1 – 9/25 = 16/25`

Hence `sin x = ± 4/5`

Since `x` lies in third quadrant, sin `x` is negative. Therefore

`sin x = – 4/5`

which also gives

`cosec x = 5/4`

Further, we have

`tan x = (sinx)/cosx = 4/3` and `cot x = (cosx)/(sinx) = 3/4`

Q 3176401376

If `cot x = – 5/12, x` lies in second quadrant, find the values of other five trigonometric functions.

Since cot `x = – 5/12` we have `tan x = - 12/5`

Now `sec^2 x = 1 + tan^2 x = 1 + 144/25 = 169/25`

Hence `sec x = ± 13/5`

Since x lies in second quadrant, sec `x` will be negative. Therefore

`sec x = – 13/5`

which also gives

`cos x = – 5/13`

Further, we have

`sin x = tan x cos x = (-12/5)xx (-5/13) = 12/13`

and `cosec x = 1/ sin x = 13/12`

From the definition of sine and cosine functions, we observe that they are defined for all real numbers.

`=>` Further, We observe that for each real number `x,`

`color(purple)(– 1 ≤ sin x ≤ 1)` and `color(purple)(– 1 ≤ cos x ≤ 1)`

`=>` Thus, `color(purple)(ul"domain")` of `color(red)(y = sin x)` and `color(red)(y = cos x)` is the `color(blue)("set of all real numbers")` and `color(purple)(ul"range")` is the interval `color(blue){([–1, 1])},` i.e., `– 1 ≤ y ≤ 1.`

`=>` Since `color(red)(cosec x =1/(sin x ))` , the domain of `y = cosec x` is the set `{ x : x ∈ R` and `x ≠ n π, n ∈ Z}` and range is the set `{y : y ∈ R, y ≥ 1` or `y ≤ – 1}.`

Similarly, the domain of

`=>` `color(red)(y = sec x)` is the set `{x : x ∈ R` and `x ≠ (2n + 1) π/2 , n ∈ Z}` and `color(purple)("range")` is the set `{y : y ∈ R, y ≤ – 1` or `y ≥ 1}.`

`=>` The domain of `color(red)(y = tan x)` is the set `{x : x ∈ R` and `x ≠ (2n + 1) π/2 , n ∈ Z}` and `color(purple)("range")` is the `color(blue)("set of all real numbers")`.

`=>` The domain of `color(red)(y = cot x)` is the set `{x : x ∈ R` and `x ≠ n π, n ∈ Z}` and the `color(purple)("range")` is the `color(blue)("set of all real numbers")`.

`=>` Further, We observe that for each real number `x,`

`color(purple)(– 1 ≤ sin x ≤ 1)` and `color(purple)(– 1 ≤ cos x ≤ 1)`

`=>` Thus, `color(purple)(ul"domain")` of `color(red)(y = sin x)` and `color(red)(y = cos x)` is the `color(blue)("set of all real numbers")` and `color(purple)(ul"range")` is the interval `color(blue){([–1, 1])},` i.e., `– 1 ≤ y ≤ 1.`

`=>` Since `color(red)(cosec x =1/(sin x ))` , the domain of `y = cosec x` is the set `{ x : x ∈ R` and `x ≠ n π, n ∈ Z}` and range is the set `{y : y ∈ R, y ≥ 1` or `y ≤ – 1}.`

Similarly, the domain of

`=>` `color(red)(y = sec x)` is the set `{x : x ∈ R` and `x ≠ (2n + 1) π/2 , n ∈ Z}` and `color(purple)("range")` is the set `{y : y ∈ R, y ≤ – 1` or `y ≥ 1}.`

`=>` The domain of `color(red)(y = tan x)` is the set `{x : x ∈ R` and `x ≠ (2n + 1) π/2 , n ∈ Z}` and `color(purple)("range")` is the `color(blue)("set of all real numbers")`.

`=>` The domain of `color(red)(y = cot x)` is the set `{x : x ∈ R` and `x ≠ n π, n ∈ Z}` and the `color(purple)("range")` is the `color(blue)("set of all real numbers")`.