Sin3x : Formula, Derivation, Proof, and Examples

The Identity That Folds Three Angles Into One

There is something quietly surprising about sin3x: the sine of an angle three times as large can be written using nothing but the sine of the original angle. No cosine, no extra terms, just $\sin x$ raised and scaled. That compactness is why the triple-angle identity keeps appearing in equation-solving, integration, and signal work, and why it is worth knowing cold.

The formula is:

$$\sin 3x = 3\sin x - 4\sin^3 x$$

Read it carefully: the sine of the tripled angle is three times $\sin x$, minus four times $\sin x$ cubed. The cube is what makes this a triple-angle identity rather than a simple multiple, and it is the piece most first attempts leave out.

What Is The Sin3x Formula?

The sin3x formula is the triple-angle identity:

$$\sin 3x = 3\sin x - 4\sin^3 x$$

Every symbol is plain:

• $x$ is any angle, in degrees or radians.

• $3x$ is that angle tripled.

• $\sin^3 x$ means $(\sin x)^3$, the sine of $x$ cubed, not the sine of $x^3$.

The identity is often rearranged to isolate the cube, which is how the "power-reduction" form appears:

$$\sin^3 x = \frac{3\sin x - \sin 3x}{4}$$

This second form is the one calculus uses to integrate $\sin^3 x$, since a sum of plain sines is far easier to integrate than a cube. Both belong to the wider set of trigonometric identities, and sin3x sits beside its partners cos3x and tan3x in the triple-angle family.

How Do You Derive The Sin3x Formula?

The derivation writes $3x$ as $2x + x$ and then leans on the double-angle identities. Start from the sine angle-sum identity:

$$\sin 3x = \sin(2x + x) = \sin 2x \cos x + \cos 2x \sin x$$

Now substitute the sin double angle formula $\sin 2x = 2\sin x\cos x$ and the cos 2x identity $\cos 2x = 1 - 2\sin^2 x$:

$$\sin 3x = (2\sin x \cos x)\cos x + (1 - 2\sin^2 x)\sin x$$

$$= 2\sin x \cos^2 x + \sin x - 2\sin^3 x$$

Replace $\cos^2 x$ with $1 - \sin^2 x$, since that is how the cosine is expressed purely in sine:

$$= 2\sin x(1 - \sin^2 x) + \sin x - 2\sin^3 x$$

$$= 2\sin x - 2\sin^3 x + \sin x - 2\sin^3 x$$

Collect like terms:

$$\sin 3x = 3\sin x - 4\sin^3 x$$

That is the full proof. Why does everything reduce to sine alone? Because the $\cos 2x = 1 - 2\sin^2 x$ substitution converts the last cosine into sine, so no cosine survives the algebra. The building blocks come from the sum and difference identities.

Examples Of Sin3x

Example 1

Find $\sin 3x$ if $\sin x = \dfrac{1}{2}$.

Substitute $\sin x = \dfrac{1}{2}$ into the formula:

$$\sin 3x = 3 \cdot \frac{1}{2} - 4 \cdot \left(\frac{1}{2}\right)^3 = \frac{3}{2} - 4 \cdot \frac{1}{8}$$

$$= \frac{3}{2} - \frac{1}{2} = 1$$

Since $\sin x = \dfrac{1}{2}$ at $x = 30°$, this confirms $\sin 90° = 1$.

Final answer: $\sin 3x = 1$.

Example 2

Evaluate $\sin 90°$ by tripling $30°$. First the wrong instinct, then the fix.

The tempting shortcut is $\sin 3x = 3\sin x$, giving $\sin 90° = 3\sin 30° = 3 \cdot \dfrac{1}{2} = \dfrac{3}{2}$.

That is impossible: sine never exceeds $1$. The error is dropping the $-4\sin^3 x$ term and treating the triple angle as a simple triple.

The correct route keeps the cube:

$$\sin 90° = 3\sin 30° - 4\sin^3 30° = \frac{3}{2} - 4 \cdot \frac{1}{8} = \frac{3}{2} - \frac{1}{2} = 1$$

Final answer: $\sin 90° = 1$.

Example 3

Express $\sin^3 x$ in terms of $\sin x$ and $\sin 3x$.

Rearrange the formula to isolate the cube. Starting from $\sin 3x = 3\sin x - 4\sin^3 x$:

$$4\sin^3 x = 3\sin x - \sin 3x$$

$$\sin^3 x = \frac{3\sin x - \sin 3x}{4}$$

Final answer: $\sin^3 x = \dfrac{3\sin x - \sin 3x}{4}$.

Example 4

Evaluate $\sin 270°$ using $x = 90°$.

With $x = 90°$, $\sin x = 1$:

$$\sin 270° = 3\sin 90° - 4\sin^3 90° = 3(1) - 4(1)^3 = 3 - 4 = -1$$

Final answer: $\sin 270° = -1$, matching the known value.

Example 5

Integrate $\displaystyle\int \sin^3 x , dx$ using the power-reduction form.

