The unit circle and radians.

Step 1 of 7

More than 90 degrees

A right triangle has no angle bigger than $90^\circ$⁠. To talk about any angle, put the triangle inside a circle of radius $1$⁠, the unit circle. A bead turns on the circle, and the line from the centre to it is the hypotenuse, always $1$⁠ long. So the sine is simply the height of the bead and the cosine its width, since you divide by $1$⁠. The angle is written with the Greek letter $\theta$⁠, theta.

This definition still works when the bead carries on past the first quarter turn. Predict, then see the answer on the board.

Step 2 of 7

Signs in the quadrants

The axes split the circle into four quadrants. The bead’s height is negative below the horizontal axis, and its width is negative left of the vertical axis, just like coordinates. Predict where the bead will be at $240^\circ$⁠, and what signs the sine and the cosine will have there.

Step 3 of 7

Two angles, one sine

$\sin 30^\circ = 0.5$⁠: at $30^\circ$⁠ the bead is at height one half. But a horizontal line at height one half crosses the circle at two points. Move Angle and find the other one.

Step 4 of 7

Double the angle

When the angle doubles, does the height double too? The bead’s height can never pass $1$⁠, so think before you answer. Predict, then the board will turn the bead to $60^\circ$⁠.

Step 5 of 7

Radians

Degrees are one way to measure an angle: a full turn split into $360$⁠. The other way measures the arc. An angle of one radian is an angle whose arc on the unit circle is exactly as long as the radius, $1$⁠. Half a turn is half the circumference of the unit circle, $\pi \approx 3.14$⁠, so $180^\circ = \pi$⁠ radians.

The board now shows the angle in radians. Move Angle and find one radian.

Step 6 of 7

A full turn

Here the bead has made half a turn, and the board shows $\pi$⁠. Answer how many radians there are in a full turn.

Step 7 of 7

Degrees or radians on a calculator

A calculator has a degrees mode, DEG, and a radians mode, RAD. In both you type a number with no unit, and the calculator decides on its own how to read it. Answer, then check on the board: $30$⁠ radians is four full turns and about $279^\circ$⁠ more. Move Angle to $279^\circ$⁠ and read the sine.