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Basic Trigonometric Ratios

A ratio of the lengths of sides of a right-angled triangle is called a trigonometric ratio. There are total six trigonometric ratios in which 3 are basic trigonometric ratios and 3 are reciprocal trigonometric ratios. The basic trigonometric ratios are sine, cosine and tangent, which are abbreviated as sin, cos and tan, respectively.

  • Sine: In a right-angled triangle, the sine of an acute angle is defined as the 'ratio of the length of the side that is opposite to the angle and the length of hypotenuse'. In the below figure, Δ PQR is a right-angled triangle and θ is an acute angle. The sine of the acute angle θ is defined as:
  • Cosine: In a right-angled triangle, the cosine of an acute angle is defined as the 'ratio of the length of the side that is adjacent to the angle and the length of the hypotenuse'. In the above figure, in a right-angled triangle PQR, the cosine of the acute angle θ is defined as:
  • Tangent: In a right-angled triangle, the tangent of an acute angle is defined as 'the ratio of the length of the side that is opposite to the angle and the length of the side that is adjacent to the angle'. In the above figure, in a right-angled triangle PQR, the tangent of the acute angle θ is defined as:

SOH–CAH–TOA is a helpful mnemonic for learning the ratios sine, cosine and tangent using the first letter of each word in the ratios. In the mnemonic SOH–CAH–TOA, SOH stands for Sine equals Opposite over Hypotenuse, CAH stands for Cosine equals Adjacent over Hypotenuse and TOA stands for Tangent equals Opposite over Adjacent.

Note: Trigonometric ratios are related to the acute angles of a right-angled triangle, not the right angle.

Reciprocal Trigonometric Ratios

Each basic trigonometric ratio has a reciprocal ratio. The reciprocals of the sine, cosine and tangent are called cosecant, secant and cotangent respectively. These are abbreviated as csc or cosec, sec and cot respectively. The table below gives the definitions of trigonometric ratios cosecant, secant and cotangent.

Trig. ratio Abbr. Reciprocal Definition
cosecant θ csc θ
secant θ sec θ
cotangent θ cot θ

Note: Cosecant has two abbreviations – csc and cosec. While csc is in line with 3 letters used for abbreviating the other 5 trigonometric ratios, cosec is more commonly used. Both of these are used in our discussions.

Signs of the T-ratios x and y are positive in the 1st quadrant. In the 2nd quad, x is negative and y is positive. Both x and y are negative in the 3rd quad, while in the 4th, x is positive and y is negative. By definition, sin θ = y/r. Therefore, the sign of sin θ is same as that of y.
Signs of the Trigonometric Ratios

Consider an angle θ = ∠XOA as shown in the adjacent figure.
Let P be any point other than 'O' on its terminal side OA.
Let PM be perpendicular from P on x-axis.
Let OP = r, OM = x and PM = y.
We consider the length OP = r as always positive,
while x and y can be +ve or -ve depending upon the quadrant in which the terminal side of the angle lies.
From the definition of the trigonometric ratios(T-ratios)
sin θ = y/r, cos θ = x/r, tan θ = y/x
cosec θ = r/y, sec θ = r/x, cot θ = x/y

Observe that
(i) the sign of sin θ is same as sign of y
(ii) the sign of cos θ is same as sign of x
(iii) the sign of tan θ depends on the signs of x and y
Similarly signs of other trigonometric ratios depend on the signs of x and/or y.

Mnemonic: All Silver Tea Cups A simple rule to remember the signs of all trigonometric ratios.
Here
A in "All" indicates that All ratios in the first quadrant are +ve
S in the "Silver" indicates that sine and cosecant are alone +ve in 2nd quadrant
T in the "Tea" indicates that tan and cot are alone +ve in 3rd quadrant
C in the "Cups" indicates that cos and sec are alone +ve in 4th quadrant
Signs of T-ratios in the four quadrants
  • In 1st quadrant :
    x > 0, y > 0
    ∴ sin θ = y/r > 0, cos θ = x/r > 0, tan θ = y/x > 0
    Similarly, cosec θ, sec θ, cot θ are also +ve.
    Hence, in 1st quadrant all trigonometric ratios are positive.
  • In 2nd quadrant :
    x < 0, y > 0
    ∴ sin θ = y/r > 0, cos θ = x/r < 0, tan θ = y/x < 0
    cosec θ = r/y > 0, sec θ = r/x < 0, cot θ = x/y < 0
    Hence, in the 2nd quadrant all trigonometric ratios are -ve except sin θ and it's reciprocal cosec θ.
  • In 3rd quadrant :
    x < 0, y < 0
    sin θ = y/r < 0, cos θ = x/r < 0, tan θ = y/x > 0
    cosec θ = r/y < 0, sec θ = r/x < 0, cot θ = x/y > 0
    ∴ Hence, in 3rd quadrant all trigonometric ratios are -ve except tan θ and its reciprocal cot θ.
  • In 4th quadrant :
    x > 0, y < 0
    sin θ = y/r < 0, cos θ = x/r > 0, tan θ = y/x < 0
    cosec θ = r/y < 0, sec θ = r/x > 0, cot θ = x/y < 0
    Hence, in 4th quadrant all trigonometric ratios are -ve except cos θ and its reciprocal sec θ.
Trigonometric Ratios of Specific Angles
Trigonometric-ratios of allied (or related) angles

