Trigonometric Ratios of Some Specific Angles

Trigonometry is all about triangles or to be more precise the relationship between the angles and sides of a triangle (right-angled triangle). In this article, we will be discussing the ratio of sides of a right-angled triangle concerning its acute angle called trigonometric ratios of the angle and find the trigonometric ratios of specific angles: 0°, 30°, 45°, 60°, and 90°.

Consider the following triangle,

The side BA is opposite to the angle ∠BCA so we call BA the opposite side to ∠C and AC is the hypotenuse; the other side BC is the adjacent side to ∠C.

Trigonometric Ratios of Angle C

Sine: Sine of ∠C is the ratio of the side opposite to C (BA) to the hypotenuse (AC).

[Tex]sin\, C = \frac{BA}{AC}    [/Tex]  

Cosine: Cosine of ∠C is the ratio of the side adjacent to C (BC) and the hypotenuse (AC).

[Tex]cos\, C = \frac{BC}{AC}    [/Tex]  

Tangent: The tangent of ∠C is the ratio between the side opposite (BA) and adjacent to C (BC). 

[Tex]tan\, C = \frac{BA}{BC}    [/Tex]  

Cosecant: Cosecant of ∠C is the reciprocal of sin C therefore it is the ratio of the hypotenuse (AC) to the side opposite to C (BA). 

[Tex]cosec\, C = \frac{AC}{BA}    [/Tex]  

Secant: Secant of ∠C is the reciprocal of cos C therefore it is the ratio of the hypotenuse (AC) to the side adjacent to C (BC). 

[Tex]sec\, C = \frac{AC}{BC}    [/Tex]  

Cotangent: Cotangent of ∠C is the reciprocal of tan C that is the ratio of the side adjacent to C (BC) to the side opposite to C (BA). 

[Tex]cot\, C = \frac{BC}{BA}[/Tex]

Considering the length of the hypotenuse AC = a, BC = b and, BA = c.

For angles 0° and 90°

If angle A = 0°, the length of the opposite side would be zero and hypotenuse = adjacent side, and if A = 90°, the hypotenuse = opposite side. So, with the help of the above formulas for the trigonometric ratios we get – 

if A = 0°   [Tex]\\ sin A = \frac{BC}{AC} = \frac{b}{a} = 0 \\\quad\\ cos A = \frac{AB}{AC} = \frac{c}{a} =1 \\\quad\\ tan A = \frac{BC}{AB} = \frac{b}{a} = 0 \\\quad\\ cosec A = \frac{AC}{BC} = \frac{a}{b} = not\, defined \\\quad\\ sec A = \frac{AC}{AB} = \frac{a}{c}= 1 \\\quad\\ cot A = \frac{AB}{BC} = \frac{a}{b}= not\, defined \\\quad\\[/Tex]

if A = 90°   [Tex]\\ sin A = \frac{BC}{AC} = \frac{b}{a} = 1 \\\quad\\ cos A = \frac{BA}{AC} = \frac{c}{a} = 0 \\\quad\\ tan A = \frac{BC}{BA} = \frac{b}{c} = not\, defined \\\quad\\ cosec A = \frac{AC}{BC} = \frac{b}{a}= 1 \\\quad\\ sec A = \frac{AC}{BA} = \frac{a}{c}= not\, defined \\\quad\\ cot A = \frac{BA}{BC} = 0[/Tex]

Here some of the trigonometric ratios result as not defined as at the particular angle it is divided by 0 which is undefined.

For angles 30° and 60°

Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore,

∠A = ∠B = ∠C = 60°.

∆ABD is a right triangle, right-angled at D with ∠BAD = 30° and ∠ABD = 60°, 

Here ∆ADB and ∆ADC are similar as they are Corresponding parts of Congruent triangles (CPCT).

