Cos3x Formula : Derivation, Graph & Examples

What Is Cos3x?

Cos3x is the cosine of three times an angle $x$, and the cos3x formula expresses that value using only the cosine of the single angle:

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

Here $x$ is any angle, measured in degrees or radians, and $\cos^3 x$ means $(\cos x)^3$, the cube of the cosine. The identity holds for every value of $x$, which is what makes it an identity rather than an equation to solve. It belongs to the family of multiple-angle rules in the trigonometric identities toolkit.

Before the identity is useful, one distinction has to be clear, because search data shows it is the single most-asked question about the topic.

The Identity Hiding In A Distorted Note

Push a guitar amp into overdrive and the clean note sprouts a harsh third harmonic. That gritty edge is a wave oscillating three times as fast as the original, and the algebra that rewrites "three times the angle" in terms of the plain angle is exactly the cos3x identity. Overdrive, radio mixing, and the polynomials inside digital filters all lean on it.

The formula does one specific job: it converts $\cos(3x)$, a cosine of a tripled angle, into an expression that uses only $\cos x$. That conversion is what lets an engineer or a student compute a triple-angle value without a calculator, and it is the reason the identity sits in every trigonometry syllabus alongside the double angle formula.

Is Cos3x The Same As Cos Cube X?

No. They are different expressions that unluckily look alike.

• $\cos 3x$ is the cosine of the angle $3x$, so you first triple the angle, then take the cosine.

• $\cos^3 x$ is the cosine of $x$, then cubed - first take the cosine, then raise it to the third power.

They are linked, though. Rearranging the cos3x formula gives the power-reduction version:

$$\cos^3 x = \frac{\cos 3x + 3\cos x}{4}$$

So $\cos^3 x$ is built from $\cos 3x$ and $\cos x$, but the two are never equal except at isolated angles. Confusing the two is where a lot of integration and simplification work goes wrong.

How Do You Derive The Cos3x Formula?

The derivation starts by splitting the tripled angle into a sum, then applying rules the reader already has. Write $3x$ as $2x + x$ and use the cosine angle-sum identity $\cos(A + B) = \cos A \cos B - \sin A \sin B$:

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

Now substitute the double-angle forms $\cos 2x = 2\cos^2 x - 1$ (see cos 2x) and $\sin 2x = 2\sin x \cos x$ (see sin 2x):

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

Expand both products, one term per line:

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

Replace $\sin^2 x$ with $1 - \cos^2 x$ using the Pythagorean identity:

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

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

Collect like terms:

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

Every symbol traces back to $\cos x$, which is the whole point. The result is a cubic in $\cos x$, and that cubic is why the identity connects to Chebyshev polynomials later on.

What Are The Properties Of Cos3x?

Every feature of $y = \cos 3x$ can be read off the graph and the formula.

• Domain: all real numbers - any angle can go in.

• Range: $[-1, 1]$. Like every cosine, the output stays inside this band.

• Period: $\frac{2\pi}{3}$ (or $120^\circ$). Because the angle runs three times as fast, one full cycle finishes in a third of the usual distance.

• Amplitude: $1$. The coefficient in front is $1$, so the wave still peaks at $\pm 1$.

• Even function: $\cos(-3x) = \cos 3x$, so the graph is symmetric about the $y$-axis, exactly like $\cos x$.

• Zeros: wherever $3x = \frac{\pi}{2} + n\pi$, that is $x = \frac{\pi}{6} + \frac{n\pi}{3}$.

What Is The Period Of Cos3x?

The period is $\frac{2\pi}{3}$. For any function $\cos(bx)$, the period is $\frac{2\pi}{b}$; here $b = 3$, so the wave repeats every $\frac{2\pi}{3}$ radians. This is the fanout question readers ask most after the formula itself, and it is worth committing to memory: triple the angle, and the graph gets three times as busy.

Examples Of Cos3x

Example 1

Evaluate $\cos 3x$ when $x = 0$.

$$\cos 3x = 4\cos^3 0 - 3\cos 0 = 4(1)^3 - 3(1) = 4 - 3 = 1$$

Check: $\cos(3 \cdot 0) = \cos 0 = 1$.

Final answer: $\cos 0 = 1$.

Example 2

Find $\cos 135^\circ$ using $x = 45^\circ$. First instinct, then the correct route.

The tempting move is to read $\cos 3x$ as "three lots of $\cos x$" and write $\cos 135^\circ = 3\cos 45^\circ = 3 \cdot \frac{1}{\sqrt{2}} \approx 2.12$.

Stop and check that against the range. Every cosine lives in $[-1, 1]$, and $2.12$ sits far outside it. An answer bigger than $1$ is impossible, so the linear shortcut is wrong.

The rescue is the actual cos3x formula. With $\cos 45^\circ = \frac{1}{\sqrt{2}}$:

$$\cos 135^\circ = 4\left(\frac{1}{\sqrt{2}}\right)^3 - 3\left(\frac{1}{\sqrt{2}}\right)$$

$$= 4 \cdot \frac{1}{2\sqrt{2}} - \frac{3}{\sqrt{2}} = \frac{2}{\sqrt{2}} - \frac{3}{\sqrt{2}} = -\frac{1}{\sqrt{2}}$$

$$\cos 135^\circ = -\frac{\sqrt{2}}{2}$$

Final answer: $\cos 135^\circ = -\dfrac{\sqrt{2}}{2}$, which matches the trigonometric table.

Example 3

Express $\cos^3 x$ in terms of $\cos 3x$.

