Ratios of the Sum of Two Angles
A technician installs a solar panel on the roof of a school in Santiago. First he tilts it at an angle $\alpha$, and then he raises it $15^\circ$ more. What is the tangent of the total tilt, without using a calculator? A very common mistake is to believe that $\sin(\alpha+\beta)=\sin\alpha+\sin\beta$. It is not so. The correct formula is longer, but it lets us find the ratios of angles like $75^\circ$ ($=45^\circ+30^\circ$) from special angles.
The sum formulas: 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 with two special ones
$75^\circ$, $105^\circ$, or $15^\circ$ are not in the table of special angles, but they can be written as a sum of two angles that are.
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 formulas in reverse
Read from right to left, the formulas turn an expression with four factors into a single trigonometric ratio.
An application: area with the sine of the angle
The area of a parallelogram with sides $a$ and $b$ forming an angle $\theta$ is $A=ab\sin\theta$. This is because the height over side $a$ measures $b\sin\theta$ and the area is base times height. The triangle formed by the same sides and that angle has half of that: $\frac12ab\sin\theta$.
Exercise MEDIO-07 combines this with the sine of a sum formula: first recognize the total angle and then use $ab\sin\theta$.
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 know only the ratios, not the angles
Many problems give a ratio for each angle, not the angle itself. First the missing ratios are calculated with $\sin^2\theta+\cos^2\theta=1$, paying attention to the sign according to the quadrant, and then the formula is applied.
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 $\cos(\alpha + \beta)$.
Identities and proofs
The formulas also serve to verify identities: we expand the side containing $\alpha+\beta$ and simplify until we reach the other side.
Prove the following trigonometric identity by expanding the numerator using the corresponding formula and simplifying the fraction:…
Prove the following trigonometric identity by expanding the numerator using the corresponding formula and simplifying the fraction:
$\frac{\sin(\alpha+\beta)}{\cos\alpha \cos\beta} = \tan\alpha + \tan\beta$
Simplify the trigonometric expression $\cos(x + \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$ so that it is in terms of $\cos x$ and/or $\sin x$ only.
Simplify the trigonometric expression $\cos(x + \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$ so that it is in terms of $\cos x$ and/or $\sin x$ only.
Applications and equations
The formulas solve situations where two tilts or turns are combined: a solar panel, the ramp of a bridge, a robotic arm, or an equation whose unknown is the angle.
Equations
To solve $\sin(x+30^\circ)=2\cos x$, we expand the sine of the sum, group terms, and divide by $\cos x$ to obtain an equation in $\tan x$. Remember to give all the solutions in the requested interval.
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.