Sum Angle Formula

Introduction to sum angle formula:

In math trigonometric functions of the sum of two angles happen frequently in applications. There are many ways of confirming these consequences. The sum angle formulas permit you to state the accurate value of trigonometric expressions that you could not otherwise convey. This article describes the several sum angle formula and using this formula solves number of problems. Let see the important sum angle formula and its examples.

Sum Angle Formula:

Assume the sin(105°) and unlike the sin(30) this can be uttered as ½, the sin(105) cannot easily be denoted as a rational expression. However, the angle sum formula allows you to denote the accurate value of this function.

Important sum angle formula:

`sin(A+B)=sin A cos B+cos A sin B`

`cos(A+B)=cos A cos B-sin A sin B`

`tan(A+B)=(tan A + tan B)/(1-tan A tan B)`

`cot(A+B)=(cot A cot B-1)/(cot B+cot A)`

`sec(A+B)=(sec A sec B csc A csc B)/(csc A csc B-sec A sec B)`

`csc(A+B)=(sec A sec B csc A csc B)/(sec A csc B+csc A sec B)`

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Examples of Sum Angle Formula:

Example problem 1:

Find the sum of the angle of the given function sin(60+45).

Solution:

Given: sin (60+45)

Formula:

`sin(A+B)=sin A cos B+cos A sin B`

Solve:

By the formula sin(60+45) can be written as:

`sin(60+45)=sin 60 cos 45+cos 60 sin 45`

`=sqrt(3)/2*sqrt(2)/2+1/2*sqrt(2)/2`

`=(sqrt(6)+sqrt(2))/4`

Therefore, the required sum of angle is `(sqrt(6)+sqrt(2))/4`

Example problem 2:

By using the sum angle formula find the accurate value of cos(75°)

Solution:

Given: cos(75°)

here, cos(75°) can be written as cos(30°+45°)

Formula:

`cos(A+B)=cos A cos B-sin A sin B`

Solve:

`cos(30+45)=cos 30 cos 45-sin 30 sin 45`

`=sqrt(3)/2*sqrt(2)/2-1/2*sqrt(2)/2`

`=(sqrt(6)-sqrt(2))/4`

Therefore, the required sum of angle is `(sqrt(6)-sqrt(2))/4`

Example problem 3:

By using the sum angle formula find the accurate value of sin(75°)

Solution:

Given: sin(75°)

here, sin(75°) can be written as sin(30°+45°)

Formula:

`sin(A+B)=sin A cos B+cos A sin B`

Solve:

`sin(30+45)=sin 30 cos 45+cos 30 sin 45`

`=1/2*sqrt(2)/2+sqrt(3)/2*sqrt(2)/2`

`=(sqrt(2)+sqrt(6))/4`

Therefore, the required sum of angle is `(sqrt(2)+sqrt(6))/4`

Example problem 4:

By using the sum angle formula find the accurate value of cos(105°)

Solution:

Given: cos(105°)

here, cos(105°) can be written as cos(60°+45°)

Formula:

`cos(A+B)=cos A cos B-sin A sin B`

Solve:

`cos(60+45)=cos 60 cos 45-sin 60 sin 45`

`=1/2*sqrt(2)/2-sqrt(3)/2*sqrt(2)/2`

`=(sqrt(2)-sqrt(6))/4`

Therefore, the required sum of angle is `(sqrt(2)-sqrt(6))/4`