Two angles are said to be allied when their sum or difference is either zero or a multiple of 90° (or radians).
The angles – θ, 90° ± θ, 180° ± θ, 360° ± θ, etc are allied angles to the angle θ.
i.e, in radians the angles – θ, ± θ, π ± θ, 2π ± θ, etc are allied angles to the angle θ.
Using the allied angles, we can find the trigonometric ratios of angles of any magnitude.

Values of trigonometrical ratios of any angle

Procedure to find the values of trigonometrical ratios of any angle in terms of +ve acute angle:

Step i: First determine the sign of given angle α.
If it is -ve, make it +ve, by using the formulae
sin(– θ) = – sin θ, cos (– θ) = cos θ
tan(– θ) = – tan θ, sec (– θ) = sec θ, cosec(– θ) = – cosec θ
Step ii: Express the positive angle α in the form of n ± θ where θ is acute angle.
Step iii: Determine the quadrant in which the terminal side of the angle α lies.
Step iv: Determine the sign of the trigonometrical function in that quadrant.
Step v: Use the trigonometric ratios of allied angles and some specific angles.
Example: Find the value of sin(– 315°).
Sol:
Step i: sin(– 315°) = – sin(315°)   [∵ sin(– θ) = – sin θ]
Step ii: – sin(90° × 3 + 45°)
Step iii: Clearly, 315° is in the 4th quadrant.
Step iv: – sin(90° × 3 + 45°)   (∵ sin(n + θ) = cos θ if n is odd and sign of sine is –ve in the 4th quadrant)
Step v: cos 45° = 1/√2
Trigonometric Identities

An equation involving a trigonometric ratio that is true for all values of the angle measure is called a trigonometric identity. By using trigonometric identities, a complex trigonometric equation can be converted into a form that's easier to understand via standard algebra tools.

Reciprocal trigonometric identities:

The reciprocal trigonometric identities define the relationship between the 6 trig. ratios: sine, cosine, tangent, cosecant, secant and the cotangent. These identities express each trigonometric ratio via a formula that makes it equal to one over another trigonometric ratio. The six identities are:

Pythagorean trigonometric identities:

The basic relation between the sine and cosine is the Pythagorean trigonometric identity:

sin2 θ + cos2 θ = 1

Dividing this identity through by either cos2 θ or sin2 θ yields two other identities:

1 + tan2 θ = sec2 θ and

1 + cot2 θ = cosec2 θ

For variations of the above identities, refer adjacent table. These identities can also be used to simplify trigonometric expressions and equations.

Quotient Identities:

Trigonometric Ratios of Compound Angles

The algebraic sum of the two or more angles are called compound angles and the angles are known as the constituent angles. For example, If A, B, C are the 3 constituent angles then A ± B, B ± C, A + B – C, A – B + C, A + B + C etc are compound angles.
The formula to express the trigonometric ratios of the compound angles in terms of the constituent angles are given below:

Trigonometric Ratios of the sum and difference of two angles

For values of trigonometric ratios of 15°, 75°, and click

Trigonometric ratios of multiple & sub-multiple angles

If A is an angle, then its integral multiples 2A, 3A, 4A, ... are called multiple angles of A.
The multiples of A by fractions such as A/2, A/3, A/4 .... are called its sub-multiple angles.

For values of trigonometric ratios of 18°, 36°, 54° and 72° click

Conditional Identities

When the angles A, B and C satisfy a given relation, many interesting identities can be established connecting the trigonometric functions of these angles. In proving these identities, the properties of complementary and supplementary angles are used.

