Trigonometric Ratios of Standard Angles Made Simple

Trigonometric Ratios in Trigonometry are derived from the three sides of a right-angled triangle: 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 the trigonometric ratio in Math, there are three basic or primary Trigonometric Ratios also known as trigonometric identities.Here they are:

What is Sin, Cos and Tan?

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

2. Cosine: The 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 the angle to the adjacent in the triangle. 

The Different Values of Sin, Cos and Tan concerning Radians have been Listed Down in the Table Given Below 

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 of how we got the values. Let's take an acute angle θ. The values of sin θ and cos θ lie 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°:

Derivation of Values of Trigonometric Standard Angles

We will deal with the Trigonometric Ratios of 30 degrees and 60 degrees first. 

To do so, take an equilateral triangle ABC whose side is 2 units.

Then AB=BC=CA=2.

Let AD be the altitude of this aforementioned triangle we have taken.

This implies that BD=1.

From the Pythagoras theorem, we then get AD = √3.

From ΔABD, we will use the definitions of the various trigonometric quantities to find their standard values. Thus, 

• Sin 60 = √3/2 which implies that Cosec 60 = 2/ √3

• Cos 60 = 1/2 which implies that Sec 60 = 2

• Tan 60 = √3 which implies that Cot 60 = 1/√3

Again from ΔABD, we will determine the values again. 

• Sin 30 = ½ which implies that Cosec 30 = 2

• Cos 30 = √3/2 which implies that Sec 30 = 2/√3

• Tan 30 = 1/√3 which implies that Cot 30 = √3.

Let us now derive the values of 45 degrees of these trigonometric quantities. 

This time, we will take an isosceles right-angled triangle ABC with a right angle at B and AB=BC =1 unit.

Using the definitions of Trigonometric Ratios, we see that:

• Sin 45 = 1/√2 which implies that Cosec 45 = √2

• Cos 45 = 1/√2 which implies that Sec 45 = √2

• Tan 45 = 1 which implies that Cot 45 = 1.

Here are Values of the Trigonometry Standard Angles 

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 

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

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

Here is the Standard Angles Table 

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

Here is the Standard Angles Table 

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

Here is the Standard Angles Table

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)

Few Applications of Trigonometry

• Trigonometry is used in cartography, which is the creation of maps. 

• It has its applications in satellite systems. 

• It is used in the aviation industry. 

• The functions of Trigonometry are used to describe sound and light waves.

Questions to be Solved

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) =12/13

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