You are standing next to the flagpole in front of your school. You can't climb it, but you can measure its shadow on the ground, and you know how high the sun is in the sky. How do you turn those two numbers into the height of the pole? Trigonometry is the tool, and in this page you will build it for angles of any size, not just the angles inside a right triangle.
Ratios in a right triangle
Pick one acute angle $\theta$ of a right triangle. The side across from it is the opposite side, the side that touches it (other than the hypotenuse) is the adjacent side, and the longest side is the hypotenuse.
Why do these ratios depend only on the angle? Any two right triangles with the same acute angle are similar, so their sides are proportional and the ratios come out the same, no matter how big the triangle is. Also notice that $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$, because the hypotenuse cancels.
The two acute angles of a right triangle are complementary, and the opposite side of one is the adjacent side of the other. So $\sin\theta=\cos(90^\circ-\theta)$.
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Calculate the exact value of the sine, cosine, and tangent of the smallest angle of a right triangle knowing that its legs measure $5$ in and $12$ in.
Calculate the exact value of the sine, cosine, and tangent of the smallest angle of a right triangle knowing that its legs measure $5$ in and $12$ in.
To optimize solar energy capture in winter, a rectangular panel $2$ feet long is raised by resting one of its shorter sides on the flat ground, so that the opposite side…
To optimize solar energy capture in winter, a rectangular panel $2$ feet long is raised by resting one of its shorter sides on the flat ground, so that the opposite side is exactly $1.2$ feet high. Determine the exact value of the cosine and tangent of the angle of inclination of the panel.
A triangle can only hold angles smaller than $90^\circ$. To go further, place the angle in standard position: its vertex is at the origin, its initial side lies along the positive $x$-axis, and you rotate counterclockwise for positive angles (clockwise for negative ones). The unit circle has radius $1$ and is centered at the origin.
Let $P(x,y)$ be the point where the terminal side of the angle crosses the unit circle. Drop a vertical segment from $P$ to the $x$-axis. For an acute angle you get a right triangle with hypotenuse $1$, adjacent side $x$ and opposite side $y$. So $\cos\theta=\dfrac{x}{1}$ and $\sin\theta=\dfrac{y}{1}$.
The signs of the coordinates depend on the quadrant, so the signs of the ratios do too. Sine is the $y$-coordinate, cosine is the $x$-coordinate, and the sign of tangent comes from dividing them.
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Determine in which quadrant the angle $\beta$ is located if it is simultaneously true that $\sin \beta < 0$ and $\tan \beta > 0$.
Determine in which quadrant the angle $\beta$ is located if it is simultaneously true that $\sin \beta < 0$ and $\tan \beta > 0$.
A grinder disc rotates at $7200$ revolutions per minute. Express this angular velocity in radians per second and calculate the exact angle (in radians) swept by a…
A grinder disc rotates at $7200$ revolutions per minute. Express this angular velocity in radians per second and calculate the exact angle (in radians) swept by a particle on its outer edge in $0.15$ seconds.
Some angles have exact values that you can get from two simple triangles. Cut a square along its diagonal to get a $45^\circ$-$45^\circ$-$90^\circ$ triangle with legs $1$ and hypotenuse $\sqrt2$. Cut an equilateral triangle of side $2$ with its altitude to get a $30^\circ$-$60^\circ$-$90^\circ$ triangle with sides $1$, $\sqrt3$ and $2$.
The points $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$ on the unit circle give the quadrantal angles $0^\circ$, $90^\circ$, $180^\circ$ and $270^\circ$ directly.
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Evaluate the following expression by substituting the trigonometric ratios with their exact values, without using a calculator:…
Evaluate the following expression by substituting the trigonometric ratios with their exact values, without using a calculator:
Every point $P(\cos\theta,\sin\theta)$ on the unit circle is exactly $1$ unit from the origin. The Pythagorean Theorem in the triangle below gives the most used relation in trigonometry.
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Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using the fundamental trigonometric…
Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using the fundamental trigonometric identities.
