Trigonometric Ratios of Allied Angles

Introduction 

Trigonometry is the branch of mathematics that deals with angles and their measurements. Trigonometric operations can be performed on multiple angles, compound angles, and allied angles. This article explains trigonometric ratios of allied angles.

Trigonometric ratios of allied angles can be found in the trigonometric ratios of angle Ө. If Ө is any angle, the angles in the form of 90 – Ө, 90 + Ө, 180 – Ө, 180 + Ө, 270 – Ө and so on are its allied angles. 

The sum or difference of allied angles is either 0 or 90. By applying trigonometric ratios of allied angles, the trigonometric ratios for angles of any magnitude may be found.

Allied Angles

When two parallel lines are intersected by a transversal line, the angles formed by the intersection are known as allied angles. Two angles are allied if their sum or difference is either a multiple of 90 or 0. The angles Ө, 90 ± Ө, 180 ± Ө, 270 + Ө, 360 – Ө and so on are allied angles to Ө if Ө is measured in degrees. 

Periodicity of Trigonometric Function

If f(x+T) = f(x) ∀ x, then T is called a period of f(x) if T is the smallest possible positive number. 

In periodicity, after a particular value of x, the function repeats itself.

Finding Trigonometric ratios of Allied Angles

Trigonometric ratios of angle Ө 

Consider the angle Ө placed in the standard position. Then draw a circle with center O and radius r and let the circle intersect the terminal line at a point P (x, y) as given in the figure. Draw a perpendicular line PM to the x-axis from P. Then OM = x, MP = y, and OP = r.

The six trigonometric ratios for angle Ө are defined by the formulae given below.

Trigonometry formula:

sin Ө = y/r

cos Ө = x/r

tan Ө = y/x

cosec Ө = r/y

sec Ө = r/x

cot Ө = x/y

Trigonometric ratios of angle (90-Ө)

In the given figure, the point P(x, y) is reflected in the line y = x. Then, the image of P(x, y) in this reflection is P’(y, x). Join OP’ and ∠XOP = 90 – Ө.

Here,

sin (90 – θ) = cos θ = x/r

cosec (90 – θ) = sec θ = r/x

cos (90 – θ) = sin θ = y/r

sec (90 – θ) = cosec θ = r/y

tan (90 – θ) = cot θ = x/y

cot (90 – θ) = tan θ = y/x

Trigonometric Ratios of angle (90+Ө)

In the given figure, if we rotate the point P(x, y) around the origin through 90 in an anticlockwise direction, the image of P(x, y) in this rotation is P’(-y, x). When OP’ is joined, ∠XOP’ = 90 + Ө. 

Here,

sin (90 + θ) = cos θ = x/r

cosec (90 + θ) = sec θ = r/x

cos (90 + θ) = -sin θ = -y/r

sec (90 + θ) = -cosec θ = -r/y

tan (90 + θ) = -cot θ = -x/y

cot (90 + θ) = -tan θ = -y/x

Trigonometric ratios of angle (180+Ө)

In the given figure, if we rotate the point P(x, y) around the origin through 180 in an anticlockwise direction, the image of P(x, y) in this rotation is P’(-x, -y). When OP’ is joined, ∠XOP’= 180 + Ө.

Here,

sin (180 + θ) = -sin θ = -y/r

cosec (180 + θ) = -cosec θ = -r/y

cos (180 + θ) = -cos θ = -x/r

sec (180 + θ) = -sec θ = -r/x

tan (180 + θ) = tan θ = -y/-x

cot (180 + θ) = cot θ = -x/-y

Trigonometric ratios of angle (270- θ)

In the given figure, if we reflect the point P(x, y) in the line y = -x, the image of P(x, y) in this reflection is P’(-y, -x). When OP’ is joined, ∠XOP’ = 270 – Ө.

Here,

sin (270 – θ) = -cos θ = -x/r

cosec (270 – θ) = -sec θ = -r/x

cos (270 – θ) = -sin θ = -y/r

sec (270 – θ) = -cosec θ = -r/y

tan (270 – θ) = cot θ = -x/-y

cot (270 – θ) = tan θ = -y/-x

Trigonometric ratios of angle (270+Ө)

In the given figure, if we rotate the point P(x, y) around the origin through 180 in an anticlockwise direction, the image of P(x, y) in this rotation is P’(-x, -y). When OP’ is joined, ∠XOP’ = 180 + Ө.

Here,

sin (270 + θ) = -cos θ = -x/r

cosec (270 + θ) = -sec θ = -r/x

cos (270 + θ) = sin θ = y/r

sec (270 + θ) = cosec θ = r/y

tan (270 + θ) = -cot θ = -x/y

cot (270 + θ) = -tan θ = -y/x

Trigonometric ratios of angle (360- θ) 

In the given figure, if we reflect the point P(x, y) in the x-axis line, the image of P(x, y) in this reflection is P’(x, -y). When OP’ is joined, ∠XOP’ = 360 – Ө and ∠XOP’ in the clockwise direction is -Ө.

Signs of Trigonometric Ratios

Trigonometric ratios have different signs in different quadrants. The table given below shows the signs of the ratios and angles. 

Trigonometric Ratios of Standard Angles

Trigonometric ratios of higher standard angles

The angles 120, 135, 150, and 180 are called the higher standard angles in trigonometry. The trigonometric ratios of the higher standard angles are as follows:

Trigonometric Ratios of 120

sin 120 = sin (1 × 90 + 30) = cos 30 = √3/2

cosec 120 = cosec (1 × 90 + 30) = sec 30 = 2/√3

cos 120 = cos (1 × 90 + 30) = – sin 30= – ½

sec 120 = sec (1 × 90 + 30) = – cosec 30 = – 2

tan120 = tan (1 × 90 + 30) = – cot 30 = – √3

cot 120 = cot (1 × 90 + 30) = – tan30 = – 1/√3

Trigonometric Ratios of 135

sin 135 = sin (1 × 90 + 45) = cos 45 = 1/√2

cosec 135 = cosec (1 × 90 + 45) = sec 45 = √2

cos 135 = cos (1 × 90 + 45) = – sin 45 = – 1/√2

sec 135 = sec (1 × 90 + 45) = – cosec 45 = – √2

tan 135 = tan (1 × 90 + 45) = – cot 45 = – 1

cot 135 = cot (1 × 90 + 45) = – tan45 = – 1

Trigonometric Ratios of 150°

sin 150 = sin (1 × 90 + 60) = cos 60 = 1/2

cosec 150 = cosec (1 × 90 + 60) = sec 60 = 2

cos 150 = cos (1 × 90 + 60) = – sin 60 = – √3/2

sec 150 = sec (1 × 90 + 60) = – cosec 60 = – 2/√3

tan 150 = tan (1 × 90 + 60) = – cot 60 = – 1/√3

cot 150 = cot (1 × 90 + 60) = – tan 60 = – √3

Trigonometric Ratios of 180°

sin 180 = sin (1 × 90 + 90) = cos 90 = 0

cosec 180 = cosec (1 × 90 + 90) = sec 90 = ∞

cos 180 = cos (1 × 90 + 90) = – sin 90 = – 1

sec 180 = sec (1 × 90 + 90) = – cosec 90 = – 1

tan 180 = tan (1 × 90 + 90) = – cot 90 = 0

cot 180 = cot (1 × 90 + 90) = – tan 90 = ∞

Conclusion 

By using the trigonometric ratios of allied angles, the trigonometric ratio of any angle may be calculated.