Teaching objectives
Sine and cosine are defined from a point that turns on the circle of radius 1: cos θ and sin θ are its coordinates (x, y). The lab starts there and, over 25 rooms, reaches the right triangle, the notable angles, the graph as a function, frequency and amplitude, sound, the ellipse, the double angle and polar coordinates. It is a Bachillerato lab, with drawings the student moves and challenges that are solved by calculating.
There are no separate explanation screens: each room combines an interactive drawing (a draggable point, sliders, an animation) with one or more questions that must be answered to move on. There are multiple-choice questions, numerical answers (with a hint after the second mistake and the solution after the third) and two timed speed tests. Some drawings appear only as a help after a mistake (the hidden point of room 12, the double angle of room 21).
What you'll learn
The lab takes the student from “the sine is a coordinate” to using sine and cosine as a tool for waves, curves and points in the plane.
- Define cos θ and sin θ as coordinates of a point on the unit circle and see that cos²θ + sin²θ = 1 (room 1).
- Convert from degrees to radians and back (room 2).
- Recognise the legs of a right triangle as sin θ and cos θ, and scale them when the hypotenuse is not 1 (rooms 3-4).
- Know the values of sine and cosine at 0°, 30°, 45°, 60° and 90° and use them to calculate sides of triangles (rooms 5-6).
- Know the sign of sine and cosine in each quadrant (room 7).
- See sine and cosine as the height and width of a spinning point, “unrolled” into a graph, and that one curve is the other shifted by a quarter turn (rooms 8-10).
- Use symmetries (half turn, reflection, quarter turn) and reference angles to calculate values without a calculator (rooms 11-12).
- Read the graph as a function: domain, range and periodicity (rooms 13-15).
- Understand how frequency N compresses the graph and how amplitude A scales the range without depending on N (rooms 16-17).
- Apply the sine wave to a pure sound, to a computer drawing and to an ellipse (rooms 18-20).
- Work out the sign and the value of sin(2x) and cos(2x) (rooms 21-24).
- Convert from Cartesian to polar coordinates and back (room 25).
Key mathematical ideas
- The unit circle: the point of the circle of radius 1 at angle θ is (cos θ, sin θ), so cos²θ + sin²θ = 1. The angle can go past a full turn or be negative.
- Radians: one radian is the angle whose arc is as long as the radius; a full turn is 2π radians. With the rule of three, 150° = 5π/6 and 3π/2 = 270°.
- The triangle: with hypotenuse 1, the opposite leg is sin θ and the adjacent leg cos θ; with hypotenuse h, they are h · sin θ and h · cos θ. For example, with 30° and hypotenuse 4, the opposite leg is 4 · 1/2 = 2.
- Notable values: sin θ is √0/2, √1/2, √2/2, √3/2, √4/2 at 0°, 30°, 45°, 60°, 90°, and cos θ takes the same values in reverse order.
- Signs by quadrant: (+, +), (−, +), (−, −), (+, −) for (cos θ, sin θ) in quadrants I, II, III and IV.
- From the turn to the wave: sin t is the height of the spinning point and cos t its x-coordinate; cos θ = sin(θ + π/2).
- Symmetries: sin(x + π) = −sin x, sin(−x) = −sin x, cos(x + π) = −cos x, cos(x + π/2) = −sin x. With a notable reference angle and the sign of the quadrant, values such as sin 150° = 1/2 or sin 300° = −√3/2 can be worked out.
- Domain, range and period: sine and cosine are defined for every x, take values in [−1, 1] and repeat every 2π: sin(x + 2π) = sin x.
- Frequency and amplitude: in y = A · sin(Nx), N is the number of periods in [0, 2π] and A sets the range [−A, A], which does not depend on N.
- Sound and drawing: a pure tone is A · sin(2π f t) (f gives the pitch, A the volume); a circle is drawn by joining with segments the points (cx + r · cos t, cy + r · sin t), and with more steps the polygon becomes indistinguishable from the circle; (a · cos t, b · sin t) is an ellipse when a ≠ b.
- Double angle: sin(2x) = 2 · sin x · cos x and cos(2x) = cos²x − sin²x. Doubling the angle can change the quadrant, and with it the sign. With sin x = 2√6/5, sin²x + cos²x = 1 gives cos x = 1/5, and from there sin(2x) = 4√6/25 and cos(2x) = 23/25.
