This unit turns the unit circle into graphs. You'll learn to sketch and transform all six trig functions, read periods and shifts from equations, find amplitudes for sine and cosine, and use inverse trig functions to get an angle back from a ratio.
Read the overview first, then work through each topic note and try its checkpoint question before checking the answer.
A periodic function repeats its outputs on a fixed interval called the period. Trig functions are the main examples, and this unit is about graphing them and reading their features. By the end you should be able to go from an equation to a graph, from a graph to an equation, and from a ratio to an angle.
Honors Pre-Calculus Unit 6 covers the graphs of the trig functions and their inverses. You start with sine and cosine in the form . When and are nonzero, is the amplitude, is the period, is the phase shift, and is the midline. Next come tangent, cotangent, secant, and cosecant, whose graphs have vertical asymptotes wherever their denominators equal zero. The unit ends with arcsin, arccos, and arctan, which return angles from restricted ranges so each input has exactly one output.
In , sets the height, sets how fast the wave repeats, slides it sideways, and moves the midline. Once you can read those four values, you can sketch the graph or write a model for data that cycles, like tides or daily temperature.
Tangent and secant have cosine in the denominator, so they have vertical asymptotes where . Cotangent and cosecant have sine in the denominator, so their asymptotes fall where . Knowing which denominator matters tells you where the graph can't exist.
Because trig functions repeat, many angles share the same sine or cosine. To make an inverse function, restrict the original function's inputs to an interval where it is one-to-one. That's why is and not .
Sine and cosine are the coordinates of a point moving around the unit circle, which is why they repeat every . The other four functions are ratios of those coordinates, and the inverses undo them on one carefully chosen stretch. If a graph or answer seems strange, go back to the unit circle and check what the coordinates are doing.
Every sine or cosine graph is the same wave, stretched and slid around.
Tangent, cotangent, secant, and cosecant are all built from sine and cosine, so their graphs break wherever the denominator is zero.
An inverse trig function takes a ratio and gives back an angle.
Every sine or cosine graph is the same wave, stretched and slid around. In the form (or with cosine), when and are nonzero, the amplitude is , the period is , the phase shift is , and the midline is . If or is zero, the graph is constant and has no least positive period. A negative flips the wave across its midline.
The parent graphs differ only in where they start. starts on the midline at and rises, while starts at its maximum. That's because , so cosine is sine shifted to the left. Both have domain all real numbers and range .
To sketch one cycle, split the period into four equal parts. Those five x-values give you the max, the min, and the midline crossings.
Worked example. Graph . First factor the inside so it matches the form: . The amplitude is 3, the period is , the phase shift is to the right, and the midline is . The range is . One cycle runs from to in steps of . The graph is on the midline at , reaches 4 at , returns to the midline at , hits at , and ends on the midline at .
Try this. A Ferris wheel seat goes from a low of 2 m to a high of 30 m. What are the amplitude and midline of a model for its height? (Answer: The amplitude is m and the midline is m.)
Tangent, cotangent, secant, and cosecant are all built from sine and cosine, so their graphs break wherever the denominator is zero. Those breaks are vertical asymptotes.
Since , tangent has asymptotes where , at , and so on. Between asymptotes it rises from to and crosses zero at multiples of . Its period is , not . Cotangent, , has asymptotes at multiples of and falls from left to right instead of rising.
Secant and cosecant are reciprocals, and . To sketch one, lightly draw the cosine or sine curve, put asymptotes at its zeros, and then draw U-shaped branches that touch each peak and valley and open away from the x-axis. Their range is and their period is .
Transformations work as in 6.1. For nonzero and , the period of or is . For or , it is .
Worked example. Find the period and asymptotes of . The period is . Asymptotes occur where , so . That gives , and so on.
Try this. Where are the vertical asymptotes of , and why? (Answer: At , such as , because and at those points.)
| Function | Period | Vertical asymptotes | Range |
|---|---|---|---|
| $\tan x$ | $\pi$ | $x = \frac{\pi}{2} + k\pi$ | All real numbers |
| $\cot x$ | $\pi$ | $x = k\pi$ | All real numbers |
| $\sec x$ | $2\pi$ | $x = \frac{\pi}{2} + k\pi$ | $(-\infty, -1] \cup [1, \infty)$ |
| $\csc x$ | $2\pi$ | $x = k\pi$ | $(-\infty, -1] \cup [1, \infty)$ |
An inverse trig function takes a ratio and gives back an angle. asks which angle has a sine of . The catch is that infinitely many angles do, such as , , and , because sine repeats. A function can only have one output, so each inverse picks answers from a restricted range where the original function is one-to-one.
returns angles in , returns angles in , and returns angles in . Arcsin and arccos only accept inputs from to , while arctan accepts any real number. The graph of has horizontal asymptotes at .
The notation means arcsin, not . That reciprocal is .
Worked example. Evaluate . Cosine equals at and , but only is in , so the answer is . Compare that with , which lands in quadrant IV instead of at .
Worked example. Find . Let , so and is in quadrant I. Draw a right triangle with adjacent side 3 and hypotenuse 5. The opposite side is , so .
Try this. What is ? (Answer: . The inside equals , and the angle in with sine is , not .)
| Function | Allowed inputs | Output range | Possible quadrants; axis angles are also possible |
|---|---|---|---|
| $\arcsin x$ | $[-1, 1]$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$ | I and IV |
| $\arccos x$ | $[-1, 1]$ | $[0, \pi]$ | I and II |
| $\arctan x$ | All real numbers | $(-\frac{\pi}{2}, \frac{\pi}{2})$ | I and IV |
| Term | Definition |
|---|---|
| Periodic function | A function with a positive number $P$ such that, for every $x$ in its domain, $x + P$ is also in its domain and $f(x + P) = f(x)$. |
| Sinusoidal function | Any function with the smooth wave shape of sine or cosine, written as $y = a\sin(b(x - c)) + d$ or with cosine. |
| Amplitude | Half the distance between a sinusoid's maximum and minimum values, equal to $|a|$. |
| Period | The length of one full cycle. For functions in the form $y = a f(b(x - c)) + d$ with nonzero $a$ and $b$, it is $\frac{2\pi}{|b|}$ for sine, cosine, secant, and cosecant and $\frac{\pi}{|b|}$ for tangent and cotangent. |
| Phase shift | The horizontal slide of a trig graph, equal to $c$ once the function is written as $b(x - c)$ inside. |
| Midline | The horizontal line $y = d$ that a sinusoid oscillates around. |
| Vertical asymptote | A line $x = c$ where the function approaches positive or negative infinity as $x$ approaches $c$ from at least one side. For the parent tangent, cotangent, secant, and cosecant functions, these lines occur where the denominator is zero. |
| Inverse trigonometric function | A function such as arcsin, arccos, or arctan that takes a ratio as input and returns an angle from a restricted range. |
In the period is not 4. The parent period gets divided by , so the period is . A bigger means a shorter period and more cycles packed in. For tangent and cotangent, divide instead of .
For , it's tempting to say the shift is . Factor first to get , so the shift is to the right. Also watch the sign, since shifts 3 units left.
The here means inverse function, not an exponent. is the angle , while is . The confusion comes from , where the exponent really does mean squaring.
Your unit circle shows several angles with the same sine or cosine, so it's easy to pick the wrong one. Before you write an answer, check it against the range: for arcsin, for arccos, and for arctan. For example, , not .