If A + B + C = π, sin(B + C) = sin A
A + B + C = π ⇒ B + C = π – A
Apply sine on both sides
sin(B + C) = sin(π – A)
sin(B + C) = sin A (∵ sine is +ve in 2nd quadrant)

Similarly, cos B = – cos (C + A)

Other conditional identities follow as in adjacent table

For important results on conditional identities click

Properties of trigonometrical functions

Periodic function

A function f(x) is said to be periodic function with period 'T' if f(x + T) = f(x).
The least positive value of T is called the fundamental period of the function.
Note that 'T' is a real number.
All the trigonometrical functions are periodic functions.
For example: sin(x + 2π) = sin x; sin(x – 4π) = sin x; sin(x + 6π) = sin x etc.
In general,
sin(x + 2nπ) = sin x, n ∈ z
Here T = .... – 8π, – 6π, – 4π, – 2π, 0, 2π, 4π, 6π, 8π, ...
But the fundamental period is least +ve real number.
∴ T = 2π is the fundamental period
Hence, we can say that sine function is periodic function with fundamental period 2π.
The fundamental periods of other trigonometric functions are given below:

Trigonometric Function Fundamental Period
sin x, cosec x
cos x, sec x
tan x, cot x π

Note:

(i) If f(x) is periodic function with period 'T' then 1/f(x), √f(x) are also periodic functions with period 'T'

(ii) If f(x) = {x} then period of f(x) is '1' where {x} is fractional part of 'x'.

(iii) Constant function is a periodic function with no fundamental period.

(iv) Period of sinnx & cosn x → π if n is even integer
Period of sinn x & cosn x → 2π if n is odd integer
Period of tann x → π for every integer

(v) If period of f(x) is T, then periods of c + f(x), cf(x), f(x + c) are T itself

(vi) If period of f(x) is T, then period of f(ax + b) = T/| a |

(vii) If period of f(x) is T1 and period of g(x) is T2, then period of f(x) ± g(x) is L.C.M. {T1, T2}.

Examples of periodic motion

If an object repeats its motion along a certain path, about a certain point, in a fixed interval of time(T), the motion of such an object is known as periodic motion. Few common examples (with reference to adjacent images) are listed below:
(i) motion of a pendulum
(ii) motion of a spring
(iii) The vibration of a guitar string
(iv) The rotation of the earth over its axis
(v) The revolving of the earth around the sun
(vi) The revolving of the sun around the center of the Galaxy
Odd and even functions

Even Function

A function f(x) is said to be even function, if f(– x) = f(x) for all x in its domain.

Odd Function

A function f(x) is said to be odd function is f(– x) = – f(x) for all 'x' in its domain

From the allied angles concept, we can conclude that
(i) sin θ, tan θ and their reciprocals cosec θ, cot θ are odd functions
(ii) cos θ and it's reciprocal sec θ are even functions.

For domain and range of trigonometric functions refer adjacent table.

Maximum and minimum values of trigonometrical expressions

For the values of θ for which trigonometrical functions are defined, we have
– 1 ≤ sin θ ≤ 1
– 1 ≤ cos θ ≤ 1
– ∞ < tan θ < ∞
cosec θ ≥ 1 or cosec θ ≤ – 1
sec θ ≥ 1 or sec θ ≤ – 1
– ∞ < cot θ < ∞
It can be derived that the minimum and maximum values of trigonometrical expressions of the form a cos θ + b sin θ (for varying values of θ) are: and respectively.

Note:
(i) The extreme values of the function of the form a cos(αx + β) + b sin(αx + β) are also and .
(ii) The extreme values of the function of the form a cos x + b sin x + c are c and c + .

Graphs of sine and cosine functions

1. y = f(x) = sin x
Domain → R, Range → [– 1, 1]
Period → 2 π
sin2 x, |sin x| ∈ [0, 1]
sin x = 0 ⇒ x = n π, n ∈ I
sin x = 1 ⇒ x = (4n + 1), n ∈ I
sin x = – 1 ⇒ x = (4n – 1), n ∈ I
sin x = sin α ⇒ x = n π + (– 1)n α, n ∈ I
sin x ≥ 0 ⇒ x ∈ [2nπ, π + 2nπ]

2. y = f(x) = cos x
Domain → R, Range → [– 1, 1]
Period → 2 π
cos2 x, |cos x| ∈ [0, 1]
cos x = 0 ⇒ x = (2n + 1), n ∈ I
cos x = 1 ⇒ x = 2nπ, n ∈ I
cos x = – 1 ⇒ x = (2n + 1) π, n ∈ I
cos x = cos α ⇒ x = 2nπ ± α, n ∈ I
cos x ≥ 0 ⇒ x ∈ [2nπ – , 2nπ + ]

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