Trigonometric Functions and Graphs

A swinging pendulum in a temple clock moves back and forth, back and forth, repeating the same path forever. The voltage in your wall socket oscillates sixty times per second. A vibrating sitar string moves up and down in a pattern that repeats hundreds of times per second. All three are doing the same mathematical thing — and the curve that describes them all is built from sine and cosine.

Imagine a point moving counterclockwise around the unit circle at a constant speed. At each moment, the point has a height — its y-coordinate — which is \sin\theta for whatever angle \theta the point has swept out so far. When \theta = 0, the height is 0. As \theta increases toward 90°, the height climbs to 1. Then it comes back down through 0 at 180°, drops to -1 at 270°, and returns to 0 at 360°. And then the whole cycle repeats.

Now do something crucial: instead of watching the point go around the circle, plot the height on one axis and the angle on the other — like unrolling the circular motion onto a flat strip. The result is one of the most important curves in mathematics: the sine wave.

That is the sine function, seen not as a ratio inside a triangle but as a function — an input-output machine that takes any angle and returns a number between -1 and 1. It repeats every 360° (or 2\pi radians), forever, in both directions. A function that repeats like this is called periodic, and 360° is its period — the smallest interval after which the pattern recurs.

Look at the graph carefully. Notice that it crosses zero at 0°, 180°, 360°, 540°, 720° — every multiple of 180°. Notice that the peaks (value 1) occur at 90°, 450° — and the troughs (value -1) at 270°, 630°. The wave is perfectly symmetric: every peak looks the same as every other peak, every trough is identical.

Domain and range

Since the unit circle gives a well-defined point for every angle, and every point has coordinates:

• Domain of \sin\theta and \cos\theta: all real numbers. There is no angle for which sine or cosine is undefined. You can feed in \theta = 10000° or \theta = -\pi/7 or any number at all.
• Range of \sin\theta and \cos\theta: [-1, 1]. The unit circle has radius 1, so no coordinate can exceed 1 in absolute value. This means \sin\theta never outputs 1.01 or -3 — it is trapped between -1 and 1, reaching both endpoints.

Tangent is different. \tan\theta = \sin\theta / \cos\theta, so it is undefined wherever \cos\theta = 0 — that is, at \theta = 90°, 270°, 450°, \ldots, or in general \theta = (2n+1) \cdot 90° for any integer n. At those points, you are dividing by zero, and no output exists.

• Domain of \tan\theta: all real numbers except odd multiples of 90°.
• Range of \tan\theta: all real numbers, (-\infty, +\infty). As \theta approaches 90° from the left, \cos\theta is tiny and positive while \sin\theta is near 1, so \tan\theta = \sin\theta / \cos\theta shoots off to +\infty. From the right, \cos\theta is tiny and negative, so \tan\theta plunges to -\infty. There is no bound.

The reciprocal functions (\csc\theta, \sec\theta, \cot\theta) have their own domain restrictions: each is undefined wherever its corresponding primary function is zero.

Graphs of the six functions

The sine curve

You have already seen the sine wave above. Here is a summary of its key features, all visible in the graph:

• Shape: a smooth wave that starts at (0, 0), peaks at (90°, 1), returns to (180°, 0), dips to (270°, -1), and returns to (360°, 0).
• Odd function: \sin(-\theta) = -\sin\theta. The graph has rotational symmetry about the origin — if you rotate it 180° around (0, 0), it maps onto itself.
• Zeros (where it crosses the axis) occur at every multiple of 180°: 0°, 180°, 360°, 540°, \ldots
• Increasing intervals: the sine function rises from -1 to 1 on each interval like (270° + 360°k) to (90° + 360°k), and falls from 1 to -1 in between.
• Smooth curvature: the graph has no corners, no jumps, no sharp turns. It is infinitely differentiable — a fact that will matter when you study calculus.

The cosine curve

Cosine is the x-coordinate of the point on the unit circle. Its graph looks exactly like the sine curve, but shifted 90° to the left.

• Starts at (0, 1) — the cosine of 0° is 1, not 0. This is the critical difference from sine.
• Even function: \cos(-\theta) = \cos\theta. The graph is symmetric about the vertical axis — a mirror placed at \theta = 0 would show the same curve on both sides.
• Zeros at 90°, 270°, 450°, \ldots — every odd multiple of 90°. Shifted 90° from the zeros of sine.
• The relationship: \cos\theta = \sin(\theta + 90°), which is why the cosine graph looks like the sine graph slid 90° to the left. They are the same wave, started at different moments.

The tangent curve

Tangent has a fundamentally different shape. It is not bounded — it blows up at every odd multiple of 90° — and it repeats every 180°, not 360°.

