Solve Standard Angles

Introduction to solve standard angles:

The degree is the symbol used to denote the measurement of the angles. In geometry, the angles usually are measured in degrees. The standard angles of the geometric shapes can be find by the  measures of the other angles. The students will also refer to the websites for their doubt clarification. Now we prepare for solving the standard angles.

The standard angles includes 0o, 30o, 45o, 60o and 90o, called as standard as the trigonometric ratios are to obtained or the function is to be determined accurately at the standard angle values.
How to Solve Standard Angles :

Lets see how we get the values of different standard ratio angles from some standard triangles

1 . 45 - 90 - 45 triangle

std triangle

In the given triangle ,

sin 45o  =  `(opp. side) / (hyp. side)`

=  `a / (a sqrt(2))`

=  ` 1 / (sqrt (2))`

cos 45o = `(adj side) / (hyp. side)`

=  `a / (a sqrt(2))`

=   `1 / (sqrt (2))`

tan 45o = `(opp. side) / (adj. side)`

= `a/a `

= 1

2 . 30 - 90 - 60 triangle

standard angles

In the given triangle ,

sin 30o  =  `(opp. side) / (hyp. side)`

= `a / (2a)`

= `1/2 `

sin 60o  = `(a sqrt(3)) / (2a)`

= `(sqrt(3)) / 2`

cos 30o = `(adj side) / (hyp. side)`

= `(a sqrt(3)) / 2a`

= `(sqrt(3)) / 2`

cos 60o  = `a / (2a)`

= `1 /2`

tan 30o = `(opp. side) / (adj. side)`

= `a /(a sqrt(3))`

= `1 / (sqrt(3))`

tan 60o = `( a sqrt(3))/ a`

= `sqrt(3)`

 

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Examples Based on Solving Standard Angles :

Example 1 : Without makung use of the trigonometric tables , determine the exact value of the following

`( sin^2 45^o + cos^2 45^o ) / (tan^2 60^o) `

Solution : As we know the values of the standard angles as

sin 45o = `1/sqrt 2 `

cos 45o =` 1/sqrt 2 `

tan 60o= `sqrt(3)`

so by plugging in the respective values , we have

`( sin^2 45^o + cos^2 45^o ) / (tan^2 60^o) `` (((1/(sqrt(2)))^2) + ((1/sqrt(2))^2 )) / (((sqrt(3))^2))`

=  `((1/2)+(1/2))/ 3`

=`(1)/ 3`

Example 2 : Evaluate the following values from the standard angles :

20 sin 30˚ + 2 cos 60˚ - 18 tan 45˚

Solution

20 sin 30˚ + 2 cos 60˚ - 18 tan 45˚

As we know that the value of the standard angles are

sin 30o = `1/2 ` <br>



cos 60o =  `1/2 `

tan 45o= 1

By putting the respective values we have ,

= 20 x `1/2`   + 2 x `1/2`  - 18

= 10 + 1 - 18

= 11 - 18

= - 9

So , 20 sin 30˚ + 2 cos 60˚ - 18 tan 45˚ = -9 (Answer)

 

I am planning to write more post on What are Adjacent Angles and its example and, cbse 11th sample paper and its problem with solution. Keep checking my blog.

 

Practice Problems Based on Solving Standard Angles :

Problem 1 : Without makung use of the trigonometric tables , determine the exact value of the following

`( sin^2 60^o + cos^2 30^o ) / (tan^2 45^o) `

Problem 2 : Evaluate the following values from the standard angles :

20 sin 30˚ + 2 cos 60˚ - 18 tan 45˚