Easy methods to find the value of any angle of trigonometry

Easy methods to find the value of any angle of trigonometry

In this article we learn how to find the values of all trigonometry by using short tricks.

We all know the basic angle values of sin, cos and tan.

Before going to learn initially we see how we will convert radian into degree and degree into radian in simple manner without confusion.

1c = (180/π)0    and      10 = (π/180)C

Conversion of degree into radian:

To convert degree into radian we simply multiply that term by (π/180)C.

For example:

  • Convert 300 into radian.

To convert degree into radian we just multiply 300 by (π/180)C.

Hence, 300 = 30*(π/180)C = (π/6)C

 

  • Similarly, convert 900 into radian.

To convert 900 into radian we simply multiply it by (π/180)c.

Hence, 900 = 900*(π/180)C = (π/2)C

In this way, we can easily convert degree into radian.

Conversion of radian into degree:

To convert radian into degree we simply multiply that term by (180/π)0.

For example:

  • Convert (π/2)C into degree.

To convert radian into degree we just multiply (π/2)C by (180/π)0.

Hence, (π/2)C = (π/2)*(180/π)0 = 900

  • Similarly, convert (3π/2)C into degree.

To convert radian into degree we simply multiply (3π/2)C by (180/π)0.

Hence, (3π/2)C = (3π/2)*(180/π)0 = 3*90 = 2700

Now, we find the basic angle values of sin, cos and tan which we have learned already.

Angle 00

 

300

or

(π/6)C

450

Or

(π/4)C

600

Or

(π/3)C

900

Or

(π/2)C

1800

Or

(π)C

Sin o 1/2 1/√2 √3/2 1 0
Cos 1 √3/2 1/√2 1/2 0 -1
tan 0 1/√3 1 √3 infinity 0

Some direct relations to find angles of trigonometry are given below:

  • Sin (nπ) = 0, where n is any integer.

For example:

Sin (π) = 0, Sin (2π) = 0, Sin (3π) = 0 and so on.

 

  • Cos (nπ) = (-1)n , where n is any integer

Cos (nπ) = 1, when n is even

Cos (nπ) = -1, when n is odd

 

For example:

Cos (2π) = 1, since n= 2 which is even number.

Similarly, Cos (3π) = -1, since n = 3 which is odd number

 

  • Cos (nπ/2) = 0, where n is odd integer
  • Cos (nπ/2) = ±1, where n is even integer

For example:

Cos (4π/2) = Cos (2π) = +1, since n = 4 which is even number.

Cos (6π/2) = Cos (3π) = -1, since n = 6 which is even number.

 

Similarly, Cos (3π/2) = 0, since n = 3 which is odd number.

Cos (5π/2) = 0, since n = 5 which is odd number.

 

  • Sin (nπ/2) = 0, when n is even number.
  • Sin (nπ/2) = ±1, when n is odd number.

For example:

Sin (4π/2) = Sin (2π) = 0, since n = 4 which is even number.

Sin (6π/2) = Sin (3π) = 0, since n = 6 which is even number.

 

Similarly, Sin (π/2) = +1, since n = 1 which is odd number.

Sin (3π/2) =-1, since n = 3 which is odd number.

Sin (5π/2) = +1, since n = 5 which is odd number.

 

Note:

Here, we can observe that the values of sin (nπ/2) for n = odd number are alternate +1 and -1.

Sine and Cosine waves in graphical form:

  • The following figure shows the sine wave in which we can see that, the values of Sin(nπ/2) for n = even number are all zero.
  • While the values of Sin (nπ/2) for n = odd number are alternately +1and -1.

Also, we can observe that the sine wave repeats its interval after angle 2π which means that the period of sine wave is 2π.

 

  • The following figure shows the cosine wave in which we can see that, the values of Cos(nπ/2) for n = odd number all are zero.
  • While the values of Cos (nπ/2) for n = even number are alternately +1 and -1.

Also, we can observe that cosine wave repeats its interval after 2π which means that the period of cosine wave is 2π.

Note:

  • Cosine wave leads sine wave by π/2.
  • Sine wave lags cosine wave by π/2.

 

  • The following figure shows the two-dimensional cartesian coordinate system with four quadrants.
  • We know that, in first quadrant both x and y are positive and hence all the trigonometry are positive in first quadrant.
  • In second quadrant, x is negative and y is positive and hence in second quadrant sin and cosec are positive only and all the other trigonometry are negative.
  • In third quadrant, x and y both are negative and hence in third quadrant tan and cot are positive only while all the other trigonometry are negative.
  • In fourth quadrant, x is positive and y is negative and hence in fourth quadrant cos and sec are positive while all the other trigonometry are negative.

Note:

To remember above concept you may use the phrase or keyword or code as “ Add Sugar To Coffee”

Where,

A– all are positive in first quadrant

S– Sine is positive in second quadrant (hence, cosec also positive)

T– tan is positive in third quadrant (hence, cot also positive)

C– cos is positive in fourth quadrant (hence, sec is also positive)

Leave a Reply

Your email address will not be published. Required fields are marked *

4 × five =