• The graph consists of separate branches, each spanning a 180° interval.
• Within each branch, the curve rises from -\infty to +\infty, passing through 0 at the midpoint.
• The vertical lines at odd multiples of 90° are vertical asymptotes — the function approaches but never reaches them.
• Tangent is an odd function: \tan(-\theta) = -\tan\theta.

The reciprocal functions

\csc\theta = 1/\sin\theta, \sec\theta = 1/\cos\theta, and \cot\theta = 1/\tan\theta. Their graphs are derived from the primary three:

• Cosecant has vertical asymptotes wherever \sin\theta = 0 (multiples of 180°) and U-shaped branches that open upward where sine is positive and downward where sine is negative. It never takes values between -1 and 1.
• Secant has vertical asymptotes wherever \cos\theta = 0 (odd multiples of 90°) and the same U-shaped branch structure. It never takes values between -1 and 1.
• Cotangent looks like a reflected, shifted tangent — it decreases from +\infty to -\infty within each 180° interval, with asymptotes at multiples of 180°.

Understanding the reciprocal graphs is mainly about understanding their parent functions. Wherever \sin\theta is large (near \pm 1), \csc\theta is near \pm 1 too. Wherever \sin\theta is small (near 0), \csc\theta is enormous. The reciprocal magnifies small values into large ones and vice versa — that is why the U-shaped branches balloon out near the asymptotes.

Amplitude, period, and phase shift

The basic sine wave y = \sin\theta has amplitude 1 (it swings between -1 and 1) and period 360° (it repeats every 360°). You can change both of these, and also shift the wave left, right, up, or down.

The general form is:

Each parameter does exactly one thing to the graph:

Amplitude A. Multiplying sine by A stretches the wave vertically by a factor of |A|. Take y = 3\sin\theta: instead of oscillating between -1 and 1, it oscillates between -3 and 3. If A is negative, the wave also flips upside down — y = -\sin\theta is the sine curve reflected across the \theta-axis.

Period via B. Multiplying \theta by B compresses or stretches the wave horizontally. The wave y = \sin(2\theta) completes a full cycle in 180° instead of 360° — it is twice as fast. The wave y = \sin(\theta/2) takes 720° for one cycle — twice as slow.

Phase shift via C. Adding C inside the sine function shifts the wave horizontally. y = \sin(\theta - 90°) shifts the sine wave 90° to the right — which turns out to produce exactly the graph of -\cos\theta. The formula for the phase shift is -C/B (not just -C), because you must factor out B first.

Vertical shift D. Adding D outside the sine function shifts the entire wave up or down. y = \sin\theta + 2 oscillates between 1 and 3 instead of -1 and 1. The centre line of the wave moves from y = 0 to y = D.

Here is a concrete demonstration. Compare y = \sin\theta with y = 2\sin(2\theta) — the second curve has amplitude 2 (twice as tall) and period 180° (twice as fast):

Transformations: a systematic approach

Graph transformations follow the same rules as for any function. For y = A\sin(B\theta + C) + D, read the transformations from the inside out:

1. Horizontal shift by -C/B: this is the phase shift.
2. Horizontal scale by 1/|B|: this changes the period.
3. Vertical scale by |A|: this changes the amplitude.
4. Vertical shift by D: this moves the centre line.

If A < 0, add a reflection across the \theta-axis. If B < 0, add a reflection across the y-axis — but since \sin(-x) = -\sin(x), a negative B is equivalent to flipping the sign of A.

The same rules apply to cosine, tangent, and the other three — the shape of the base graph changes, but the transformation machinery is identical.

Tangent transformations. The general tangent function is y = A\tan(B\theta + C) + D. The period of \tan\theta is 180° (not 360°), so the transformed period is 180°/|B|. There is no amplitude for tangent (it is unbounded), but A still controls the vertical stretching and D still shifts the centre. The asymptotes shift by the same phase shift -C/B.

How to identify each parameter from a graph. When you see a sinusoidal curve and need to extract the equation:

1. Read the maximum and minimum y-values. The amplitude is half their difference; the vertical shift is their average.
2. Measure the distance between two consecutive identical points (e.g., peak to peak) — that is the period. Then B = 360°/\text{period}.
3. Find where the first peak (or first zero-crossing) occurs. Compare that to where it would occur for the unshifted curve. The difference is the phase shift.

Reading a graph: two worked examples

Common confusions

• "The period of \sin(3\theta) is 3 \times 360° = 1080°." It is the opposite: multiplying \theta by 3 makes the wave faster, not slower. The period is 360°/3 = 120°. Think of it this way: the input to sine reaches 360° (one full cycle) when 3\theta = 360°, i.e., \theta = 120°.

• "The phase shift of \sin(2\theta - 60°) is 60°." It is 30°. You must factor out the B: \sin(2\theta - 60°) = \sin(2(\theta - 30°)). The phase shift is 30°, not 60°.

