Trigonometric Functions: Sine, Cosine, and Tangent
You're riding the Ferris wheel at the county fair. Up, over the top, down to the bottom, up again, and your height repeats every few minutes. Tides, sound waves, the electricity in a wall outlet, and your own breathing follow the same kind of repeating pattern. We call these periodic phenomena. Can one short formula tell you your height at any moment on the ride? By the end of this page it will.
Angles in radians and the unit circle
To graph sine and cosine we measure angles in radians. A half-turn is $\pi$ radians, so $\pi \text{ rad} = 180^\circ$ and a full turn is $2\pi$.
The unit circle is the circle of radius $1$ centered at the origin. Pick an angle $x$ and walk around the circle from the point $(1, 0)$ counterclockwise. You land on a point $(\cos x,\ \sin x)$:
$\cos x$ is the x-coordinate of the point.
$\sin x$ is the y-coordinate of the point.
$\tan x = \dfrac{\sin x}{\cos x}$ is the y-coordinate divided by the x-coordinate, the slope of the line from the origin to the point.
Memorize this table. Everything else in this page is built from it.
Angle $x$ (degrees)
$0$ ($0^\circ$)
$\tfrac{\pi}{6}$ ($30^\circ$)
$\tfrac{\pi}{4}$ ($45^\circ$)
$\tfrac{\pi}{3}$ ($60^\circ$)
$\tfrac{\pi}{2}$ ($90^\circ$)
$\sin x$
$0$
$\tfrac12$
$\tfrac{\sqrt2}{2}$
$\tfrac{\sqrt3}{2}$
$1$
$\cos x$
$1$
$\tfrac{\sqrt3}{2}$
$\tfrac{\sqrt2}{2}$
$\tfrac12$
$0$
The four points on the axes are even easier. Read them straight off the circle:
Angle $x$
$0$
$\tfrac{\pi}{2}$
$\pi$
$\tfrac{3\pi}{2}$
Point
$(1, 0)$
$(0, 1)$
$(-1, 0)$
$(0, -1)$
$\cos x$
$1$
$0$
$-1$
$0$
$\sin x$
$0$
$1$
$0$
$-1$
For angles in other quadrants, use the point's mirror image. The angle $\dfrac{5\pi}{6}$ is $\pi - \dfrac{\pi}{6}$. Its point sits at the same height as the point for $\dfrac{\pi}{6}$, so $\sin\dfrac{5\pi}{6} = \dfrac12$. The angle $\dfrac{7\pi}{6}$ is the same distance below the x-axis, so $\sin\dfrac{7\pi}{6} = -\dfrac12$. Only the sign changes. The signs come from the quadrant: $\sin$ is positive above the x-axis, and $\cos$ is positive to the right of the y-axis.
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Evaluate the function $g(x) = 2\cos(x) - 1$ at the values $x = 0$, $x = \frac{\pi}{2}$, $x = \pi$ and $x = \frac{3\pi}{2}$.
Evaluate the function $g(x) = 2\cos(x) - 1$ at the values $x = 0$, $x = \frac{\pi}{2}$, $x = \pi$ and $x = \frac{3\pi}{2}$.
Plot the height $\sin x$ of the point on the unit circle against the angle $x$, and you get the sine wave.
The wave rises to $1$, falls to $-1$, and then repeats forever, in both directions. The cosine graph is the same wave starting from its top at $(0, 1)$.
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Find all the points of intersection with the x-axis of the function $y = \sin(x)$ in the closed interval $[0, 2\pi]$.
Find all the points of intersection with the x-axis of the function $y = \sin(x)$ in the closed interval $[0, 2\pi]$.
What if the wave is taller, or faster? Two numbers control that in $f(x) = A\sin(Bx)$.
Move the sliders. A bigger $A$ stretches the wave vertically. A bigger $B$ squeezes it horizontally.
The same two steps work for any function of this form. For $f(x) = 6\sin(2x)$, read $A = 6$ and $B = 2$: the amplitude is $6$ and the period is $\dfrac{2\pi}{2} = \pi$.
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Determine the amplitude and period of the trigonometric function $f(x) = 4\sin(3x)$.
Determine the amplitude and period of the trigonometric function $f(x) = 4\sin(3x)$.