Knowing that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
Knowing that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
Two angles are coterminal if they have the same terminal side. They differ by whole turns, so they have identical ratios. Add or subtract $360^\circ$ (or $2\pi$) until the angle is in $[0^\circ,360^\circ)$. For a negative angle, add $360^\circ$.
The reference angle $\theta'$ is the acute angle between the terminal side and the $x$-axis. Its ratios equal the ratios of $\theta$ up to a sign, and the sign comes from the quadrant.
Angles of the form $\pi\pm x$, $\dfrac{\pi}{2}\pm x$ and so on
Expressions like $\sin(\pi-x)$ or $\cos\left(\dfrac{\pi}{2}+x\right)$ are reference-angle problems with a variable. Imagine $x$ is a small acute angle, and see which quadrant the angle lands in. That gives the sign. If the angle is built from $\dfrac{\pi}{2}$ or $\dfrac{3\pi}{2}$, the function swaps with its cofunction ($\sin\leftrightarrow\cos$, $\tan\leftrightarrow\cot$), because the reference triangle is turned on its side. If it is built from $\pi$ or $2\pi$, the function stays the same.
Angle
$\sin$
$\cos$
$\tan$
$\pi-x$
$\sin x$
$-\cos x$
$-\tan x$
$\pi+x$
$-\sin x$
$-\cos x$
$\tan x$
$2\pi-x$
$-\sin x$
$\cos x$
$-\tan x$
$\frac{\pi}{2}-x$
$\cos x$
$\sin x$
$\cot x$
$\frac{\pi}{2}+x$
$\cos x$
$-\sin x$
$-\cot x$
$\frac{3\pi}{2}-x$
$-\cos x$
$-\sin x$
$\cot x$
$\frac{3\pi}{2}+x$
$-\cos x$
$\sin x$
$-\cot x$
Don't memorize the table; rebuild each entry with the quadrant rule.
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Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and its…
Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and its cotangent.
Simplify the following trigonometric expression as much as possible, assuming that $x$ is an angle in the first quadrant. Pay special attention to the signs when…
Simplify the following trigonometric expression as much as possible, assuming that $x$ is an angle in the first quadrant. Pay special attention to the signs when reducing to this quadrant:
An angle of elevation is measured up from the horizontal. To use trigonometry in a word problem: draw the right triangle, mark the angle, label opposite, adjacent and hypotenuse, pick the ratio that links the known and the unknown, and solve.
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A building casts a shadow of $24$ feet when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height…
A building casts a shadow of $24$ feet when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height of the building and determine its value. If the angle of elevation is halved, does the length of the shadow double? Justify your answer based on the non-linearity of the tangent ratio.
A pendulum of length $80$ in is displaced from its vertical equilibrium position until it forms an angle of $18^\circ$ with the vertical. Calculate the exact horizontal…
A pendulum of length $80$ in is displaced from its vertical equilibrium position until it forms an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance the mass of the pendulum has been displaced and the height it has risen with respect to its lowest point. Express the results in terms of the trigonometric ratios of $18^\circ$.
Join the center of a regular $n$-gon inscribed in a circle of radius $R$ to its vertices. You get $n$ congruent isosceles triangles with two sides of length $R$ and an angle of $\dfrac{360^\circ}{n}$ between them.
To find the area of one triangle with two sides $R$ and included angle $\varphi$, take one $R$ as the base. The height is the altitude from the other vertex: in the right triangle it forms, the height is the side opposite to $\varphi$, so $h=R\sin\varphi$. Therefore the area is $\dfrac12\cdot R\cdot R\sin\varphi$. Multiplying by $n$ gives the formula for a regular polygon.
A crossed belt around two pulleys
A belt that crosses between two pulleys (radii $R$ and $r$, centers $d$ apart) touches the circles at $T_1$ and $T_2$. The belt crosses the line of centers at a point $X$ and makes an angle $\alpha$ with it. The radii $O_1T_1$ and $O_2T_2$ are perpendicular to the belt, so they create two right triangles.
