Cos (a+ b) Formula

Cos (a + b) is one of the important trigonometric identities, cos (a + b) is also called the cosine addition formula in trigonometry. Cos(a+b) is given as, cos (a + b) = cos a cos b - sin a sin b. In this article, we will learn about, cos(a + b), Proof of this Identity, How to Apply cos(a + b) Formula, and Others in detail.

Formula for Cos (a + b) is,

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

Trigonometry Identities

Trigonometric identities are equations involving trigonometric functions that are true for all possible values of the variables within their domains.

These identities are fundamental in trigonometry and are used to simplify expressions, solve equations, and establish relationships between different trigonometric functions.

What is Cos(a + b)?

The trigonometric identity for the cosine of a sum of two angles is expressed as cos(a + b) = cos(a)cos(b) - sin(a)sin(b). This identity is used to find the cosine of a compound angle, where the angle is given as the sum of two separate angles. In this context, (a + b) represents the compound angle.

Cos(a + b) Formula

Cos(A + B) is a trigonometric identity for compound angles. We use this identity when the angle for which we want to calculate the cosine function is given as the difference of two angles, such as (90° + 30°) or (45° + 15°). Angle (A + B) represents the compound angle.

Cos(a + b) Compound Angle Formula

Formula for cos(A + B) is given by,

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

This formula can express the cosine of a compound angle in terms of the sine and cosine functions of the individual angles.

Some Similar Formulas to Cos(a+b)

Some similar formulas to Cos (a+ b) Formula are:

• 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) Formula Proof

Proof of cos(A + B) formula can be done using various methods, such as Geometrical Construction Method, Using Complex Numbers, etc. Proof of cos(A + B) using complex numbers is discussed below,

Cos(A + B) formula can be derived using the complex numbers as,

e^ix = cos x + i.sin x

Let us assume x = (A + B)

e^i(A+B) = cos (A+B) + i.sin (A+B)

Now, applying exponent rule in e^i(A+B)

e^i(A+B) = e^i(A). e^i(B)

cos (A+B) + i.sin (A+B)

= {cos A + i.sin A}.{cos (B) + i.sin (B)}

= cos A.cos B + i.cos A.sin B + isin A.cos B + i²sinA.sinB

[ Here i = √-1 and i² = -1 ]

= cos A.cos B - sinA.sinB + i(cos A.sin B +sin A.cos B)

Comparing Real and Imaginary Parts,

• cos (A + B) = cos A.cos B - sin A.sin B
• sin (A + B) = sin A.cos B + cos A.sin B

Thus, cos (A+B) formula is derived.

How to Apply Cos(a + b) Formula?

Cos(A + B) formula can be used to find the value of cosine function for angles that can be expressed as the sum of standard or simpler angles. For example, we can use this formula to find cos(75°) which is not directly available in the trigonometric table.

To apply the cos(A + B) formula, we can follow simple three steps given below:

Step 1: First we need to figure out the angles A and B in the given expression, so that A + B is equal to the required angle say theta.

Step 2: Substitute the values of sin and cos of A and B from trigonometric table or using other identities.

Step 3: Simplify the expression to get the final answer.

Cos(a + b) Examples

Some examples of using cos(A + B) formula are,

Example 1: Find the value of cos(75°).

Solution:

75° can be expressed as a sum of 45° and 30°.

Therefore, a = 45° and b = 30°.

Using cos(a + b) formula, cos(a+ b) = cos a.cos b - sin a.sin b

cos(75°) = cos(45° + 30°) = cos 45°.cos 30° - sin 45°.sin 30°

Using Trigonometric Table

cos 45° = 1/√2, cos 30° =√3/2, sin 45° = 1/√2, sin 30° = 1/2

Substituting,

cos(75°) = (1/√2).(√3/2) - (1/√2).(1/2) = (√3−1)/(2√2)

Thus, exact value of cos(75°) = (√3−1)/(2√2)

Example 2: Prove that cos(60°) is equal to 1/2.

Solution:

60° can be expressed as a sum of 30° and 30°.

Therefore, a = 30° and b = 30°.

Using cos(a + b) formula, cos(a+ b) = cos a.cos b - sin a.sin b

cos(60°) = cos(30° + 30°) = cos 30°.cos 30° - sin 30°.sin 30°

Using Trigonometric Table

cos 30° = √3/2, sin 30° = 1/2

Substituting,

cos(60°) = (√3/2).(√3/2)-(1/2).(1/2)

= 3/4 - 1/4

= 2/4 = 1/2

Therefore , cos(60°) = 1/2

Hence proved.

Example 3: Calculate cos(120°).

Solution:

We know that 120° = 60° + 60°

We can apply the cos(a+b) formula i.e. cos(a+b) = cos a cos b - sin a sin b

Now we will put the values known to us: cos(120°) = cos(60°) cos(60°) - sin(60°) sin(60°)

= (1/2)(1/2) - (√3/2)(√3/2) = 1/4 - 3/4 = -1/2

Therefore , cos(120°) = -1/2

Hence proved.

Example 4: Express cos(x + π/6) in terms of cos x and sin x

Solution:

We will apply the cos(a+b) formula: cos(a+b) = cos a cos b - sin a sin b

Now let's suppose a = x and b = π/6:

cos(x + π/6) = cos x cos(π/6) - sin x sin(π/6)

Now we will put the known values for cos(π/6) and sin(π/6):

cos(x + π/6) = cos x (√3/2) - sin x (1/2).

Hence proved.

Example 5: Calculate cos(225°).

Solution:

We can write 225° as the sum of two angles: 225° = 180° + 45°

Now let's apply the cos(a+b) formula: cos(a+b) = cos a cos b - sin a sin b

Put a = 180° and b = 45°: cos(225°) = cos(180°) cos(45°) - sin(180°) sin(45°)

Put the known values:

cos(180°) = -1

cos(45°) = 1/√2

sin(180°) = 0

sin(45°) = 1/√2

Let's just Calculate:

cos(225°) = (-1)(1/√2) - (0)(1/√2)

= -1/√2

= -√2/2

Therefore, cos(225°) = -√2/2.

Hence Proved.

Practice Questions on Cos (a + b)

Some practice questions on cos(A + B) formula are given below:

Q1: Find the value of cos(105°).

Q2:Prove that cos(45°) is equal to 1/√2.

Q3: Find the value of cos(135°).

Q4: Find the value of cos(2x + y).

Q5: Find the value of cos(π/4 + π/6).

Q6: Find the value of cos(240°).

Q7: Prove that cos(60° + 30°) is equal to 1/4.

Q8: Find the value of cos(315°).

Q9: Express cos(x + π/3) in terms of cos x and sin x.

Q10: Calculate the value of cos(5π/12).

Q11: Find the value of cos(π/3 + π/4).

Q12: Prove that cos(75°) = (√6 - √2)/4.

Q13: Express cos(a - b) in terms of cos a, cos b, sin a, and sin b.

Q14: Find the value of cos(7π/6).

Q15: Calculate cos(π/6 + π/4) without using a calculator.

Conclusion

The cos(a+b) formula is a major fiduciary tool in trigonometry, giving the calculation of cosine values for compound angles. This simplifies complex trigonometric problems and relates different trigonometric functions. Only then will it be possible to understand and apply such a formula for the solving of most trigonometric equations and real situations about angles and periodic functions.