[Tex]In\, \Delta ABD\;,AB=a\,,BD=\frac{a}{2} \\and\,AB^2=BD^2+AD^2\\ \quad\\\implies AD^2=AB^2-BD^2 \\ \quad\\\implies AD^2=a^2-(\frac{a}{2})^2\\ \quad\\ \implies AD^2=a^2-\frac{a^2}{4} \\ \quad\\ \implies AD^2=\frac{3a^2}{4} \\\quad\\ \implies AD= \frac{\sqrt{3} a}{2}[/Tex]

Now we know the values of AB, BD, and AD, So the trigonometric ratios for angle 30° are,

[Tex]sin\ 30=\frac{BD}{AB}= \frac{a/2}{a}=\frac{1}{2} \\ \quad\\cos\ 30=\frac{AD}{AB}=\frac{\sqrt{3}a/2}{a} =\frac{\sqrt{3}}{2} \\ \quad\\tan\ 30=\frac{BD}{AD}=\frac{a/2}{\sqrt{3}a/2}=\frac{1}{\sqrt{3}} \\\quad\\cosec\ 30=\frac{AB}{BD}=\frac{a}{a/2}=2 \\\quad\\sec\ 30=\frac{AB}{AD}=\frac{a}{\sqrt{3}a/2} =\frac{2}{\sqrt{3}}  \\\quad\\cot\ 30=\frac{AD}{BD}=\frac{\sqrt{3}a/2}{a/2}= \sqrt{3}[/Tex]

For angle 60°

[Tex]sin\ 60=\frac{AD}{AB}= \frac{\sqrt{3}a/2}{a}=\frac{\sqrt{3}}{2} \\ \quad\\cos\ 60=\frac{BD}{AB}=\frac{a/2}{a}=\frac{1}{2} \\\quad\\tan\ 60=\frac{AD}{BD}=\frac{\sqrt{3}a/2}{a/2}=\sqrt{3} \\\quad\\cosec\ 60=\frac{AB}{AD}=\frac{a}{\sqrt{3}a/2}=\frac{2}{\sqrt{3}} \\\quad\\sec=\frac{AB}{BD}=\frac{a}{a/2}=2 \\\quad\\cot\ 60=\frac{BD}{AD}=\frac{a/2}{\sqrt{3}a/2}=\frac{1}{\sqrt{3}}[/Tex]

For angle 45°

In a right-angled triangle if one angle is 45° then the other angle is also 45° thus, making it an isosceles right-angle triangle.

If the length of side BC = a then length of AB = a and length of AC(hypotenuse) is a√2 using Pythagoras Theorem, then

[Tex]sin\ A = \frac{BC}{AC} = \frac{a}{a\sqrt2} = \frac{1}{\sqrt2}\\ \quad\\ cos\ A = \frac{AB}{AC} = \frac{a}{a\sqrt2} = \frac{1}{\sqrt2}\\ \quad\\ tan\ A = \frac{BC}{AB} = \frac{a}{a} = 1\\ \quad\\ cosec\ A = \frac{1}{sin\ A}= \sqrt2\\ \quad\\ sec\ A = \frac{1}{cos\ A} = \sqrt2\\ \quad\\ cot\ A = \frac{1}{tan\ A} = 1\\[/Tex]

Some of the common values of trigonometric ratios are listed in the following table:

Related Articles

• Trigonometry in Maths
• Trigonometric Ratios 
• Trigonometry Table
• Trigonometry Formulas and Identities
• Some Applications of Trigonometry

What are trigonometric ratios?

Trigonometric ratios are ratios of the lengths of two sides in a right-angled triangle. They are used to relate the angles of a triangle to the lengths of its sides.

What are the main trigonometric ratios?

The main trigonometric ratios are sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively. Additionally, there are their reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot).

How are these ratios defined?

• Sine (sin θ) = Opposite / Hypotenuse
• Cosine (cos θ) = Adjacent / Hypotenuse
• Tangent (tan θ) = Opposite / Adjacent
• Cosecant (csc θ) = 1 / sin θ
• Secant (sec θ) = 1 / cos θ
• Cotangent (cot θ) = 1 / tan θ

What is the unit circle?

The unit circle is a circle centered at the origin (0,0) with a radius of 1. It is used to define trigonometric functions for all real numbers.