A pure sound wave emitted by a tuning fork is modelled by the function $P(t) = 0.02 \sin(880\pi t)$, where $P$ represents the pressure variation in pascals and $t$ the…
A pure sound wave emitted by a tuning fork is modelled by the function $P(t) = 0.02 \sin(880\pi t)$, where $P$ represents the pressure variation in pascals and $t$ the time elapsed in seconds. Determine the amplitude of the wave and its frequency (the number of complete cycles it performs per second).
A function is even if $f(-x) = f(x)$. Its graph is symmetric about the y-axis. It is odd if $f(-x) = -f(x)$. Its graph looks the same after a $180^\circ$ turn around the origin.
To decide whether a graph is sine or cosine, look at the y-axis. If it passes through $(0, 0)$ and is odd, it is a sine curve. If it passes through $(0, 1)$ and is even, it is a cosine curve. Look back at the sine graph in the previous section: it passes through $(0, 0)$, and its left half is its right half turned $180^\circ$ about the origin. That is the signature of an odd function, so it is a sine curve. Justify a graph this way: name the y-intercept, then name the symmetry.
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Observe the following graph and determine whether it corresponds to the sine function or the cosine function. Justify your answer by indicating the y-axis intercept and…
Observe the following graph and determine whether it corresponds to the sine function or the cosine function. Justify your answer by indicating the y-axis intercept and the type of symmetry it presents.
Classify the elementary functions $f(x) = \sin(x)$, $g(x) = \cos(x)$ and $h(x) = \tan(x)$ according to the type of symmetry they present (even or odd).
Classify the elementary functions $f(x) = \sin(x)$, $g(x) = \cos(x)$ and $h(x) = \tan(x)$ according to the type of symmetry they present (even or odd).
Since $\tan x = \dfrac{\sin x}{\cos x}$, the tangent breaks wherever $\cos x = 0$. Near those angles it shoots up or down without limit.
Real situations show asymptotes too. Picture a lighthouse beam sweeping along a straight shoreline. The distance along the shore to the lit spot is a tangent function of the beam's angle. When the beam points parallel to the shore, the spot would be infinitely far away, so the beam never hits the shore. That moment is a vertical asymptote.
In a real problem, the domain is only the part of the math domain that makes physical sense. Start at the first instant of the problem and stop before the first asymptote.
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Calculate the exact period of the function $y = \tan\left(\frac{x}{3}\right)$.
Calculate the exact period of the function $y = \tan\left(\frac{x}{3}\right)$.
The light beam of a rotating lighthouse illuminates a long, straight cliff. The distance $d$ (in feet) from the closest point of the cliff to the illuminated point at…
The light beam of a rotating lighthouse illuminates a long, straight cliff. The distance $d$ (in feet) from the closest point of the cliff to the illuminated point at the instant $t$ (in seconds) is given by the function $d(t) = 200 \tan\left(\frac{\pi}{4}t\right)$. Determine the domain of this function in the physical context of the problem (for the first rotation) and interpret what happens to the light beam when $t$ approaches $2\text{ seconds}$.
Now slide a wave up or down. Adding $D$ to a function moves its whole graph $D$ units up, the same translation you used in Math 8. A negative $D$ moves the graph down. The wave then oscillates around a new center line called the midline, $y = D$.
Try a negative $A$. The wave flips upside down: it starts by going down instead of up.
Notice the shortcut: when $B = \dfrac{\pi}{k}$, the period is $\dfrac{2\pi}{\pi/k} = 2k$. So $\cos\left(\dfrac{\pi}{3}t\right)$ has period $6$, and $\cos\left(\dfrac{\pi}{5}t\right)$ would have period $10$.
Back to the Ferris wheel from the start: this one formula gives your height at every moment. It repeats every $6$ minutes, and you never go higher than $75$ ft or lower than $5$ ft.
A negative midline is perfectly fine. It happens when you measure a position from a fixed reference point and the object is on the negative side.
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Indicate the domain and range of the trigonometric function $h(x) = 5\sin(x) + 2$.
Indicate the domain and range of the trigonometric function $h(x) = 5\sin(x) + 2$.
Describe what geometric transformation the graph of the function $f(x) = \cos(x)$ undergoes to become the graph of $g(x) = \cos(x) - 3$. What is the new range of the…
Describe what geometric transformation the graph of the function $f(x) = \cos(x)$ undergoes to become the graph of $g(x) = \cos(x) - 3$. What is the new range of the function $g(x)$?
