The Formula Behind A Projectile's Longest Throw
Ask an artillery officer or a long-jumper what launch angle sends something farthest, and the answer is $45°$. The reason is buried inside one identity: a projectile's range depends on $\sin 2\theta$, and $\sin 2\theta$ hits its maximum when the doubled angle equals $90°$. That is the sin double angle formula doing quiet work, and it is one of the most-used identities in all of trigonometry.
The formula is short:
$$\sin 2x = 2\sin x \cos x$$
It says the sine of twice an angle is not twice the sine, but twice the product of the sine and cosine of the original angle. That distinction, product rather than double, is the whole idea, and it is exactly where most first attempts go wrong.
What Is The Sin Double Angle Formula?
The sin double angle formula is the trigonometric identity:
$$\sin 2x = 2\sin x \cos x$$
It is called a double angle identity because the angle on the left, $2x$, is double the angle on the right, $x$. The symbols are plain:
$x$ is any angle, in degrees or radians.
$2x$ is that angle doubled.
The right side pairs $\sin x$ and $\cos x$, the sine and cosine of the single angle.
There is also a form written purely in terms of the tangent, which is useful when only $\tan x$ is known:
$$\sin 2x = \frac{2\tan x}{1 + \tan^2 x}$$
This one is a rearrangement of the same identity, not a new fact. It sits in the same family as the cos 2x and tan 2x double-angle identities, and all three come from the sum and difference identities.
How is this different from sin A cos A? The single-angle product sin A cos A equals $\frac{1}{2}\sin 2A$, and its doubled cousin 2 sin A cos A equals $\sin 2A$ exactly. The sin double angle formula is the general statement $\sin 2x = 2\sin x\cos x$ read from left to right: given the single angle, produce the doubled one.
How Do You Derive The Sin Double Angle Formula?
The proof takes one line of setup. Start from the sine angle-sum identity:
$$\sin(A+B) = \sin A \cos B + \cos A \sin B$$
Now set $A = B = x$, since $x + x = 2x$:
$$\sin(x+x) = \sin x \cos x + \cos x \sin x$$
The two terms on the right are identical, so they add:
$$\sin 2x = 2\sin x \cos x$$
That is the derivation. The tangent form comes from dividing top and bottom by $\cos^2 x$ and using $\sin x = \tan x \cos x$, which trades the product for a ratio in $\tan x$.
Examples Of The Sin Double Angle Formula
Example 1
Given $\sin x = \dfrac{3}{5}$ and $\cos x = \dfrac{4}{5}$, find $\sin 2x$.
Substitute both values into the formula:
$$\sin 2x = 2\sin x \cos x = 2 \cdot \frac{3}{5} \cdot \frac{4}{5}$$
$$= \frac{24}{25}$$
Final answer: $\sin 2x = \dfrac{24}{25}$.
Example 2
Evaluate $\sin 120°$ with the double angle formula. First the wrong instinct, then the fix.
The tempting move is to read $\sin 2x = 2\sin x$ and write $\sin 120° = 2\sin 60° = 2 \cdot \dfrac{\sqrt{3}}{2} = \sqrt{3} \approx 1.73$.
That result is impossible: sine never exceeds $1$. The error is treating $\sin 2x$ as twice the sine instead of twice the product of sine and cosine.
The correct route uses $120° = 2 \cdot 60°$:
$$\sin 120° = 2\sin 60° \cos 60° = 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}$$
Final answer: $\sin 120° = \dfrac{\sqrt{3}}{2} \approx 0.866$.
Example 3
Write $2\sin 3\theta \cos 3\theta$ as a single trigonometric term.
This is the formula read right to left, with the single angle equal to $3\theta$:
$$2\sin 3\theta \cos 3\theta = \sin(2 \cdot 3\theta) = \sin 6\theta$$
Final answer: $2\sin 3\theta \cos 3\theta = \sin 6\theta$.
Example 4
If $\tan x = \dfrac{5}{12}$, find $\sin 2x$ using the tangent form.
Apply $\sin 2x = \dfrac{2\tan x}{1 + \tan^2 x}$ with $\tan x = \dfrac{5}{12}$:
$$\sin 2x = \frac{2 \cdot \frac{5}{12}}{1 + \frac{25}{144}} = \frac{\frac{10}{12}}{\frac{169}{144}}$$
$$= \frac{5}{6} \cdot \frac{144}{169} = \frac{120}{169}$$
Final answer: $\sin 2x = \dfrac{120}{169}$.
