Trigonometric Addition Formulas

The trigonometric addition formulas can be applied to simplify a complicated expression or find an exact value when you are with only some trigonometric values. For instance, if you want the Sine of 15 degrees, you can use a subtraction formula to calculate sin(15) as sin(45-30). In the same way, you can also find the value of cos 15 and tan 15. But to calculate the value of some angles such as sin 120, tan 120 or cos 120, we have to guess such angles whose values can be easily remembered. Like, for sin 120, we can write it as, sin (180 – 60). Here one more identity of a trigonometric function is used, i.e. sin(180-θ). We are going to find the value for this, here in this article. But before that let us know some addition formulas for trigonometry.

Trigonometry Addition Formulas

The following are the trigonometric addition identity formulas for sine, cosine and tangent. These formulas are very helpful in solving many trigonometry problems.

• sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

• sin(a-b)= sin(a)cos(b)-cos(a)sin(b)

• cos(a+b)=cos(a)cos(b)-sin(a)sin(b)

• cos(a-b)=cos(a)cos(b)+sin(a)sin(b)

• \(\begin{array}{l}\tan (a+b) = \frac{\tan a + \tan b}{1 – \tan a \tan b}\end{array} \)

• \(\begin{array}{l}\tan (a+b) = \frac{\tan a – \tan b}{1 + \tan a \tan b}\end{array} \)

Value of Sin 120

Sin is a trigonometric ratio whose value is equal to the ratio of perpendicular to the hypotenuse of a right-angled triangle, i.e.

Sin θ = (Perpendicular/Hypotenuse)

where θ is the angle opposite to the longest side of the triangle.

Some of the angles are commonly used while trigonometric problems such as 0, 30, 45, 60 and 90 degrees. Check the below table to find the values of trigonometric angles for sin, cos and tan ratio.

To find the value of sin 120, we will use the addition formula and values of these angles.

sin 120 = sin ( 90+30)

Now using the formula,

sin(a+b)= sin(a)cos(b)+cos(a)sin(b)

We can write;

sin ( 90+30) = sin 90 cos 30 + cos 90 sin 30

Now putting the values of sin 90, cos 30, cos 90 and cos 30 from the table above, we get;

sin 120 = (1) (√3/2) – (0) (1/2)

sin 120 = √3/2

Now as we know, sin (90+θ ) = cos θ

So, sin (90+30) = cos 30 = √3/2

This proves that addition formulas are correct.

In the same way, you can find the value of cos 120 and tan 120.

Examples

Example 1: Verify the identity cos (a+b). cos (a-b) = cos² a – sin² b

Solution:

cos(a+b) cos(a-b) = (cos(a)cos(b)-sin(a)sin(b)) (cos(a)cos(b) + sin(a)sin(b))

This gives,

cos(a+b) cos(a-b) = cos² a cos² b – sin² a sin² b …………(i)

But,

sin² a.sin² b = (1- cos² a).(1- cos² b) = 1 – cos² a – cos² b + cos² a. cos² b ………(ii)

Equating the value of (ii) in (i), we have;

cos(a+b) cos(a-b)= cos² a cos² b – 1 + cos² a + cos² b – cos² a cos² b

cos(a+b) cos(a-b)= cos² a + cos² b – 1

We know, cos² b = 1 – sin² b

Therefore, we get;

cos(a + b) cos(a-b) = cos²(a) – sin²(b)

Example 2:  Calculate the exact value of cos 165°.

Solution: By applying the addition formulas for the cosine function, we get

cos(165) = cos(120+45) = cos(120)cos(45) – sin(120)sin(45)

Since we know that,

cos(120) = -½

sin(120) = √3/2

cos(45) = √2/2

sin(45) = √2/2

Therefore, we get

cos(165) = -(1/2).(2√2) – (√3/2) (√2/2)

cos 165°= -(√2+√6)/4

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