A weight hanging from the end of a spring oscillates vertically. Its position with respect to the ceiling is given by the equation $y(t) = 8\cos(4t) - 20$, measured in…
A weight hanging from the end of a spring oscillates vertically. Its position with respect to the ceiling is given by the equation $y(t) = 8\cos(4t) - 20$, measured in inches. Indicate at what distance from the ceiling the weight is at the initial instant ($t=0$), what its lowest position is (greatest distance from the ceiling) and what the period of oscillation of the spring is.
Sliding a wave left or right is a horizontal translation, called a phase shift. Put everything together and you get the general form.
When the input is a day of the year or a time, the phase shift tells you when the cycle "starts". A shift of $80$ days means the wave crosses its midline going up on day $80$.
Two facts help with this kind of model. First, the period is $365$ days, so half a period is about $182.5$ days. Half a period after the wave crosses its midline going up, it crosses the midline going down. That is why the daylight is again about $12$ hours half a year after day $80$. Second, type the whole inside in parentheses on your calculator, in radian mode.
Sine and cosine are the same shape in different positions. The next example proves it with translations.
In the same way, $\cos\left(x - \dfrac{\pi}{2}\right) = \sin x$.
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The number of hours of sunlight in a certain city throughout the year can be approximated by the function $H(d) = 12 + 3\sin\left(\frac{2\pi}{365}(d-80)\right)$, where…
The number of hours of sunlight in a certain city throughout the year can be approximated by the function $H(d) = 12 + 3\sin\left(\frac{2\pi}{365}(d-80)\right)$, where $d$ represents the day of the year ($d=1$ is January 1). How many hours of daylight are there exactly on day $d=80$? And on day $d=262$ (half a year later)? Interpret the meaning of both results.
Draw a sketch of the function $g(x) = 3\sin\left(x - \frac{\pi}{2}\right)$. Explain step by step how this graph is obtained by applying translations and stretches to the…
Draw a sketch of the function $g(x) = 3\sin\left(x - \frac{\pi}{2}\right)$. Explain step by step how this graph is obtained by applying translations and stretches to the original graph of the function $f(x) = \sin(x)$.
Prove by analysing horizontal translations that the function $f(x) = \sin\left(x + \frac{\pi}{2}\right)$ is equivalent to the function $g(x) = \cos(x)$. Use the concepts…
Prove by analysing horizontal translations that the function $f(x) = \sin\left(x + \frac{\pi}{2}\right)$ is equivalent to the function $g(x) = \cos(x)$. Use the concepts of phase shift, periodicity and the shape of their graphs to justify your answer.
Often you're handed a graph or a description of a repeating pattern and must write the formula. Follow these steps.
When the exercise asks for the form $h(t) = A\sin(Bt) + D$, the graph must start on its midline.
Graphs of yearly data work the same way, but the period is $12$ months.
When a problem describes the motion in words, translate each phrase. "Maximum displacement" is the amplitude $A$. "One full back-and-forth cycle every $T$ seconds" means the period is $T$, so $B = \dfrac{2\pi}{T}$.
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The water level in a port varies periodically due to the tides. The following graph shows the height of the water in feet over $24\text{ hours}$. Determine the…
The water level in a port varies periodically due to the tides. The following graph shows the height of the water in feet over $24\text{ hours}$. Determine the amplitude, the period and the mean height of the water. From this data, propose a trigonometric expression of the type $h(t) = A\sin(Bt) + D$ that fits the graph shown.
The pendulum of a wall clock oscillates from side to side. Its horizontal displacement in inches from the center is modelled by the function $x(t) = A \cos(Bt)$. If it…
The pendulum of a wall clock oscillates from side to side. Its horizontal displacement in inches from the center is modelled by the function $x(t) = A \cos(Bt)$. If it is known that the pendulum reaches a maximum displacement of $12\text{ in}$ and completes a full back-and-forth cycle every $2\text{ seconds}$, determine the exact values of the parameters $A$ and $B$. Furthermore, calculate at what exact instants during the first $4\text{ seconds}$ the pendulum passes right through the central equilibrium position.
The mean monthly temperature of a city is modelled with a trigonometric function. Observe the graph representing the temperature $T$ in degrees Fahrenheit in terms of…
The mean monthly temperature of a city is modelled with a trigonometric function. Observe the graph representing the temperature $T$ in degrees Fahrenheit in terms of the month $m$ (where $m=0$ corresponds to the beginning of January, $m=1$ to the beginning of February, etc.). Determine the values of the parameters $A, B, C$ and $D$ so that the function $T(m) = A \cos(B(m - C)) + D$ accurately represents this curve.
