What are trig graphs, really?

Up until now you've probably only used sin, cos and tan for right-angled triangles, where the angle has to be somewhere between 0° and 90° (it can't be obtuse or you don't have a triangle anymore!). Trig graphs free the angle from that restriction. Here, x is just a number that represents an angle — and it can be obtuse (90°–180°), reflex (180°–360°), or even negative or bigger than 360°. The graphs show you what sin, cos and tan do across the whole range of angles, not just the acute ones.

The key thing that makes these graphs special is that they're periodic — the shape repeats itself forever in both directions. Think of a running track: once you've gone all the way round, you're back exactly where you started, and the next lap looks identical to the last one. That's exactly what sin and cos do every 360°.

The graph of y = sin x

y = sin x is a smooth wave. It passes through the origin (0, 0), rises to a maximum height of 1 at x = 90°, comes back down through 0 at x = 180°, dips to a minimum of −1 at x = 270°, and returns to 0 at x = 360° — then the whole pattern repeats.

The graph of y = cos x

y = cos x is exactly the same wave shape as sin x — same amplitude, same period — just shifted. Instead of starting at 0, it starts at its maximum: the y-intercept is (0, 1). It dips to −1 at x = 180°, and comes back up to 1 at x = 360°.

A really useful way to remember the relationship: cos x is just sin x shifted 90° to the left. If you slide the sin graph left by a quarter of a cycle, you get the cos graph exactly. This connection comes up a lot in later topics, so it's worth internalising now rather than just memorising two separate shapes.

The graph of y = tan x

Tan behaves completely differently from sin and cos, and this trips a lot of students up. It is not a smooth wave — it's made of separate curved "branches." Each branch starts down near −∞, climbs up through the origin (or through a repeat of it), and shoots up towards +∞ — then suddenly a new branch starts again.

The vertical dashed lines the branches never quite touch are called asymptotes. They occur at x = 90°, x = 270°, and so on (every 180°, since tan repeats twice as fast as sin and cos). The curve gets closer and closer to these lines but mathematically can never touch them — this is because tan x = sin x / cos x, and when cos x = 0 the fraction is undefined (you can't divide by zero!). That's exactly why the asymptotes sit at 90°, 270°, etc. — those are the angles where cos x = 0.

What are oscillating graphs?

Sine and cosine are called oscillating graphs because they swing back and forth (oscillate) between a highest point and a lowest point, over and over, forever in both directions along the x-axis. A full "swing" from start back to the same point is called one cycle.

Amplitude

The amplitude is half the vertical distance between the maximum point and the minimum point. Picture the horizontal line running exactly through the middle of the wave (often the x-axis itself) — the amplitude is simply how far the wave reaches above (or below) that middle line.

For the basic graph y = sin x, the wave goes up to 1 and down to −1, so the amplitude is 1 (it reaches exactly 1 unit above and below the middle line, y = 0).

Period

The period is the horizontal distance you need to travel along the x-axis before the graph starts repeating itself exactly. You can measure it several equivalent ways: from one maximum point to the next maximum point, from one minimum to the next minimum, or from one x-intercept to the x-intercept two crossings later (skipping the one in between).

For y = cos x, the period is 360° — go 360° along the x-axis from any point on the curve, and you land on an identical point.

Sketching graphs from their equation

Oscillating functions are usually written in the form f(x) = a sin(bx) or f(x) = a cos(bx). The two letters a and b each control a completely different feature of the graph — don't mix them up!

What "a" does — controls amplitude (height)

a is the amplitude directly. The bigger |a| is, the taller the wave — it stretches the whole graph vertically. For example, in y = 5cos x, the amplitude is 5: the wave now swings between 5 and −5 instead of between 1 and −1.

If there's a number added on outside the trig function too — like y = 2 + (1/5)cos x — that number (2) shifts the whole wave up or down, and it becomes the new "middle line" that the amplitude is measured from. So here the amplitude is still just 1/5, but it's measured from y = 2 rather than from y = 0.

What "b" does — controls period (width/speed)

b controls how quickly the graph repeats. The relationship is:

Example: for y = sin(2x), period = 360°/2 = 180°. The graph has been squashed horizontally by a factor of 2, so it now completes two full cycles in the space that used to fit just one (360°).

Example: for y = sin(½x), period = 360° ÷ ½ = 360 × 2 = 720°. The graph has been stretched out, so it now only completes half a cycle in 360° — you need a full 720° to see one complete wave.

Why trig equations have multiple solutions

Here's the crucial thing that makes trig equations different from normal equations: because the graphs repeat and are symmetric, one equation can have several correct answers within a given range. Your calculator will only ever give you one of them (via the inverse function — sin⁻¹, cos⁻¹, tan⁻¹) — it's your job to use the shape of the graph to hunt down the rest.

Every question will give you an interval to search in, e.g. 0° ≤ x ≤ 360°. Any solutions outside that range don't count, even if they're mathematically valid.

Solving sin x = ...

Step 1: Use your calculator's inverse sin function to get the first solution.
Step 2: Sketch the sine graph over the given interval, and mark that first solution on it.
Step 3: Use the symmetry of the wave to spot the second solution.

Worked example: Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

First solution: x = sin⁻¹(0.5) = 30°. Because sine is symmetric about x = 90° (the peak), the second solution is found by 180° − 30° = 150°. Both angles give the same sin value because they're mirror images of each other across the peak of the wave.

Solving cos x = ...

The method is the same, but the symmetry rule is different because cos x is symmetric about x = 0° and x = 360° (not about 90°).

Worked example: Solve cos x = 0.5 for 0° ≤ x ≤ 360°.

First solution: x = cos⁻¹(0.5) = 60°. By symmetry, the second solution is 360° − 60° = 300°.

Solving tan x = ...

Tan doesn't work by mirror symmetry — because it's not a wave, it doesn't have a peak to reflect around. Instead, you use its periodic nature: every branch is identical, just shifted along by 180°. So once you have one solution, you just keep adding (or subtracting) 180° to find the rest within your interval.

Worked example: Solve tan x = 1 for 0° ≤ x ≤ 360°.

First solution: x = tan⁻¹(1) = 45°. Adding 180° (one full period): 45° + 180° = 225°. Both are valid solutions within 0°–360°.

Rearranging trig equations first

Equations aren't always handed to you neatly as "sin x = 0.5." Sometimes you'll need to rearrange first. For example, 2 sin x − 1 = 0 rearranges (add 1, then divide by 2) to sin x = ½ — then you solve exactly as normal.

What if the first calculator answer is negative?

Sometimes your calculator spits out a negative angle for the first solution — that's completely normal, it just means the "natural" first solution sits to the left of 0° on the graph. Don't panic: extend your sketch to the left of the y-axis to see it, then use symmetry/periodicity as usual to find the solutions that actually fall inside your given range.