Cos(a + b)
In trigonometry, cos(a + b) is one of the important trigonometric identities involving compound angle. It is one of the trigonometry formulas and is used to find the value of the cosine trigonometric function for the sum of angles.
cos (a + b) is equal to cos a cos b - sin a sin b. This expansion helps in representing the value of cos trig function of a compound angle in terms of sine and cosine trigonometric functions. Let us understand the cos(a+b) identity and its proof in detail in the following sections.
| 1. | What is Cos(a + b)? |
| 2. | Cos(a + b)Formula |
| 3. | Proof of Cos(a + b) Formula |
| 4. | How to Apply Cos(a + b)? |
| 5. | FAQs on Cos(a + b) |
What is Cos(a + b)?
Cos(a+b) is the trigonometry identity for compound angles given in the form of a sum of two angles. It says cos (a + b) = cos a cos b - sin a sin b. It is therefore applied when the angle for which the value of the cosine function is to be calculated is given in the form of the sum of angles. The angle (a+b) here represents the compound angle.
Cos(a + b) Formula
Cos(a + b) formula is generally referred to as the cosine addition formula in trigonometry. The cos(a+b) formula can be given as,
cos (a + b) = cos a cos b - sin a sin b
where a and b are the given angles.
Proof of Cos(a + b) Formula
The verification of the expansion of cos(a+b) formula can be done geometrically. Let us see the stepwise derivation of the formula for the cosine trigonometric function of the sum of two angles in this section. In the geometrical proof of cos(a+b) formula, let us initially assume that 'a', 'b', and (a+b) are positive acute angles, such that (a+b) < 90. But this formula, in general, stands true for any positive or negative value of a and b.
To prove: cos (a + b) = cos a cos b - sin a sin b
Construction: Assume a rotating line OX and let us rotate it about O in the anti-clockwise direction till it reaches Y. OX makes out an acute angle with Y given as, ∠XOY = a, from starting position to its final position. Again, this line rotates further in the same direction and starting from the position OY till it reaches Z, thus making out an acute angle given as, ∠YOZ = b. ∠XOZ = a + b < 90°.
On the bounding line of the compound angle (a + b) take a point P on OZ, and draw PQ and PR perpendiculars to OX and OY respectively. Again, from R draw perpendiculars RS and RT upon OX and PQ respectively.
Now, from the right-angled triangle PQO we get,
cos (a + b) = OQ/OP
= (OS - QS)/OP
= OS/OP - QS/OP
= OS/OP - TR/OP
= OS/OR ∙ OR/OP + TR/PR ∙ PR/OP
= cos a cos b - sin ∠TPR sin b
= cos a cos b - sin a sin b, (since we know, ∠TPR = a)
Therefore, cos (a + b) = cos a cos b - sin a sin b.
How to Apply Cos(a + b)?
The expansion of cos(a + b) can be used to find the value of the cosine trigonometric function for angles that can be represented as the sum of standard angles in trigonometry. We can follow the steps given below to learn to apply cos(a + b) identity. Let us evaluate cos(30º + 60º) to understand this better.
- Step 1: Compare the cos(a + b) expression with the given expression to identify the angles 'a' and 'b'. Here, a = 30º and b = 60º.
- Step 2: We know, cos (a + b) = cos a cos b - sin a sin b.
⇒ cos(30º + 60º) = cos 30ºcos 60º - sin 30ºsin 60º
since, sin 60º = √3/2, sin 30º = 1/2, cos 60º = 1/2, cos 30º = √3/2
⇒ cos(30º + 60º) = (√3/2)(1/2) - (1/2)(√3/2) = √3/4 - √3/4 = 0
Also, we know that cos 90º = 0. Therefore the result is verified.
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Let us have a look a few solved examples to understand cos(a+b) formula better.
FAQs on Cos(a + b)
What is Cos(a + b) Formula?
Cos(a+b) is one of the important trigonometric identities also called cosine addition formula in trigonometry. Cos(a+b) can be given as, cos (a + b) = cos a cos b - sin a sin b, where 'a' and 'b' are angles.
What is the Formula of Cos a Plus b?
The cos(a+b) formula is used to express the cos compound angle formula in terms of sine and cosine of individual angles. Cos(a+b) formula in trigonometry can be given as, cos (a + b) = cos a cos b - sin a sin b.
What is Expansion of Cos(a + b)
The expansion of cos a plus b formula is given as, cos (a + b) = cos a cos b - sin a sin b. Here, a and b are the measures of angles.
How to Prove Cos (a + b) Formula?
The proof of cos(a + b) formula can be given using the geometrical construction method. We initially assume that 'a', 'b', and (a+b) are positive acute angles, such that (a+b) < 90. Click here to understand the stepwise method to derive cos a plus b formula.
What are the Applications of Cos (a + b) Formula?
Cos(a+b) can be used to find the value of cosine function for angles that can be represented as the sum of standard or simpler angles. Thus, it makes the deduction easier while calculating the values of trig functions. It can also be used in finding the expansion of other double and multiple angle formulas.
How to Find the Value of Cos 15º Using Cos a Plus b Identity.
The value of cos 15º using (a + b) identity can be calculated by first writing it as cos[(45º+(-30º)] and then applying cos(a+b) identity and using the trigonometric table.
⇒cos[(45º+(-30º)] = cos 45ºcos(-30)º - sin(-30)ºsin 45º = (1/√2)(√3/2) - (-1/2)(1/√2) = (√3/2√2) + (1/2√2) = (√3+1)/2√2 = (√6+√2)/4
How to Find Cos(a + b + c) using Cos (a + b)?
We can express cos(a+b+c) as cos((a+b)+c) and expand using cos(a+b) and sin(a+b) formula as, cos(a+b+c) = cos(a+b).cos c - sin(a+b).sin c = cos c.(cos a cos b - sin a sin b) - sin c.(sin a cos b + cos a sin b) = cos a cos b cos c - sin a sin b cos c - sin a cos b sin c - cos a sin b sin c.