Once you have a formula, you can read facts straight from it.
Back to the Ferris wheel: a model like $h(t) = 40 - 35\cos\left(\dfrac{\pi}{3}t\right)$ lets you answer "How many trips around in $30$ minutes?" Divide: $30 \div 6 = 5$ rotations.
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The height in feet of one of the cabins on a Ferris wheel in terms of time in minutes is given by the expression $h(t) = 20 - 15\cos\left(\frac{\pi}{2}t\right)$.…
The height in feet of one of the cabins on a Ferris wheel in terms of time in minutes is given by the expression $h(t) = 20 - 15\cos\left(\frac{\pi}{2}t\right)$. Calculate the maximum and minimum height reached by the Ferris wheel, and determine the exact time it takes to complete one full revolution.
The voltage of an alternating current in a household electrical system can be expressed by the function $V(t) = 325 \sin(100\pi t)$, where $V$ is the voltage in volts…
The voltage of an alternating current in a household electrical system can be expressed by the function $V(t) = 325 \sin(100\pi t)$, where $V$ is the voltage in volts and $t$ the time in seconds. Calculate the maximum voltage reached and find out how many times per second the voltage takes exactly the value of $0\text{ voltios}$.
The volume of air present in the lungs of a resting person, measured in liters, is modelled by the function $V(t) = 2.5 + 0.5\cos\left(\frac{\pi}{2}t\right)$, where $t$…
The volume of air present in the lungs of a resting person, measured in liters, is modelled by the function $V(t) = 2.5 + 0.5\cos\left(\frac{\pi}{2}t\right)$, where $t$ is measured in seconds. Calculate the maximum and minimum volume of air in the lungs. From the period of the function, determine how many complete breaths (complete cycles of inhalation and exhalation) this person takes in exactly one minute.
Sometimes you know the output and want the input, such as "When is the tide above $9$ feet?" Use the graph and the unit circle together.
Equations. To solve $\cos x = \dfrac12$ on $[0, 2\pi]$, find where the x-coordinate on the unit circle equals $\dfrac12$. That happens at two angles, mirror images across the x-axis: $x = \dfrac{\pi}{3}$ and $x = 2\pi - \dfrac{\pi}{3} = \dfrac{5\pi}{3}$.
Equations with both sine and cosine. If $\sin x$ and $\cos x$ appear together, divide by $\cos x$ to turn the equation into one about $\tan x$. This is only allowed if $\cos x \neq 0$, so first check that $\cos x = 0$ can't be a solution.
Inequalities. Find the endpoints with the equation, then use the graph to see which side is above or below.
When the variable is time in months, say what the numbers mean in the calendar. If $t = 0$ is January 1, then $t = 1$ is February 1, $t = 2$ is March 1, and so on.
Cycles that line up. A wave $\sin\left(\dfrac{2\pi}{T}t\right)$ has period $T$ and crosses zero going up at $t = 0, T, 2T, \dots$ To find when two such waves both cross zero going up at the same time, find a common multiple of their periods. The first time after $0$ is the LCM.
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Find analytically all the points of intersection ($x$ and $y$ coordinates) of the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ in the closed interval…
Find analytically all the points of intersection ($x$ and $y$ coordinates) of the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ in the closed interval $[0, 2\pi]$. Justify your answer using the properties of trigonometric functions.
According to biorhythm theory, a person's physical capacity follows a sinusoidal cycle of $23\text{ days}$ and their emotional capacity a cycle of $28\text{ days}$. If…
According to biorhythm theory, a person's physical capacity follows a sinusoidal cycle of $23\text{ days}$ and their emotional capacity a cycle of $28\text{ days}$. If both capacities are modelled with sine functions of amplitude $1$ starting from zero on the day of birth ($t=0$), write the functions $F(t)$ and $E(t)$. How many exact days must pass from birth for both capacities to cross the zero value simultaneously again, with both passing to have positive values?
The rabbit population in a nature reserve fluctuates annually according to the model $P(t) = 800 + 300\sin\left(\frac{\pi t}{6}\right)$, where $t$ represents the time in…
The rabbit population in a nature reserve fluctuates annually according to the model $P(t) = 800 + 300\sin\left(\frac{\pi t}{6}\right)$, where $t$ represents the time in months ($t=0$ is January 1). Set up and solve the corresponding trigonometric inequality to determine exactly during which months of the first year the population remains strictly above $950\text{ conejos}$.