Double-angle ratios
A cannon fires a projectile: its range depends on $\sin(2\theta)$, not on $\theta$ alone. An industrial roof has twice the slope of another. In engineering, the double angle turns up all the time, and it is handy to have its trigonometric ratios already "packaged" in a formula, without having to add the angle to itself every time.
Where the formulae come from
You already know the formulae for the sine and cosine of a sum. The double angle is just the special case $2\alpha = \alpha+\alpha$: all you need to do is replace $\beta$ with $\alpha$.
The cosine of the double angle has three equivalent forms: they are all equal, so use whichever fits best with the data you have (only $\cos\alpha$, only $\sin\alpha$, or both).
Simplify the following trigonometric expression to a single exact numerical value: $4\sin(15^\circ)\cos(15^\circ)$
Simplify the following trigonometric expression to a single exact numerical value:
$4\sin(15^\circ)\cos(15^\circ)$
Given that $\cos(x) = 0,6$, determine the value of $\cos(2x)$ using exclusively the identity relating the cosine of a double angle to the cosine of a single angle.
Given that $\cos(x) = 0,6$, determine the value of $\cos(2x)$ using exclusively the identity relating the cosine of a double angle to the cosine of a single angle.
With only one known ratio
If you only know $\sin\alpha$ or $\cos\alpha$ (and the quadrant), you can find the other ratio using the fundamental identity $\sin^2\alpha+\cos^2\alpha=1$, and then apply the double-angle formula.
Given that $\sin(\alpha) = \frac{4}{5}$ and that $\alpha$ is an angle in the first quadrant, calculate the exact value of $\sin(2\alpha)$.
Given that $\sin(\alpha) = \frac{4}{5}$ and that $\alpha$ is an angle in the first quadrant, calculate the exact value of $\sin(2\alpha)$.
Given that $\cos(\alpha) = -\frac{1}{3}$ and that $\alpha$ belongs to the second quadrant, determine the exact value of $\cos(2\alpha)$.
Given that $\cos(\alpha) = -\frac{1}{3}$ and that $\alpha$ belongs to the second quadrant, determine the exact value of $\cos(2\alpha)$.
If $\tan(\alpha) = \frac{3}{2}$, calculate the value of $\tan(2\alpha)$, expressing the result as an irreducible fraction.
If $\tan(\alpha) = \frac{3}{2}$, calculate the value of $\tan(2\alpha)$, expressing the result as an irreducible fraction.
Simplifying expressions
Many expressions involving a product of sine and cosine, or differences of squares, are really a double-angle formula in disguise. Recognising the pattern saves you using a calculator.
Calculate the exact value of the following expression without using a calculator: $\cos^2\left(\frac{\pi}{8}\right) - \sin^2\left(\frac{\pi}{8}\right)$
Calculate the exact value of the following expression without using a calculator:
$\cos^2\left(\frac{\pi}{8}\right) - \sin^2\left(\frac{\pi}{8}\right)$
Write the following expression in terms of a single trigonometric ratio of the angle $x$, simplifying as much as possible: $\frac{\sin(2x)}{2\sin(x)}$
Write the following expression in terms of a single trigonometric ratio of the angle $x$, simplifying as much as possible:
$\frac{\sin(2x)}{2\sin(x)}$
Find the exact value of the following operation using double-angle identities: $\frac{2\tan(22,5^\circ)}{1 - \tan^2(22,5^\circ)}$
Find the exact value of the following operation using double-angle identities:
$\frac{2\tan(22,5^\circ)}{1 - \tan^2(22,5^\circ)}$
Equations with the double angle
When an equation mixes $\sin(2x)$ (or $\cos(2x)$) with ratios of $x$ that are not doubled, the first step is to substitute the double-angle formula so that everything is in terms of the same angle.
Solve the following trigonometric equation, finding all solutions in the interval $[0, 2\pi)$: $\sin(2x) + \sqrt{3}\cos(x) = 0$
Solve the following trigonometric equation, finding all solutions in the interval $[0, 2\pi)$:
$\sin(2x) + \sqrt{3}\cos(x) = 0$
Solve the following trigonometric equation and give the solutions in degrees for the interval $[0^\circ, 360^\circ)$: $\cos(2x) - 3\sin(x) = 2$
Solve the following trigonometric equation and give the solutions in degrees for the interval $[0^\circ, 360^\circ)$:
$\cos(2x) - 3\sin(x) = 2$
Technical applications
The double angle appears in the range of projectiles, in roof slopes that double a reference inclination, and in articulated mechanisms.
In the construction of an industrial warehouse, two adjacent roofs have been designed. The first roof has an inclination $\alpha$ to the horizontal such that…
In the construction of an industrial warehouse, two adjacent roofs have been designed. The first roof has an inclination $\alpha$ to the horizontal such that $\tan(\alpha) = 0,4$. For water drainage reasons, the second roof has been designed with an angle of inclination that is exactly twice that of the first, that is, $2\alpha$. Calculate the exact slope (tangent) of the second roof.
In physics, the maximum range $R$ of a projectile launched over horizontal ground is modelled using the formula $R = \frac{v_0^2 \sin(2\theta)}{g}$, where $v_0$ is the…
In physics, the maximum range $R$ of a projectile launched over horizontal ground is modelled using the formula $R = \frac{v_0^2 \sin(2\theta)}{g}$, where $v_0$ is the initial velocity, $\theta$ is the launch angle, and $g \approx 10 \text{ m/s}^2$ is the acceleration due to gravity. If a test cannon fires a projectile with an initial velocity of $30 \text{ m/s}$ and it is desired to hit a target situated $45 \text{ m}$ away, find the possible launch angles $\theta$ in degrees.