Trigonometric ratios of the sum of two angles
$\sin(\alpha+\beta)$ is not $\sin\alpha+\sin\beta$: it is one of the most common traps in trigonometry. The real formula is longer, but extremely useful: it lets you calculate the ratios of angles such as $75º$ ($=45º+30º$) without a calculator, by breaking them down into special angles, and it appears in physics when adding waves or successive rotations.
The sum formulae: sine, cosine and tangent
Express $\cos(x + \frac{\pi}{2})$ in terms of the trigonometric ratios of the angle $x$ by expanding the corresponding sum formula and simplifying the result.
Express $\cos(x + \frac{\pi}{2})$ in terms of the trigonometric ratios of the angle $x$ by expanding the corresponding sum formula and simplifying the result.
Calculating non-special angles by combining two special ones
$75º$, $105º$ and $15º$ are not in the table of special angles, but they can be written as the sum of two that are: $75º=45º+30º$, $105º=60º+45º$...
Calculate the exact value of $\sin(75^\circ)$ by decomposing the angle as the sum of two special angles ($45^\circ$ and $30^\circ$).
Calculate the exact value of $\sin(75^\circ)$ by decomposing the angle as the sum of two special angles ($45^\circ$ and $30^\circ$).
Determine the exact value of $\cos(105^\circ)$ using the formula for the cosine of a sum, given that $105^\circ = 60^\circ + 45^\circ$.
Determine the exact value of $\cos(105^\circ)$ using the formula for the cosine of a sum, given that $105^\circ = 60^\circ + 45^\circ$.
Find the exact value of $\tan(75^\circ)$ by applying the formula for the tangent of a sum using the angles of $45^\circ$ and $30^\circ$. Rationalise the result if…
Find the exact value of $\tan(75^\circ)$ by applying the formula for the tangent of a sum using the angles of $45^\circ$ and $30^\circ$. Rationalise the result if necessary.
Using the formulae in reverse: from an expression to a single value
The same formulae, read from right to left, turn an expression with four terms into a single trigonometric ratio.
Simplify the following trigonometric expression to a single exact numerical term by applying the addition formulas in reverse:…
Simplify the following trigonometric expression to a single exact numerical term by applying the addition formulas in reverse:
$\sin(22^\circ)\cos(8^\circ) + \cos(22^\circ)\sin(8^\circ)$
Reduce the following expression to an exact numerical value: $\cos(40^\circ)\cos(20^\circ) - \sin(40^\circ)\sin(20^\circ)$
Reduce the following expression to an exact numerical value:
$\cos(40^\circ)\cos(20^\circ) - \sin(40^\circ)\sin(20^\circ)$
Calculate the exact value of the expression $\sin(15^\circ)\cos(30^\circ) + \cos(15^\circ)\sin(30^\circ)$. Use this geometric result to find the exact area of a…
Calculate the exact value of the expression $\sin(15^\circ)\cos(30^\circ) + \cos(15^\circ)\sin(30^\circ)$. Use this geometric result to find the exact area of a parallelogram whose adjacent sides measure $8$ cm and $10$ cm, given that the angle they form is precisely the sum of $15^\circ$ and $30^\circ$.
When you do not know the angles, only their ratios
Many problems do not give the value of $\alpha$ or $\beta$, only some of their ratios. You must first calculate the missing ratios (using the fundamental identity, paying attention to the quadrant) and then apply the sum formula.
Given that $\sin \alpha = \frac{4}{5}$ and $\cos \beta = \frac{12}{13}$, and that both $\alpha$ and $\beta$ are acute angles (belonging to the first quadrant), calculate…
Given that $\sin \alpha = \frac{4}{5}$ and $\cos \beta = \frac{12}{13}$, and that both $\alpha$ and $\beta$ are acute angles (belonging to the first quadrant), calculate the exact value of $\sin(\alpha + \beta)$.
Given two angles $\alpha$ and $\beta$ such that $\tan \alpha = 3$ and $\tan \beta = 2$, determine the exact value of $\tan(\alpha + \beta)$.
Given two angles $\alpha$ and $\beta$ such that $\tan \alpha = 3$ and $\tan \beta = 2$, determine the exact value of $\tan(\alpha + \beta)$.
Given the angles $\alpha$ and $\beta$ such that $\sin \alpha = \frac{3}{5}$ with $\alpha \in [\frac{\pi}{2}, \pi]$ and $\cos \beta = \frac{5}{13}$ with…
Given the angles $\alpha$ and $\beta$ such that $\sin \alpha = \frac{3}{5}$ with $\alpha \in [\frac{\pi}{2}, \pi]$ and $\cos \beta = \frac{5}{13}$ with $\beta \in [\frac{3\pi}{2}, 2\pi]$, calculate the exact value of $\cos(\alpha + \beta)$.
Applications: solar panels, gears and equations
The sum formulae solve situations where two rotations or inclinations are combined: a solar panel tilted by an extra angle, a robotic arm with two links, or an equation in which the angle sum is the unknown.
In a solar engineering project, a photovoltaic panel has an initial inclination $\alpha$ such that $\tan \alpha = \frac{3}{4}$. To optimise collection in winter, a…
In a solar engineering project, a photovoltaic panel has an initial inclination $\alpha$ such that $\tan \alpha = \frac{3}{4}$. To optimise collection in winter, a mechanism increases the inclination by an additional angle of $15^\circ$. Calculate the exact value of $\tan(\alpha + 15^\circ)$ without using decimal approximations, given that $\tan(15^\circ) = 2 - \sqrt{3}$.
Given that $\tan \alpha = \frac{1}{2}$ and that the tangent of the sum angle is $\tan(\alpha + \beta) = 3$, determine the exact value of $\tan \beta$ by setting up the…
Given that $\tan \alpha = \frac{1}{2}$ and that the tangent of the sum angle is $\tan(\alpha + \beta) = 3$, determine the exact value of $\tan \beta$ by setting up the corresponding equation from the addition formula.
Solve the trigonometric equation $\sin(x + 30^\circ) = 2\cos x$ for the interval $x \in [0^\circ, 360^\circ)$. Express the solutions exactly in degrees.
Solve the trigonometric equation $\sin(x + 30^\circ) = 2\cos x$ for the interval $x \in [0^\circ, 360^\circ)$. Express the solutions exactly in degrees.