Ratios of the Sum of Two Angles
Ximena wants the exact value of $\operatorname{sen}(75^\circ)$ on her admissions exam, without a calculator. That angle is not in the table of special angles. Can you get it from the ones you do know? Yes, and besides, $\operatorname{sen}(\alpha+\beta)$ is not $\operatorname{sen}\alpha+\operatorname{sen}\beta$: that is one of the most common traps in trigonometry. On this page you will see the correct formula and how to use it.
The Sum Formulas: Sine, Cosine, and Tangent
When you chain two rotations, first $\alpha$ and then $\beta$, the total angle is $\alpha+\beta$. Its ratios depend on the ratios of both angles, combined like this:
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.
Angles in Radians
Several exercises use radians ($\frac\pi2$, $\frac\pi3$, $2\pi$). The formulas are the same; only the unit changes. Remember that a full turn is $360^\circ=2\pi\text{ rad}$, so $180^\circ=\pi\text{ rad}$.
To convert: from degrees to radians, multiply by $\frac{\pi}{180}$; from radians to degrees, multiply by $\frac{180}{\pi}$.
| Degrees | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|---|---|
| Radians | $\frac\pi6$ | $\frac\pi4$ | $\frac\pi3$ | $\frac\pi2$ | $\pi$ | $\frac{3\pi}{2}$ | $2\pi$ |
The quadrants are as follows: first $\left(0,\frac\pi2\right)$, second $\left(\frac\pi2,\pi\right)$, third $\left(\pi,\frac{3\pi}{2}\right)$, and fourth $\left(\frac{3\pi}{2},2\pi\right)$. You need them to choose the sign of the ratios.
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.
Calculating Non-Special Angles with Two Special Ones
The angles $75^\circ$, $105^\circ$, or $15^\circ$ are not in the table of special angles, but they can be written as the sum of two that are: $75^\circ=45^\circ+30^\circ$, $105^\circ=60^\circ+45^\circ$.
Calculate the exact value of $\sin(75^\circ)$ by decomposing the angle as the sum of two special angles ($45^\circ$ and $30^\circ$).
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 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 Only Know the Ratios, Not the Angles
Many problems give only one ratio for each angle. First you calculate the missing ones with the fundamental identity $\operatorname{sen}^2\theta+\cos^2\theta=1$, being careful about the quadrant, and then you apply the 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 $\cos(\alpha + \beta)$.
A cannon fires a projectile at an angle of elevation $\alpha$ relative to the slope of a mountain, satisfying $\sin \alpha = 0.6$ (where $\alpha$ is an acute angle). The…
A cannon fires a projectile at an angle of elevation $\alpha$ relative to the slope of a mountain, satisfying $\sin \alpha = 0.6$ (where $\alpha$ is an acute angle). The slope itself has an inclination $\beta$ relative to the horizontal with $\cos \beta = 0.96$ (where $\beta$ is acute). Find the exact value of the sine of the total launch angle relative to the horizontal, that is, $\sin(\alpha + \beta)$.
Identities and Equations with the Sum Formula
The formulas are useful for proving identities (you expand one side using the angle sum and simplify) and for solving equations (you expand and get an equation in $\operatorname{sen}x$ and $\cos x$).
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:
$\frac{\sin(\alpha+\beta)}{\cos\alpha \cos\beta} = \tan\alpha + \tan\beta$
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.
Find all exact solutions to the equation $\sin(x + \frac{\pi}{6}) - \cos x = 0$ in the interval $[0, 2\pi)$.
Find all exact solutions to the equation $\sin(x + \frac{\pi}{6}) - \cos x = 0$ in the interval $[0, 2\pi)$.
Applications: Inclinations, Headings, and Unknown Angles
When two inclinations or rotations are combined (a solar panel that tilts further, a heading that drifts because of the wind, a two-link robotic arm), the total angle is a sum and its ratios come from these formulas.
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.
In an optics experiment, a light beam passes through two consecutive prisms that deviate its path in the same plane. The first prism produces a deviation $\alpha$ such…
In an optics experiment, a light beam passes through two consecutive prisms that deviate its path in the same plane. The first prism produces a deviation $\alpha$ such that $\tan \alpha = \frac{1}{3}$, and the second an additional deviation $\beta$ such that $\tan \beta = \frac{1}{2}$. Prove that the total deviation $\alpha + \beta$ is exactly $45^\circ$ by calculating the tangent of the sum of the angles.
A planar robotic arm consists of two links. The first link forms an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac{\sqrt{2}}{2}$ (lying in the first…
A planar robotic arm consists of two links. The first link forms an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac{\sqrt{2}}{2}$ (lying in the first quadrant). The second link forms an angle $\beta$ of relative rotation with respect to the extension of the first link. If reaching a critical part requires the total inclination of the second link with respect to the horizontal to be such that $\tan(\alpha + \beta) = 2$, calculate the exact value of $\tan \beta$.
A surveyor observes the roof of a building at an angle of elevation $\alpha$, satisfying $\cos \alpha = \frac{4}{5}$. Behind this building, in the same line of sight,…
A surveyor observes the roof of a building at an angle of elevation $\alpha$, satisfying $\cos \alpha = \frac{4}{5}$. Behind this building, in the same line of sight, there is a telecommunications tower. The additional angle of elevation from the roof of the building to the top of the tower, measured from the previous line of sight, is $\beta$, with $\sin \beta = \frac{7}{25}$. Given that both angles are acute, calculate the exact value of the cosine of the total angle of elevation $\alpha + \beta$ towards the top of the tower.
A reconnaissance aircraft initially flies on a heading forming an angle $\alpha$ north of east, satisfying $\tan \alpha = \frac{3}{4}$. A strong crosswind deflects its…
A reconnaissance aircraft initially flies on a heading forming an angle $\alpha$ north of east, satisfying $\tan \alpha = \frac{3}{4}$. A strong crosswind deflects its trajectory by an additional angle $\beta$ towards the north, so that the effective ground track forms an angle $\alpha + \beta$ with the east. If the control tower measures that the tangent of the effective heading is $\frac{12}{5}$, find the exact values of $\sin \beta$ and $\cos \beta$ without calculating the angles using a calculator.
Three Angles and Systems: Challenges
The formula can be chained: group two angles as one and apply it twice.
Expand the expression $\cos(\alpha+\beta+\gamma)$ purely in terms of the sines and cosines of the single angles $\alpha$, $\beta$ and $\gamma$. Group the terms…
Expand the expression $\cos(\alpha+\beta+\gamma)$ purely in terms of the sines and cosines of the single angles $\alpha$, $\beta$ and $\gamma$. Group the terms appropriately.
In any non-right triangle $ABC$, given that the sum of its interior angles is $A+B+C = 180^\circ$, mathematically prove the following fundamental relation:…
In any non-right triangle $ABC$, given that the sum of its interior angles is $A+B+C = 180^\circ$, mathematically prove the following fundamental relation:
$\tan A + \tan B + \tan C = \tan A \tan B \tan C$
Solve the following system of trigonometric equations for $x, y \in [0, \frac{\pi}{2}]$ and give the exact values of $x$ and $y$ in radians: $\sin(x+y) = 1$…
Solve the following system of trigonometric equations for $x, y \in [0, \frac{\pi}{2}]$ and give the exact values of $x$ and $y$ in radians:
$\sin(x+y) = 1$
$\cos x \sin y = \frac{1}{2}$
Back to the beginning: $\operatorname{sen}75^\circ=\operatorname{sen}(45^\circ+30^\circ)=\frac{\sqrt6+\sqrt2}{4}$, and Ximena gets it without a calculator.