A wave is a point going round a circle. Its height over time is sine, and its distance across is cosine. Watch the point ride the circle while Pip surfs the wave it draws, then change the wave with the sliders.
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A radian measures a turn with the circle’s own radius. Lay the radius along the rim again and again: six fit, with a small gap of about 0.283 left over.
So a full turn is 2π ≈ 6.283 radians, and half a turn is π. Thinking that π is “180” as a number is the usual slip: π radians is 180 degrees.
One radian is one radius laid along the rim: a full turn takes 2π of them, about 6.28.
On a circle of radius 1, the point at angle θ has coordinates (cos θ, sin θ). Sine is the height of the point, and cosine is how far across it is.
At the top of the circle, 90° or π/2, the point is straight up: sin = 1 and cos = 0. Swapping sine and cosine is the usual mix-up. In a right triangle, SOH CAH TOA gives the same ratios.
Sine is the height of the point on the circle and cosine is how far across.
Track the height of the point as it goes round. Draw that height as time passes and you get the sine wave.
After one full turn, 2π, the pattern repeats, so the period of sine is 2π. The wave does not stop after one turn. Cosine is the same wave, starting at the top.
A wave is a point going round a circle: its height over time is sine.
Read three things off a wave: the middle line (midline), how far it rises above that line (amplitude), and how long one repeat takes (period).
Then y = A sin(Bx) + D, where A is the amplitude, D is the midline, and B = 2π ÷ period. Forgetting the midline, or measuring amplitude from 0, are the usual slips.
Middle, height, rhythm: three numbers write the whole wave.
A way to measure an angle using the circle’s own radius: one radian is the angle that cuts off an arc as long as the radius. A full turn is 2π radians, and 180° is π radians.
On the unit circle the point at angle θ is (cos θ, sin θ): cosine across, sine up. In a right triangle, SOH CAH TOA: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, tangent = opposite ÷ adjacent.
The amplitude is how far a wave rises above its middle line. The period is the length of one full repeat. For y = A sin(Bx) the period is 2π ÷ B.
Multiply by π ÷ 180. So 180° = π, 90° = π ÷ 2 and 360° = 2π radians.
World 8 of Pip: Math Adventure is about waves. You surf through the stars, measure angles in radians, and learn to write the equation of any wave.
It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.