• "Amplitude can be negative." The amplitude is |A|, always non-negative. If A = -3, the amplitude is 3 and the wave is flipped vertically. The flip changes the direction of the initial swing, not the amplitude.

• "The tangent graph has amplitude." Tangent has no amplitude because it is unbounded — it takes values from -\infty to +\infty. The concept of amplitude applies only to bounded oscillating functions.

• "Cosine is just a shifted sine, so it is not a separate function." Algebraically, \cos\theta = \sin(\theta + 90°), so yes, one can be expressed in terms of the other. But they are still treated as separate functions because they behave differently (one is even, the other odd) and play different roles in formulas, physics, and geometry.

Going deeper

Why sinusoidal curves appear everywhere

The sine wave is not just a mathematical curiosity. It appears in sound (every musical note is a combination of sine waves), in light (electromagnetic waves), in alternating current (the voltage in your wall socket is V = V_0 \sin(\omega t)), and in the motion of a pendulum for small swings. This is not a coincidence — it is a theorem. Any quantity that oscillates under a restoring force proportional to displacement follows a sine curve. The mathematical name for this is simple harmonic motion, and it is why sine and cosine dominate physics.

Madhava of Sangamagrama, a 14th-century Indian mathematician, was the first to write down an infinite series for sine:

(where \theta is in radians). This series converges for every real number \theta — a fact that makes the sine function computable to any desired accuracy and places it at the foundation of analysis.

The graph of cosecant and secant, traced

Since \csc\theta = 1/\sin\theta, the cosecant graph inherits structure from sine. Wherever \sin\theta = 1, \csc\theta = 1. Wherever \sin\theta = -1, \csc\theta = -1. These are the only points where the two graphs touch. Between these touch-points, cosecant curves away from the \theta-axis, forming U-shaped branches that open upward when sine is positive and downward when sine is negative. At every zero of sine, cosecant has a vertical asymptote.

Secant relates to cosine in exactly the same way: \sec\theta = 1/\cos\theta has U-shaped branches opening away from the \theta-axis, with asymptotes at the zeros of cosine.

Period of tangent and cotangent

Tangent repeats every 180° (not 360°) because \tan(\theta + 180°) = \tan\theta. The proof: the point at \theta + 180° is diametrically opposite the point at \theta on the unit circle, so both coordinates change sign. The ratio y/x = (-y)/(-x) is unchanged, giving \tan(\theta + 180°) = \tan\theta.

This means the general tangent function y = A\tan(B\theta + C) + D has period 180°/|B| (or \pi/|B| in radians), not 360°/|B|.

Phase shift as time delay

In physics, the phase shift -C/B often corresponds to a time delay. Two waves with the same frequency but different phase shifts represent the same oscillation starting at different times. The quantity C is called the phase angle, and it determines where in the cycle the wave begins at \theta = 0. When two waves are "in phase," their phase angles differ by 0 (or a multiple of 360°). When they are "out of phase" by 180°, they are mirror images — one peaks when the other troughs. This is the basis of interference in physics.

Radians and the natural period

Throughout this article, angles have been in degrees. In calculus and higher mathematics, the natural unit is the radian, where one full revolution is 2\pi radians instead of 360°. In radians, the period of \sin\theta is 2\pi (about 6.28), and the general sinusoidal is y = A\sin(B\theta + C) + D with period 2\pi/|B|.

The radian version is not just a different convention — it is the version that makes the derivative work cleanly. The derivative of \sin\theta is \cos\theta only when \theta is in radians. In degrees, you pick up an extra factor of \pi/180, which is messy. This is why every calculus textbook uses radians, and why you should get comfortable with both.

Sine and cosine as building blocks

One of the deepest results in mathematics: every periodic function — square waves, sawtooth waves, even the complex vibration patterns of a tabla — can be written as a sum of sine and cosine waves of different frequencies. This is the field now called harmonic analysis, and it is the reason that sine and cosine are not just two functions among many, but the building blocks of all periodic behaviour. When you decompose sound into frequencies, you are decomposing it into sines and cosines. When you compress a digital image, similar transforms are doing the work.

The seeds of this idea go back to Indian mathematicians of the Kerala school, including Madhava of Sangamagrama, who worked with infinite trigonometric series in the 14th century — three centuries before the same ideas appeared in Europe.

Where this leads next

With the graphs of all six functions in hand, you can move on to the algebraic relationships between them.

• Trigonometric Identities — the fundamental identities and strategies for proving new ones. These are the algebraic backbone of trigonometry.
• Trigonometric Ratios of Any Angle — the unit-circle foundation that makes these graphs possible.
• Periodic Functions — a general treatment of functions that repeat, of which sine and cosine are the most important examples.
• Graph Transformations: Translations — the general framework for horizontal and vertical shifts, applied here to specific functions.
• Domain and Range — the formal treatment of what goes in and what comes out, which is especially interesting for the reciprocal trig functions.