In the right triangles $O_1T_1X$ and $O_2T_2X$ the radii are opposite $\alpha$: $\sin\alpha=\dfrac{R}{O_1X}=\dfrac{r}{O_2X}$. Hence $O_1X=\dfrac{R}{\sin\alpha}$ and $O_2X=\dfrac{r}{\sin\alpha}$. Since $O_1X+O_2X=d$, we get
$$\sin\alpha=\frac{R+r}{d}$$
The straight belt is $T_1T_2=XT_1+XT_2$. In each right triangle the side along the belt is adjacent to $\alpha$, so $XT_1=O_1X\cos\alpha$ and $XT_2=O_2X\cos\alpha$. Therefore $T_1T_2=d\cos\alpha$.
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Show that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression…
Show that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression $A = \frac{n \cdot R^2}{2} \sin\left(\frac{360^\circ}{n}\right)$. Then, apply the formula to calculate the exact area of a regular octagon inscribed in a circle of radius $10$ in.
Two circular pulleys of radii $R = 15$ in and $r = 5$ in are connected by a tight belt that crosses between them. The distance between the centers of the pulleys is $40$…
Two circular pulleys of radii $R = 15$ in and $r = 5$ in are connected by a tight belt that crosses between them. The distance between the centers of the pulleys is $40$ in. If $\alpha$ is the angle formed by the straight section of the belt with the imaginary line joining the centers, calculate the exact value of $\sin \alpha$, $\cos \alpha$ and the total length of the straight section of the belt between the two points of tangency.
An identity is an equation that is true for every angle where both sides are defined. To simplify or prove one, rewrite the more complicated side step by step until it matches the other.
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Simplify the following trigonometric expression as much as possible using the fundamental identities, expressing the result in terms of a single trigonometric ratio:…
Simplify the following trigonometric expression as much as possible using the fundamental identities, expressing the result in terms of a single trigonometric ratio:
Prove the following trigonometric identity, indicating the algebraic steps and the fundamental relationships used. Assume that $\alpha$ is an angle for which all ratios…
Prove the following trigonometric identity, indicating the algebraic steps and the fundamental relationships used. Assume that $\alpha$ is an angle for which all ratios are defined:
Prove that if $\alpha$ and $\beta$ are two complementary acute angles (i.e., $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relationship is…
Prove that if $\alpha$ and $\beta$ are two complementary acute angles (i.e., $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relationship is always verified:
Two situations come up often. In the first, a relation links $\sin\theta$ and $\cos\theta$ (for example $\sin\theta=2\cos\theta$), and you want their values. In the second, an equation mixes $\sin^2$ and $\cos$, and you rewrite it with the Pythagorean identity so that only one ratio remains.
Equations like these also appear in physics. Snell's law says that when light goes from a medium with index $n_1$ into one with index $n_2$, $n_1\sin\theta_1=n_2\sin\theta_2$. The angles are measured from the normal to the surface and are acute, so cosines are positive.
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Given the relationship $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasoning…
Given the relationship $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasoning in which quadrants the angle $\theta$ can be found.
Find all angles $\alpha \in [0^\circ, 360^\circ)$ that satisfy the following trigonometric equation: $2\cos^2 \alpha - 3\sin \alpha = 0$ (Hint: express the entire…
Find all angles $\alpha \in [0^\circ, 360^\circ)$ that satisfy the following trigonometric equation:
$2\cos^2 \alpha - 3\sin \alpha = 0$
(Hint: express the entire equation in terms of a single trigonometric ratio using the fundamental identity).
A ray of light passes from a liquid medium with refractive index $n_1 = 1.33$ to a crystal with index $n_2 = 1.52$. Snell's law states that…
A ray of light passes from a liquid medium with refractive index $n_1 = 1.33$ to a crystal with index $n_2 = 1.52$. Snell's law states that $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction measured with respect to the normal to the surface. If the ray is incident at an angle of $\theta_1 = \frac{\pi}{4}$ rad, determine the exact value of the cosine of the angle of refraction $\theta_2$.