Replace $\sin^3 x$ with $\dfrac{3\sin x - \sin 3x}{4}$ from Example 3:

$$\int \sin^3 x , dx = \frac{1}{4}\int (3\sin x - \sin 3x), dx$$

Integrate each term:

$$= \frac{1}{4}\left(-3\cos x + \frac{\cos 3x}{3}\right) + C$$

$$= -\frac{3\cos x}{4} + \frac{\cos 3x}{12} + C$$

Final answer: $\displaystyle\int \sin^3 x , dx = -\frac{3\cos x}{4} + \frac{\cos 3x}{12} + C$.

Example 6

Given $\sin x = \dfrac{3}{5}$ with $x$ in the first quadrant, find $\sin 3x$.

Substitute $\sin x = \dfrac{3}{5}$:

$$\sin 3x = 3 \cdot \frac{3}{5} - 4 \cdot \left(\frac{3}{5}\right)^3 = \frac{9}{5} - 4 \cdot \frac{27}{125}$$

$$= \frac{225}{125} - \frac{108}{125} = \frac{117}{125}$$

Final answer: $\sin 3x = \dfrac{117}{125}$.

Where Is The Sin3x Formula Used?

The identity earns its keep whenever a tripled angle needs to be handled in terms of the single angle, or a cube of sine needs to be broken down. Why prefer the formula to a calculator? Because it turns an unwieldy expression into something you can integrate, differentiate, or solve symbolically, which a decimal cannot do.

• Integration. The power-reduction form $\sin^3 x = \dfrac{3\sin x - \sin 3x}{4}$ is the standard way to integrate $\sin^3 x$, as Example 5 showed, and the same trick extends to differentiation of trigonometric functions.

• Solving trigonometric equations. Equations mixing $\sin 3x$ and $\sin x$ collapse to a single-variable cubic in $\sin x$ once the identity is applied.

• Signal processing. A pure tone driven through a non-linear device produces a third harmonic, and the $\sin 3x$ term is the mathematics of that harmonic appearing.

Common Mistakes With Sin3x

Mistake 1: Writing $\sin 3x = 3\sin x$

Where it slips in: Tripling the angle and reflexively tripling the sine.

Don't do this: $\sin 3x = 3\sin x$.

The correct way: $\sin 3x = 3\sin x - 4\sin^3 x$. The $-4\sin^3 x$ term is essential, and dropping it can produce values above $1$, which sine can never reach.

The range test catches this instantly: if a "sine" comes out larger than $1$, the cubed term went missing. The memoriser who learned "triple it" without the cube is exactly who this trap catches.

Mistake 2: Misreading $\sin^3 x$

Where it slips in: Confusing $\sin^3 x$ with $\sin x^3$ or $\sin 3x$.

Don't do this: Treating $\sin^3 x$ as $\sin(x^3)$.

The correct way: $\sin^3 x = (\sin x)^3$, the sine value cubed. It is not the sine of $x$ cubed, and it is not $\sin 3x$. These three read almost alike but are entirely different quantities.

Mistake 3: Sign Slip When Collecting Terms

Where it slips in: During the derivation, when expanding $2\sin x(1 - \sin^2 x)$.

Don't do this: Writing $2\sin x - 2\sin^3 x + \sin x + 2\sin^3 x$ and cancelling the cubes to get $3\sin x$.

The correct way: Both $\sin^3 x$ terms are negative: $-2\sin^3 x - 2\sin^3 x = -4\sin^3 x$. The second-guesser who redoes the algebra without tracking signs often lands on the wrong coefficient.

A dropped or flipped harmonic term is not only a homework issue. In audio engineering, the third harmonic that $\sin 3x$ describes is what gives total harmonic distortion its character, and mis-modelling it changes how a circuit is judged to sound.

Key Takeaways

• The sin3x formula is $\sin 3x = 3\sin x - 4\sin^3 x$, a triple-angle identity in sine alone.

• It is derived by writing $3x = 2x + x$ and substituting the double-angle identities.

• The rearranged form $\sin^3 x = \dfrac{3\sin x - \sin 3x}{4}$ is the key to integrating $\sin^3 x$.

• The most common error is writing $\sin 3x = 3\sin x$, which drops the cube and breaks the range.

To work through triple-angle identities with a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for board and entrance prep, or live math classes online with peers from 20+ countries.

A Practical Next Step

Practice these to solidify your understanding: find $\sin 3x$ when $\sin x = \dfrac{1}{\sqrt{2}}$, then use the power-reduction form to integrate $\sin^3 2x$. If you get stuck, come back to the derivation and rebuild it from $3x = 2x + x$. Want a live Bhanzu trainer to walk through the proof with you? Book a free demo class.

Read More

• Sin 2x Formula — the double angle that feeds the derivation.

• Double Angle Formula — sine, cosine, and tangent double angles in one place.

• Sin A Cos B — the product-to-sum identity from the same sum-and-difference machinery.

• Sin A Cos A — the single-angle product that equals half of sin 2A.

• Trigonometry Formulas — the full identity reference in one place.