Sum Formulas for Sine, Cosine, and Tangent
Emma builds a skate ramp that rises at $30^\circ$ and sets it on a hill that already slopes up at $30^\circ$. Is the sine of the total angle $\sin 30^\circ + \sin 30^\circ = \tfrac12 + \tfrac12 = 1$? That would mean the ramp points straight up, since $\sin\theta = 1$ only at $90^\circ$. Something is off. In this section you will see why $\sin(\alpha+\beta)$ is not $\sin\alpha + \sin\beta$, what it equals instead, and how the correct formula gives exact values for angles like $75^\circ$ and $105^\circ$ without a calculator.
The Sum Formulas
Imagine a point on the unit circle at angle $\alpha$. Rotate it another $\beta$ degrees and it lands at angle $\alpha+\beta$. Its coordinates are $(\cos(\alpha+\beta),\ \sin(\alpha+\beta))$.
Rotating by $\beta$ sends the point $(1,0)$ to $(\cos\beta, \sin\beta)$ and the point $(0,1)$ to $(-\sin\beta, \cos\beta)$. The point at angle $\alpha$ is $\cos\alpha\,(1,0) + \sin\alpha\,(0,1)$, so after the rotation it becomes
$$\cos\alpha\,(\cos\beta, \sin\beta) + \sin\alpha\,(-\sin\beta, \cos\beta).$$Reading off the two coordinates gives the formulas below.
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.
Simplify the trigonometric expression $\cos(x + \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$ so that it is exclusively in terms of $\cos x$ and/or $\sin x$.
Simplify the trigonometric expression $\cos(x + \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$ so that it is exclusively in terms of $\cos x$ and/or $\sin x$.
Exact Values of Non-Special Angles
The angles $75^\circ$, $105^\circ$, and $15^\circ$ are not on the special-angle table, but you can write them as sums of angles that are: $75^\circ = 45^\circ + 30^\circ$ and $105^\circ = 60^\circ + 45^\circ$.
Keep this table of special values handy. Radians are included because many problems use them.
| Angle | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|
| $30^\circ = \frac{\pi}{6}$ | $\frac12$ | $\frac{\sqrt3}{2}$ | $\frac{\sqrt3}{3}$ |
| $45^\circ = \frac{\pi}{4}$ | $\frac{\sqrt2}{2}$ | $\frac{\sqrt2}{2}$ | $1$ |
| $60^\circ = \frac{\pi}{3}$ | $\frac{\sqrt3}{2}$ | $\frac12$ | $\sqrt3$ |
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, knowing that $105^\circ = 60^\circ + 45^\circ$.
Determine the exact value of $\cos(105^\circ)$ using the formula for the cosine of a sum, knowing 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 from the angles of $45^\circ$ and $30^\circ$. Rationalize the denominator if…
Find the exact value of $\tan(75^\circ)$ by applying the formula for the tangent of a sum from the angles of $45^\circ$ and $30^\circ$. Rationalize the denominator if necessary.
Using the Formulas Backward
Read right to left, the same formulas turn a four-term expression into a single trigonometric value.
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 exactly the 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 exactly the 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 $8$ in and $10$ in, knowing that the angle they form is precisely the sum of $15^\circ$ and $30^\circ$.
When You Only Know the Ratios
Many problems do not give $\alpha$ or $\beta$, only a ratio of each. The sum formulas need all four values: $\sin\alpha$, $\cos\alpha$, $\sin\beta$, $\cos\beta$. Find the missing ones with the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ or with a right triangle, and use the quadrant to choose the sign.
Signs by quadrant (for the angle's terminal side):
| Quadrant | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|
| I | $+$ | $+$ | $+$ |
| II | $+$ | $-$ | $-$ |
| III | $-$ | $-$ | $+$ |
| IV | $-$ | $+$ | $-$ |
Knowing 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),…
Knowing 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 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 surveyor observes the roof of a building with 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 with 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 $\sin \beta = \frac{7}{25}$. Knowing that both angles are acute, calculate the exact value of the cosine of the total angle of elevation $\alpha + \beta$ to the top of the tower.
