Sum and Difference Formulas in Trigonometry

In this article, we will discuss about the sum and difference formulas in various parts of mathematics. First we see the sum and difference formulas in trigonometry for various trig functions. The sum & difference formulas include two angles which will be defined and the angles are applied to the various fundamental trig functions. The formulas show the relationship between the two angles and trig functions. These formulas are very useful to solve the problems in trigonometry.

First we discuss about sum and difference formulas for sine function. Suppose we have two angles named as (a) and (b), then for the two angles we write the relationship as sin (a+b) =sin (a) cos (b) +cos (a) sin (b). the  difference formula is expressed as sin(a-b)=sin(a)cos(b)-cos(a)sin(b). To prove these formulas we have to use geometry calculus. Now we take cosine function, suppose we have same angles, then for the two angles we expressed sum formula as cos(a+b)=cos(a)cos(b)-sin(a)sin(b) and difference formula expressed as cos(a-b)=cos(a)cos(b)+sin(a)sin(b).

The sum & difference formulas for third trigonometric function mean Sum and difference formulas for tangent function. This formula is valid for all values where tan a, tan b and tan (a+b) are used. Where (a) and (b) are the two angles. The formula can be expressed as tan (a+b) = (tan a+tan b/1-tan a*tan b) and difference formula is tan (a-b) = (tan a-tan b/1+tan a*tan b). The formulas for tangent function also used for finding the angle between two lines. But the question is how to find the angle, so for finding angle we have to calculate the slope of both lines. The equations can be written as tan θ= (m1-m2/1+m1*m2), where m1 is slope of first line and m2 is slope of second line.

Now sum and difference formulas examples. First we take example for sine and cosine function then tangent function. First problem is, suppose we have to calculate exact value of sin (75°). For this we use sum angle formula such as sin(75°)=sin(30°+45°)=sin(30°)*cos(45°)+cos(30°)*sin(45°) and we know the value of sin(30) and sin(45). Second problem is, suppose we have cos x=1/2 and cos y=1/3 then we calculate the value of cos(x+y) and cos(x-y). So using sum and difference formula cos(x+y) = (1/2*1/3-1/3*1/2) =cos 0=1.

Now problem based on tangent formulas. First problem is, find the exact value of tan (105). Using sum formula tan (105°) =tan (60°) +tan (45°)/1-tan (60°)*tan (45°) and we well know the value of tan (60°) and tan (45°). Second problem is, suppose given data is y=3x-5 and y=-2x+2. From these two data we write slope of both lines m1=3 and m2=-2. Then we use angle formula tanθ= [3-(-2)/1+ (3)*(-2)]. After simplifying we calculate the angle.