Sin3x Formula in terms of sinx [with Proof]

 Sin3x formula in terms of sinx is as follows: 3sinx – 4sin³x. The formula can be expressed as sine 3x = 3 sine x – 4 sine cube x. In this post, we will learn how to prove sin3x formula along with some examples. This is very useful in the area of Trigonometry to solve various trigonometric equations, identities, etc.

Sin3x Formula

Sin3x is the sine trigonometric function of a triple angle 3x. The formula of sin 3x in terms of sinx is given below.

sin3x=3sinx – 4sin³x.

sin3x in Terms of sin x

To derive the sin3x formula, we will use the following trigonometric formulas:

1. sin(a+b) = sin acos b +cos a sin b
2. sin 2x=2sin x cos x
3. cos 2x=1-2sin²x
4. sin²x+cos²x=1

Below are the steps to be followed in order to establish the formula of sin 3x.

Step 1: At first, we will write 3x as 2x+x, and then we will apply the above formula 1. By doing so, we get that

sin 3x=sin(2x+x)

= sin 2x cos x +cos 2x sin x

Step 2: Now, we will apply the formulas of sin2x and cos2x which are given in (2) and (3) above.  Thus, we have

sin 3x=(2sin x cos x) cos x +(1-2sin²x)sin x

= 2sin x cos² x+sin x-2sin³x

= 2sin x (1-sin²x)+sin x-2sin³x as we know that cos^2x=1-sin^2x follows from the above formula (4).

= 2sin x -2sin³x +sin x -2sin³x

= 3sin x -4sin³x .

Hence, the formula of sin 3x is 3sin x -4sin³x.

Also Read: 

Simplify 1-sin2(x)

Values of sin15, cos15, tan15

Values of sin75, cos75, tan75

Simplify cos(sin-1x)

Simplify sin(cos-1x)

Question: Find the value sin270 degree.

Answer:

In the above formula of sin3x, that is, sin3x=3sinx-4sin^3x, we put x=90 degree. Thus, we obtain that

sin 270°=3sin 90° – 4sin³90°

= 3×1 – 4×1³ as we know that sin90°=1

= 3-4

= -1

Thus, the value of sin270 degree is -1.

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