The Tangent Formula and Unknown Tangents
When a problem gives tangents (slopes), use the tangent formula directly. You do not need sine and cosine.
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 $\tan(\alpha + \beta)$.
Knowing 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…
Knowing 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.
A planar robotic arm consists of two links. The first link forms an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac{\sqrt{2}}{2}$ (being in the first…
A planar robotic arm consists of two links. The first link forms an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac{\sqrt{2}}{2}$ (being in the first quadrant). The second link forms an angle $\beta$ of relative rotation with respect to the extension of the first link. If to reach a critical part it is required that the total inclination of the second link with respect to the horizontal is such that $\tan(\alpha + \beta) = 2$, calculate the exact value of $\tan \beta$.
Simplifying and Proving Identities
To prove an identity, start with one side, usually the more complicated one, and transform it step by step until it matches the other side. Do not treat it like an equation and move terms across the equal sign.
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:
$\frac{\sin(\alpha+\beta)}{\cos\alpha \cos\beta} = \tan\alpha + \tan\beta$
Expand the expression $\cos(\alpha+\beta+\gamma)$ exclusively in terms of the sines and cosines of the simple angles $\alpha$, $\beta$ and $\gamma$. Group the terms…
Expand the expression $\cos(\alpha+\beta+\gamma)$ exclusively in terms of the sines and cosines of the simple angles $\alpha$, $\beta$ and $\gamma$. Group the terms appropriately.
In any non-right triangle $ABC$, knowing that the sum of its interior angles is $A+B+C = 180^\circ$, prove mathematically the following fundamental relationship:…
In any non-right triangle $ABC$, knowing that the sum of its interior angles is $A+B+C = 180^\circ$, prove mathematically the following fundamental relationship:
$\tan A + \tan B + \tan C = \tan A \tan B \tan C$
Solving Equations with a Sum Angle
When an equation contains something like $\sin(x + 45^\circ)$ next to $\sin x$ or $\cos x$, expand it. The equation then involves only $\sin x$ and $\cos x$. Collect terms and divide by $\cos x$ to get a tangent.
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 of the equation $\sin(x + \frac{\pi}{6}) - \cos x = 0$ in the interval $[0, 2\pi)$.
Find all exact solutions of the equation $\sin(x + \frac{\pi}{6}) - \cos x = 0$ in the interval $[0, 2\pi)$.
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:
$\sin(x+y) = 1$
$\cos x \sin y = \frac{1}{2}$
Applications: Slopes and Combined Angles
The slope of a ramp or a line equals the tangent of its angle of inclination. When one inclination is added to another, the tangent sum formula gives the new slope without ever finding the angles.
In a solar engineering project, a photovoltaic panel has an initial inclination $\alpha$ such that $\tan \alpha = \frac{3}{4}$. To optimize capture in winter, a…
In a solar engineering project, a photovoltaic panel has an initial inclination $\alpha$ such that $\tan \alpha = \frac{3}{4}$. To optimize capture 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, knowing that $\tan(15^\circ) = 2 - \sqrt{3}$.
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 $\tan \beta = \frac{1}{2}$. Show that the total deviation $\alpha + \beta$ is exactly $45^\circ$ by calculating the tangent of the sum angle.
A reconnaissance plane initially flies on a heading that forms an angle $\alpha$ north of east, verifying that $\tan \alpha = \frac{3}{4}$. A strong crosswind deviates…
A reconnaissance plane initially flies on a heading that forms an angle $\alpha$ north of east, verifying that $\tan \alpha = \frac{3}{4}$. A strong crosswind deviates its path by an additional angle $\beta$ to the north, so that the effective heading over the ground 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 resorting to calculating the angles with a calculator.