What Is Cos A

     

Cos A - Cos B, an important identity in trigonometry, is used to find the difference of values of cosine function for angles A & B. It is one of the difference to product formulas used to lớn represent the difference of cosine function for angles A & B into their hàng hóa form. The result for Cos A - Cos B is given as 2 sin ½ (A + B) sin ½ (B - A).

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Let us understand the Cos A - Cos B formula & its proof in detail using solved examples.

1.What is Cos A - Cos B Identity in Trigonometry?
2.Cos A - Cos B Difference to sản phẩm Formula
3.Proof of Cos A - Cos B Formula
4.How to lớn Apply Cos A - Cos B Formula?
5.FAQs on Cos A - Cos B

The trigonometric identity Cos A - Cos B is used to lớn represent the difference of cosine of angles A & B, Cos A - Cos B in the product size using the compound angles (A + B) & (A - B). We will study the Cos A - Cos B formula in detail in the following sections.


The Cos A - Cos B difference to hàng hóa formula in trigonometry for angles A và B is given as,

Cos A - Cos B = - 2 sin ½ (A + B) sin ½ (A - B)

or

Cos A - Cos B = 2 sin ½ (A + B) sin ½ (B - A)

Here, A & B are angles, & (A + B) và (A - B) are their compound angles.

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We can give the proof of Cos A - Cos B trigonometric formula using the expansion of cos(A + B) & cos(A - B) formula. As we stated in the previous section, we write Cos A - Cos B = 2 sin ½ (A + B) sin ½ (B - A).

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Let us assume two compound angles A & B, given as A = X + Y & B = X - Y,

⇒ Solving, we get,

X = (A + B)/2 và Y = (A - B)/2

We know, cos(X + Y) = cos X cos Y - sin X sin Y

cos(X - Y) = cos X cos Y + sin X sin Y

cos(X + Y) - cos(X - Y) = -2 sin X sin Y

⇒ Cos A - Cos B = - 2 sin ½ (A + B) sin ½ (A - B)

⇒ Cos A - Cos B = 2 sin ½ (A + B) sin ½ (B - A)

Hence, proved.


We can apply the Cos A - Cos B formula as a difference to the sản phẩm identity. Let us understand its application using an example of cos 60º - cos 30º. We will solve the value of the given expression by 2 methods, using the formula & by directly applying the values, and compare the results. Have a look at the below-given steps.

Compare the angles A and B with the given expression, cos 60º - cos 30º. Here, A = 60º, B = 30º.Solving using the expansion of the formula Cos A - Cos B, given as, Cos A - Cos B = 2 sin ½ (A + B) sin ½ (B - A), we get,Cos 60º - Cos 30º = 2 sin ½ (60º + 30º) sin ½ (30º - 60º) = - 2 sin 45º sin 15º = - 2 (1/√2) ((√3 - 1)/2√2) = (1 - √3)/2.Also, we know that Cos 60º - Cos 30º = (1/2 - √3/2) = ( 1- √3)/2.

Hence, the result is verified.

Related Topics on Cos A + Cos B:

Let us have a look at a few examples lớn understand the concept of cos A - cos B better.


Example 1: Find the value of cos 165º - cos 15º.

Solution:

We know, Cos A - Cos B = 2 sin ½ (A + B) sin ½ (B - A)

Here, A = 165º, B = 15º

cos 165º - cos 15º = -2 sin ½ (165º + 15º) sin ½ (165º - 15º)

= -2 sin 90º sin 75º

= -2 sin 75º

= -2 sin(45º + 30º) = -2(sin 45º cos 30º + sin30º cos45º)

= -2((1/√2) (√3/2) + (1/2)(1/√2))

= -(√3 + 1)/√2


Example 3: Solve the given expression, (cos x - cos 5x)/(cos 2x - cos 4x).

Solution:

We have,

(cos x - cos 5x)/(cos 2x - cos 4x) = <-2 sin ½ (x + 5x) sin ½ (x - 5x)>/<-2 sin ½ (2x + 4x) sin ½ (2x - 4x)>

= /

= (-sin 3x sin 2x)/(-sin 3x sin x)

= sin 2x cosec x


Example 4: Verify the given expression using expansion of Cos A - Cos B: cos 70º - sin 70º = √2 sin 25º

Solution:

We have, L.H.S. = cos 70º - sin 70º

SInce sin 70º = sin(90º - 20º) = cos 20º

⇒ cos 70º - sin 70º = cos 70º - cos 20º

Using Cos A - Cos B = 2 sin ½ (A + B) sin ½ (B - A)

⇒ cos 70º - cos 20º = -2 sin ½ (70º + 20º) sin ½ (70º - 20º)

= -2 sin 45º sin 25º

= -√2 sin 25º

Hence, verified.


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