Rearrange the identity for $\cos^3 x$:

$$4\cos^3 x = \cos 3x + 3\cos x$$

$$\cos^3 x = \frac{\cos 3x + 3\cos x}{4}$$

Final answer: $\cos^3 x = \dfrac{\cos 3x + 3\cos x}{4}$. This form is what makes $\int \cos^3 x , dx$ tractable.

Example 4

Given $\cos x = \frac{1}{2}$ (so $x = 60^\circ$), find $\cos 3x$.

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

Check: $\cos(3 \cdot 60^\circ) = \cos 180^\circ = -1$.

Final answer: $\cos 3x = -1$.

Example 5

Verify the formula at $x = 30^\circ$.

With $\cos 30^\circ = \frac{\sqrt{3}}{2}$:

$$4\left(\frac{\sqrt{3}}{2}\right)^3 - 3\left(\frac{\sqrt{3}}{2}\right) = 4 \cdot \frac{3\sqrt{3}}{8} - \frac{3\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} - \frac{3\sqrt{3}}{2} = 0$$

Check: $\cos 90^\circ = 0$.

Final answer: the identity holds; $\cos 90^\circ = 0$.

Example 6

Solve $\cos 3x = 0$ for the smallest positive $x$ in radians.

$\cos 3x = 0$ when the inside angle is a quarter-turn:

$$3x = \frac{\pi}{2} + n\pi$$

$$x = \frac{\pi}{6} + \frac{n\pi}{3}$$

The smallest positive value comes from $n = 0$.

Final answer: $x = \dfrac{\pi}{6}$.

Where Does Cos3x Earn Its Keep? - "The third harmonic in one clean cubic"

Cos3x exists to answer a precise need: how do you write a wave that runs at triple frequency using only the base wave? That question is not abstract - it is the mathematics of harmonics.

• Audio and signal harmonics. A distorted or clipped signal contains overtones at multiples of the base frequency. The third harmonic is $\cos 3x$, and the identity is how it gets folded back into the base tone during synthesis and analysis.

• Chebyshev polynomials. The right-hand side, $4\cos^3 x - 3\cos x$, is the third Chebyshev polynomial $T_3(\cos x)$. These polynomials sit inside the filter designs that keep radio channels from bleeding into each other.

• Solving cubic equations. Because the formula turns a cubic in $\cos x$ into a single cosine, it gives a trigonometric method for solving certain cubic equations that resist ordinary factoring.

What competitor explainers usually skip is why the answer is a cubic and not a quadratic. Tripling the angle stacks three rotations, and each rotation multiplies through the base cosine one more time, so the highest power that survives is $\cos^3 x$. The pattern generalises: $\cos nx$ is always a degree-$n$ polynomial in $\cos x$.

Common Mistakes With Cos3x

Mistake 1: Treating Cos3x As 3cos X

Where it slips in: Any time the angle is tripled and the reader reaches for the fastest-looking move.

Don't do this: Writing $\cos 3x = 3\cos x$, as if the $3$ could slide out front.

The correct way: Cosine is not linear. Use $\cos 3x = 4\cos^3 x - 3\cos x$. The quick test that catches this every time: $3\cos x$ can reach $3$, but $\cos 3x$ never leaves $[-1, 1]$. The memorizer who learned "coefficients come out" from algebra applies a rule that simply does not hold for trigonometric functions.

Mistake 2: Confusing Cos3x With Cos³x

Where it slips in: Integration and simplification problems where both expressions appear on the same page.

Don't do this: Swapping $\cos 3x$ for $\cos^3 x$ mid-solution because they read almost the same.

The correct way: Keep the operations in order. $\cos 3x$ triples first, then takes cosine; $\cos^3 x$ takes cosine first, then cubes. The bridge between them is $\cos^3 x = \frac{\cos 3x + 3\cos x}{4}$, not equality.

Mistake 3: Getting The Period Wrong

Where it slips in: Graphing $y = \cos 3x$ or predicting when the wave repeats.

Don't do this: Assuming the period is $2\pi$ because "it is still cosine."

The correct way: The period is $\frac{2\pi}{b} = \frac{2\pi}{3}$. A wrong period is not a harmless slip. In audio work, a synthesiser that models the third harmonic on a $2\pi$ period instead of $\frac{2\pi}{3}$ reproduces the wrong overtone, and the reconstructed sound drifts audibly out of tune - the digital version of the resonance failure that periodic-motion errors cause in physical systems.

Key Takeaways

• The cos3x formula is $\cos 3x = 4\cos^3 x - 3\cos x$, a triple-angle identity.

• It is derived by writing $3x = 2x + x$ and applying the angle-sum and double-angle rules.

• $\cos 3x$ (tripled angle) and $\cos^3 x$ (cosine cubed) are different; they link through $\cos^3 x = \frac{\cos 3x + 3\cos x}{4}$.

• The graph has period $\frac{2\pi}{3}$, amplitude $1$, and range $[-1, 1]$.

• The right-hand side is the third Chebyshev polynomial, which is why the identity shows up in harmonics and filter design.

To work through the cos3x formula with a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for identity practice, or live math classes online with peers from 20+ countries.

A Practical Next Step

Practice these to solidify your understanding: derive $\cos 3x$ from scratch without looking, then use it to evaluate $\cos 135^\circ$ and confirm the sign against the quadrant. If you get stuck on the substitution step, come back to the derivation section above. Want a live Bhanzu trainer to walk through the triple-angle family with you? Book a free demo class.

Read More

• Tan3x — the triple-angle identity for tangent, built the same way.

• Cot2x — the double-angle identity for cotangent.

• Sin(a + b) — the angle-sum rule that seeds every multiple-angle formula.

• Trigonometric ratios — the sine, cosine, and tangent definitions underneath it all.

• Half angle formula — the companion that halves an angle instead of tripling it.