Trigonometric Ratios of Standard Angles

What are trigonometric Ratios?

Trigonometric ratios in trigonometry are derived from the three sides of a right- angled triangle basically the hypotenuse, the base (adjacent) and the perpendicular (opposite).These trigonometric formulas and trigonometric identities are used widely in all sciences related to geometry, mechanics and many others. Trigonometric ratios help us to find missing angles and missing sides of a triangle. To be more specific, they are used in right- angled triangles, the triangles with one angle equal to 90°. These trigonometric ratios help us to find the values of trigonometric standard angles. 


According to trigonometric ratio in maths, there are three basic or primary trigonometric ratios also known as trigonometric identities.


Here they are!


NAME

ABBREVIATION

RELATIONSHIP

Sine

Sin

Sin (θ)= \[\frac{Opposite}{Hypotenuse}\]

Cosine

Cos

Cos (θ)= \[\frac{Adjacent}{Hypotenuse}\]

Tangent

Tan

Tan (θ) =\[\frac{ Opposite}{Adjacent}\]


What is Sin, Cos and Tan?

1. Sine: Sine of an angle is defined as the ratio of the side opposite to the angle (to the hypotenuse (longest side) in the triangle.

2. Cosine: Cosine of an angle is defined as the ratio of the side which is adjacent to the angle to the hypotenuse (longest side) in the triangle.

3. Tangent: Tangent of an angle is defined as the ratio of the side which is opposite to the angle to the adjacent in the triangle. 


The Different Values of Sin, Cos and Tan with Respect to Radians have been Listed Down in the Table Given Below ( Trigonometry Standard Angles)

Angle

    30°

    45°

60° 

90°

 180°

 270°

360°

Radian

0

\[\frac{\pi}{6}\]

\[\frac{\pi}{4}\]

\[\frac{\pi}{3}\]

\[\frac{\pi}{2}\]

\[\pi\]

\[\frac{3\pi}{2}\]

\[2\pi\]


Tricks to Remember the Above Values:

Step 1: Divide the numbers 0, 1, 2, 3 and 4 by 4,

Step 2: Take the positive square roots.

Step 3: These numbers give the values of sin 0°, sin 30°, sin 45°, sin 60° and sin 90° respectively.

Step 4: Write down the values of sin 0°, sin 30°, sin 45°, sin 60° and sin 90° in reverse order and now you will get the values of cos , tan , cosec , sec and cot ratios respectively.

Here’s a little description about how we got the values,

Let’s take an acute angle θ , the values of sin θ and cos θ lies between 0 and 1 .

The sin of the standard angles 0°, 30°, 45°, 60° and 90° are the positive square roots of 0/4,1/4, 2/4,3/4 and 4/4 respectively.

The sine value of the standard angle table 0°, 30°, 45°, 60° ,90°, 180° and 360°:


Here are Values of the Trigonometry Standard Angles -

Sin Value  

0

\[\frac{1}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{\sqrt{3}}{2}\]

  1

0

-1

0


Similarly, we will find the cosine values of the values of other trigonometric ratios of standard angles are respectively the positive square roots of 4/4, 3/4, 2/4, 1/4, 0/4.

The cos value of the standard angles 0°, 30°, 45°, 60°,90°, 180° and 360°:


Here are Values of the Trigonometry Standard Angles -

Cos Value

    1

\[\frac{\sqrt{3}}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{1}{2}\]

  0

-1

0

1


Now, we know the sin and cos values of other trigonometric ratios of standard angles can easily be found.

The tangent value of the standard angles 0°, 30°, 45°, 60°,90, 180° and 360°:


Here are Values of the Trigonometry Standard Angles -

Tan Value 

0

\[\frac{1}{\sqrt{3}}\]

1

3

  ∞

0

0


Here are the cosecant value of the standard angles 0°, 30°, 45°, 60°, 90°, 180° and 360°:


Here is the Standard Angles Table -

Cosec Value

2

\[\sqrt{2}\]

\[\frac{2}{\sqrt{3}}\]

1

-1


Here are the secant value of the standard angles 0°, 30°, 45°, 60°, 90°, 180° and 360°:


Here is the Standard Angles Table -

Sec  Value

    1

\[\frac{2}{\sqrt{3}}\]

\[\sqrt{2}\]

2

-1

1


Here, are the cotangent value of the standard angles 0°, 30°, 45°, 60°,90°, 180° and 360° are listed below:


Here is the Standard Angles Table -

Cot Value

\[\sqrt{3}\]

1

\[\frac{1}{\sqrt{3}}\]

0

0


The trigonometric ratios of standard angles are listed below 0°, 30°, 45°, 60° and 90°.The values of trigonometric ratios of standard angles are very helpful and important to solve the trigonometric problems. Therefore, it is necessary for you to remember the value of the trigonometric ratios of standard angles. Here’s the trigonometric ratios of the standard angles table.


Table Showing the Value of Each Ratio with Respect to Different Angles ( Trigonometric Ratios of Standard Angles Table)

Angle

30°

45°

60° 

90°

180°

270°

360°

Sin 

0

\[\frac{1}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{\sqrt{3}}{2}\]

1

0

-1

0

Cos 

1

\[\frac{\sqrt{3}}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{1}{2}\]

0

-1

0

1

Tan 

0

\[\frac{1}{\sqrt{3}}\]

1

\[\sqrt{3}\]

0

0

Cot 

\[\sqrt{3}\]

1

\[\frac{1}{\sqrt{3}}\]

0

0

Cosec 

2

\[\sqrt{2}\]

\[\frac{2}{\sqrt{3}}\]

1

-1

Sec 

1

\[\frac{2}{\sqrt{3}}\]

\[\sqrt{2}\]

2

-1

1


A Few Applications of Trigonometry :

  1. Trigonometry is used in cartography which is the creation of maps.

  2. It has its applications in satellite systems.

  3. It is used in aviation industries.

  4. The functions of trigonometry are used to describe the sound and light waves.


Questions to be Solved –

Question 1) Calculate cos(A) from the triangle given below.

(image will be uploaded soon)

                       

Solution) We know the formula of cos (A) = \[\frac{Adjacent}{Hypotenuse}\]

In the given question, 

Adjacent = 12

Hypotenuse = 13

Then, cos (A) = \[\frac{12}{13}\]


Question 2) Evaluate the value of Sin 90 + Cos 90.

Solution) As we know that the value of Sin 90 = 1

And the value of Cos 90 = 0

Substituting the values of Sin 90 and Cos 90 ,

Therefore, Sin 90 + Cos 90 = 1 + 0

= 1

FAQ (Frequently Asked Questions)

Question 1) What are Standard Angles?

Answer) An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is known as the initial side and the other ray is known as the terminal side.