Unit circle and the sine, cosine, tangent graphs
MathematicsTrigonometryAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Trigonometry
Drag point M on the unit circle and the matching point moves in sync on the graph of y = sin x, cos x or tan x. The line y = m crossing the graph shows the solutions of a basic trigonometric equation, marked on both the circle and the graph together with the general solution formula. The Reciprocal screen graphs sec x, cosec x and cot x with their vertical asymptotes; the Inverse screen restricts the domain of sin, cos and tan and reflects it in y = x to give arcsin, arccos and arctan, their principal values and equations in context. The Transform screen graphs y = A sin(B(x − C)) + D (or cos) with sliders, showing the amplitude, period, phase shift and midline linked to a point on a circle of radius |A|, with tide, day-length and Ferris-wheel models.
Lesson: Trigonometric values, trigonometric functions, basic trigonometric equations
What it shows
On the unit circle, the point M at angle x has coordinates (cos x; sin x), and tan x = sin x / cos x where cos x ≠ 0. A line y = m meets each graph at the solutions of f(x) = m. The reciprocal functions sec x = 1/cos x, cosec x = 1/sin x (also written csc x) and cot x = cos x/sin x are undefined where the denominator is zero, which gives vertical asymptotes. Restricting sin to [−π/2, π/2], cos to [0, π] and tan to (−π/2, π/2) makes them one-to-one, and reflecting in y = x gives arcsin, arccos and arctan. On Transform, y = A sin(B(x − C)) + D has amplitude |A|, period 2π/B, phase shift C and midline y = D; the tide, day-length and Ferris-wheel models are idealised.
How to use
On Circle and graphs, drag M or press Run and read sin x, cos x and tan x, then set m to see the solutions of f(x) = m. On Reciprocal, pick sec x, cosec x or cot x and compare it with the dashed original graph near its zeros. On Inverse, choose arcsin, arccos or arctan, drag across the graph to change m, untick Restrict the domain to see why the restriction is needed, and read the problem in context. On Transform, move A, B, C and D, pick a Context and try to fit the day-length dots.
Parameters you can change
- Function y = sin x, y = cos x, y = tan x
- Right-hand side m of f(x) = m -2–2
- Angle unit Radian, Degree
- Number of 2π periods shown on the graph 1–3
- Screen Circle and graphs, Reciprocal: sec x, cosec x, cot x, Inverse: arcsin, arccos, arctan, Transform: y = A sin(B(x − C)) + D
- Transform screen: A (amplitude |A|) -3–3
- Transform screen: B (period 2π/B) 0.25–4
- Transform screen: phase shift C (radians) -3.14–3.14
- Transform screen: D (midline y = D) -3–3
- Transform screen: context None (x–y graph), Tide, Day length (London), Ferris wheel
Questions to explore
- Why does sin x = 0.5 have two families of solutions, while tan x = 1 has only one?
- What is the period of y = tan x, and how does the unit circle explain it?
- Which points on the unit circle give cos x = 0, and what happens to tan x there?