Example 5
Evaluate $\sin 90°$ by doubling $45°$.
Since $90° = 2 \cdot 45°$ and $\sin 45° = \cos 45° = \dfrac{1}{\sqrt{2}}$:
$$\sin 90° = 2\sin 45° \cos 45° = 2 \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = 2 \cdot \frac{1}{2} = 1$$
Final answer: $\sin 90° = 1$, which matches the known value.
Example 6
A projectile launched at angle $\theta$ has range $R = \dfrac{v^2 \sin 2\theta}{g}$. Which launch angle gives the longest range?
The launch speed $v$ and gravity $g$ are fixed, so $R$ is largest when $\sin 2\theta$ is largest. Sine peaks at $1$ when its input is $90°$:
$$2\theta = 90° \implies \theta = 45°$$
Final answer: the range is maximised at $\theta = 45°$, which is why $45°$ is the textbook "farthest throw" angle.
Where Is The Sin Double Angle Formula Used?
The identity earns its keep whenever an angle naturally appears doubled, letting you rewrite a $2x$ expression in terms of the $x$ you already know. Why not just compute $\sin 2x$ directly? Because in most problems the single angle is the one you have information about, and the formula is the bridge from what you know to what you need.
Projectile motion. Range, as Example 6 showed, runs on $\sin 2\theta$, so the formula pins down the optimal launch angle.
Integration. The product $2\sin x \cos x$ is awkward, but recognising it as $\sin 2x$ makes $\int 2\sin x \cos x , dx = -\frac{1}{2}\cos 2x + C$ immediate.
Simplifying identities. Doubling and half-angle work throughout trigonometry rests on collapsing $2\sin x\cos x$ into $\sin 2x$, which keeps longer proofs from ballooning.
Common Mistakes With The Sin Double Angle Formula
Mistake 1: Writing $\sin 2x = 2\sin x$
Where it slips in: Doubling the angle and reflexively doubling the sine with it.
Don't do this: $\sin 2x = 2\sin x$.
The correct way: $\sin 2x = 2\sin x \cos x$. The cosine factor is essential, and dropping it can produce values above $1$, which sine can never reach.
The check that catches this every time is a range test: if a "sine" comes out bigger than $1$, the cosine factor went missing. The memoriser who learned "double it" without the product is exactly the student this trap is built for.
Mistake 2: Forgetting To Match The Angles
Where it slips in: Using the formula when the two angles inside a product are not equal.
Don't do this: Writing $2\sin 3\theta \cos 5\theta = \sin 8\theta$ as if it were a double angle.
The correct way: $\sin 2x = 2\sin x \cos x$ only applies when the sine and cosine share the same angle. When they differ, the sin A cos B product-to-sum identity is the right tool instead.
Mistake 3: Misapplying The Tangent Form
Where it slips in: Using $\sin 2x = \dfrac{2\tan x}{1+\tan^2 x}$ without noting the sign of $\tan x$ in the given quadrant.
Don't do this: Plugging in a positive $\tan x$ when the angle actually sits in a quadrant where tangent is negative.
The correct way: Determine the sign of $\tan x$ from the quadrant first, then substitute. The tangent form inherits its sign from $\tan x$, so a wrong sign there flows straight through to $\sin 2x$.
Key Takeaways
The sin double angle formula is $\sin 2x = 2\sin x \cos x$, twice the product of sine and cosine.
It is derived by putting $A = B = x$ in the sine angle-sum identity.
The tangent form $\dfrac{2\tan x}{1+\tan^2 x}$ is the same identity rewritten for when only $\tan x$ is known.
The most common error is writing $\sin 2x = 2\sin x$, which drops the cosine and breaks the range.
To master double-angle work with a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for board and entrance prep, or live math tutoring with peers from 20+ countries.
A Practical Next Step
Practice these to solidify your understanding: given $\cos x = \dfrac{5}{13}$ in the first quadrant, find $\sin 2x$, then verify $\sin 60°$ by doubling $30°$. If you get stuck, come back to the derivation and rebuild the formula from the angle-sum identity. Want a live Bhanzu trainer to walk through these? Book a free demo class.
Read More
Sin 2x Formula — the same identity with more solved values.
Trigonometric Identities — the full identity reference in one place.
Pythagorean Identities — the $\sin^2\theta + \cos^2\theta = 1$ family behind the tan form.
Sin3x — the triple-angle identity built from the double angle.
Double Angle Formula — sine, cosine, and tangent double angles in one place.
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