Sin 3x Formula: Derivation, Examples, FAQs – Mindspark
Sin 3x Formula
The trigonometric function \sin 3x formula equals \left(3 \sin x-4 \sin ^{3} x\right).In trigonometry, \sin 3x is a triple angle identity. We can derive this formula using the angle addition identity of the sin function.
Derivation of Sin 3x Formula
For deriving the formula, we will write the angle 3x as (2x + x). Apart from this, we will use a few trigonometric identities given below:
- \sin(a+b)=(\sin a.\cos b)+(\cos a.\sin b)
- \sin 2x=2\sin x\cos x
- \cos 2 x=1-2 \sin ^{2} x
- \sin ^{2} x+\cos ^{2} x=1
Now, let’s derive the formula using the above details.
\sin 3x=\sin(2x+x)
=(\sin 2x\cos x)+(\cos 2x\sin x) \text { [Using } \sin(a+b) \text { formula]}
=(2 \sin x \cos x) \cos x+\left(1-2 \sin ^{2} x\right) \sin x \text { [Using } \sin 2 x \text { and } \cos 2 x \text { formula] }
=\left(2 \cos ^{2} x \sin x\right)+(\sin x)-\left(2 \sin ^{3} x\right)
=2\left(1-\sin ^{2} x\right) \sin x+(\sin x)-\left(2 \sin ^{3} x\right)\left[\text { Since } \sin ^{2} x+\cos ^{2} x=1, \text { Hence } \cos ^{2} x=1-\sin ^{2} x\right]
=2 \sin x-2 \sin ^{3} x+(\sin x)-\left(2 \sin ^{3} x\right)
=(2 \sin x+\sin x)-2 \sin ^{3} x-2 \sin ^{3} x
=(2 \sin x+\sin x)-\left(2 \sin ^{3} x+2 \sin ^{3} x\right)
=\left(3 \sin x-4 \sin ^{3} x\right)
Thus we proved the formula of \sin 3x using the angle addition identity.
Examples
1. Determine the value of sin 180° using the sin 3x identity?
Assume 3x = 180°
⇒ x = 180°/3 = 60°
We know that the formula of \sin 3 x=3 \sin x-4 \sin ^{3} x
Substituting the values of 3x and x
\operatorname{Sin} 180^{\circ}=3 \sin 60^{\circ}-4 \sin ^{3}\left(60^{\circ}\right) =3(\sqrt{3} / 2)-4(\sqrt{3} / 2)^{3}= 3(√3/2) – 4 {(3(√3)/8}
= 3(√3/2) – 3(√3/2)
= 0
Therefore, the value of sin 180° is 0 using the sin 3x identity.
2. Calculate the value of sin 270° using the sin 3x formula
We know that the formula of \sin 3 x=3 \sin x-4 \sin ^{3} x.
Assume 3x = 270°
⇒ x = 270°/3 = 90°
Substituting the values of 3x and x
\sin 270^{\circ}=3 \sin \left(90^{\circ}\right)-4 \sin ^{3}\left(90^{\circ}\right)
=(3 \times 1)-4 \times(1)^{3}
= 3 – 4
= (-1)
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Frequently Asked Questions
1. What is the formula of sin 3x in trigonometry?
Ans: We use the sin 3x formula to determine the value of the sine function for an angle that is three times angle x in measurement. The formula is given by \sin 3 x=(3 \sin x)-\left(4 \sin ^{3} x\right).
2. Are sin 3x and 3 (sinx) the same?
Ans: No, these are not the same as sin 3x is the value of the sine function for an angle that is three times angle x in measurement and 3 (sin x) is three times the value of sin x.
3. How can we derive the formula of sin 3x?
Ans: We use the angle addition identity to derive the formula of sin 3x. First write the angle 3x as (2x + x). After that, we use some of the trigonometric identities given below to prove the sin 3x identity:
\sin (a+b)=(\sin a.\cos b)+(\cos a.\sin b)
\sin 2x=2\sin x\cos x
\cos 2 x=1-2 \sin ^{2} x
\sin ^{2} x+\cos ^{2} x=1