Sin a

     

Sin A - Sin B is an important trigonometric identity in trigonometry. It is used khổng lồ find the difference of values of sine function for angles A and B. It is one of the difference to sản phẩm formulas used to represent the difference of sine function for angles A và B into their product form. The result for Sin A - Sin B is given as 2 cos ½ (A + B) sin ½ (A - B).

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

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

The trigonometric identity Sin A - Sin B is used lớn represent the difference of sine of angles A and B, Sin A - Sin B in the product size with the help of the compound angles (A + B) & (A - B). Let us study the Sin A - Sin B formula in detail in the following sections.


The Sin A - Sin B difference to product formula in trigonometry for angles A và B is given as,

Sin A - Sin B = 2 cos ½ (A + B) sin ½ (A - B)

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

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

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

⇒ Solving, we get,

X = (A + B)/2 và Y = (A - B)/2

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

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

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

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

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

Hence, proved.


Sin A - Sin B trigonometric formula can be applied as a difference to the product identity lớn make the calculations easier when it is difficult khổng lồ calculate the sine of the given angles. Let us understand its application using an example of sin 60º - sin 30º. We will solve the value of the given expression by 2 methods, using the formula and 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, sin 60º - sin 30º. Here, A = 60º, B = 30º.Solving using the expansion of the formula Sin A - Sin B, given as, Sin A - Sin B = 2 cos ½ (A + B) sin ½ (A - B), we get,Sin 60º - Sin 30º = 2 cos ½ (60º + 30º) sin ½ (60º - 30º) = 2 cos 45º sin 15º = 2 (1/√2) ((√3 - 1)/2√2) = (√3 - 1)/2.Also, we know that Sin 60º - Sin 30º = (√3/2 - 1/2) = (√3 - 1)/2.

Hence, the result is verified.

Topics Related to Sin A - Sin B:


Example 1: Find the value of sin 145º - sin 35º using sin A + sin B identity.

Solution:

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

Here, A = 145º, B = 35º

sin 145º - sin 35º = 2 cos ½ (145º + 35º) sin ½ (65º - 35º)

= 2 cos 90º cos 15º

= 0 < cos 90º = 0>


Example 3: Solve the given expression, (sin x - sin 5x)/(sin x + sin 5x).

Solution:

We have,

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

= /

= -cos 3x sin 2x/sin 3x cos 2x

= - tung 2x cot 3x


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

Solution:

We have, L.H.S. = sin 70º - cos 70º

Since, cos 70º = cos(90º - 20º) = sin 20º

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

Using Sin A - Sin B = 2 cos ½ (A + B) sin ½ (A - B)

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

= 2 cos 45º sin 25º

= √2 sin 25º

= R.H.S.

Hence, verified.


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