Sin3x is the trigonometric sine function of a triple angle 3x. The formula of sin3x in terms of sinx is given by sin3x = 3sinx – 4sin3x. That is, the formula of sine 3x can be expressed as 3 sine x – 4 sine cube x.
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The sin3x identity in terms of sin x is given as follows:
sin3x = 3sinx – 4sin3x.
Table of Contents
Sin3x Formula in Terms of sinx
The formula of sin3x is sin3x = 3sinx – 4sin3x. |
Proof:
The following steps have to be followed in order to prove the above formula of sin3x in terms of sinx.
STEP 1:
First we write 3x=2x+x, and then apply the formula sin(a+b) = sina cosb + cosa sinb. Therefore, we deduce that
sin 3x = sin(2x+x)
⇒ sin3x = sin2x cosx + cos2x sinx.
STEP 2:
In the above, let us now apply the trigonometric formulae given below.
- sin2x = 2sinx cosx
- cos2x = 1-2sin2x
So we obtain that
sin3x = 2sinx cosx cosx + (1-2sin2x) sinx
⇒ sin3x = 2sinx cos2x + sinx – 2sin3x
⇒ sin3x = 2sinx (1- sin2x) + sinx – 2sin3x as we know that sin2x+cos2x=1.
⇒ sin3x = 2sinx – 2sin3x + sinx – 2sin3x
⇒ sin3x = 3sinx – 4sin3x.
CONCLUSION: So the sin3x formula in terms of sinx is given by sin3x = 3sinx – 4sin3x.
Sin3x Formula in Terms of cosx
The following formulas will be used in order to express sin3x in terms of cosx.
- sin2x = 2sinx cosx.
- cos2x = 2cos2x -1.
As sin3x = sin(2x+x), we have that
sin3x = sin2x cosx + cos2x sinx.
⇒ sin3x = 2sinx cosx cosx + cos2x sinx.
⇒ sin3x = 2sinx cos2x + (2cos2x -1) sinx
⇒ sin3x = sinx (2cos2x + 2cos2x -1)
⇒ sin3x = $\sqrt{1-\cos^2x}$ (4cos2x -1) as sin2x = 1-cos2x.
So the formula of sin3x in terms of cosx is given by sin3x = $\sqrt{1-\cos^2x}$ (4cos2x -1).
Question-Answer
Question 1: Find the value of sin135 degree. |
Answer:
Applying the sin3x formula sin3x = 3sinx – 4sin3x with x=45°, we get that
sin135° = sin(3⋅45)°
⇒ sin135° = 3sin45° – 4sin345°
⇒ sin135° = 3 × $(\dfrac{1}{\sqrt{2}})$ – 4 × $(\dfrac{1}{\sqrt{2}})^3$
⇒ sin135° = $\dfrac{3}{\sqrt{2}}$ – $\dfrac{4}{2\sqrt{2}}$ = $\dfrac{3}{\sqrt{2}}$ – $\dfrac{2}{\sqrt{2}}$
⇒ sin135° = $\dfrac{1}{\sqrt{2}}$.
So the value of sin135 degree is equal to 1/√2. That is, sin135° is equal to 1/√2 in fraction form.
Question 1: Find the value of sin270 degree. |
Answer:
Apply sin3x formula sin3x = 3sinx – 4sin3x with x=90°. Therefore,
sin270° = sin(3⋅90)°
⇒ sin270° = 3sin90° – 4sin390°
⇒ sin270° = 3sin90° – 4sin390° as we know sin90° = 1.
⇒ sin270° = 3×1 – 4×13
⇒ sin270° = 3-4 = -1.
So the value of sin270 degree is equal to -1.
FAQs
Answer: The formula of sin3x in terms of sinx is given by sin3x = 3sinx -4sin3x.
Answer: sin3x = $\sqrt{1-\cos^2x}$ (4cos2x -1) is the formula of sin3x in terms of cosx.
Answer: The value of sin270° is equal to -1.
This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.