- Polar coordinates: x = r · cos θ, y = r · sin θ. The point (3√3, −3) has r = 6 and θ = 330°; the unit circle is the case r = 1.
Level: Bachillerato (ages 16-17). It helps to have seen the Pythagorean theorem and the basic trigonometric ratios, although room 3 recalls them from the circle. In class, rooms that work well projected are room 1 (drag the point and ask students to predict the signs), room 8 (pause the animation and ask) and room 16 (predict how many periods there will be before changing N). Room 18 needs sound in the classroom.
Room-by-room contents
Room 1 · Sine and cosine are born from a spinning point
A circle of radius 1 with a point the student drags. cos θ and sin θ are defined as the coordinates (x, y) of the point. The screen shows the angle θ (which keeps growing past a full turn, with no jumps), the values of cos θ and sin θ, their squares and the sum cos²θ + sin²θ, which is always 1: the Pythagorean identity seen as “the point is at distance 1 from the centre”.
Student tasks
- Drag the point around the circle and watch how cos θ, sin θ and the sum of their squares change.
- Choose which of four pairs (cos θ, sin θ) can be a real point of the unit circle: (0.6, 0.8).
Room 2 · Degrees and radians
A circle with an angle slider that shows the value in degrees and in radians at the same time, and the arc covered. Below it, the rule of three in both directions (degrees to radians and radians to degrees). One radian is the angle whose arc is as long as the radius, and a full turn is 2π radians. It ends with a speed test of conversions.
Student tasks
- Move the slider and compare the two measures of the same angle.
- Convert 120° to radians (2π/3).
- Convert 5π/4 rad to degrees (225°).
- Speed test: 10 conversions between degrees and radians with the angles 30°, 45°, 60°, 90°, 180° and 270°, in both directions, with four options and a time limit that shortens after a correct answer and lengthens after a wrong one. Missed questions go back to the end of the queue, so all 10 must be answered correctly; if the final score is negative, the test starts again.
Room 3 · The triangle you already knew
A perpendicular is dropped from the point on the circle to the x-axis, and a right triangle with hypotenuse 1 appears. The student moves the angle with a slider and sees that its legs are sin θ (the opposite one) and cos θ (the adjacent one): triangle trigonometry is a special case of the circle.
Student tasks
- Move the angle between 5° and 85° and watch the triangle.
- Say which segment of the triangle represents sin θ (the opposite leg).
- Say which segment represents cos θ (the adjacent leg).
Room 4 · When the triangle grows: the hypotenuse is no longer 1
The same triangle with two sliders, angle and hypotenuse (from 1 to 5). The legs can be read off the drawing and follow the formula opposite leg = hypotenuse × sin θ, adjacent leg = hypotenuse × cos θ. Both challenges use numbers different from the drawing, so the student has to calculate with the formula rather than read the figure.
Student tasks
- Move the angle and the hypotenuse and check the formula for the legs.
- Calculate the opposite leg with angle 30° and hypotenuse 4 (2).
- Calculate sin θ in a right triangle with opposite leg 3 and hypotenuse 5 (0.6).
Room 5 · Speed test: sine and cosine
A table with sin θ and cos θ at 0°, 30°, 45°, 60° and 90° (0, 1/2, √2/2, √3/2, 1) and the observation that sine grows and cosine decreases but they run through the same values. Then two phases of 12 questions, first with sine and then with cosine: “sin(θ) = ?” appears and the student must choose the value before time runs out. The table stays visible for the first correct answers and then disappears. A correct answer shortens the time and a wrong one lengthens it; after a mistake the right answer blinks and the question goes back to the end of the queue, so each phase ends when all 12 have been answered correctly. If a phase ends with a negative score, it is repeated.
Student tasks
- Read the table of notable values.
- Sine phase: answer 12 questions of the form sin(θ) = ? with θ between 0° and 90°.
- Cosine phase: answer 12 questions of the form cos(θ) = ?.
Room 6 · The angles worth memorising
The pattern √0/2, √1/2, √2/2, √3/2, √4/2 that gives the sine (and, in reverse order, the cosine) at 0°, 30°, 45°, 60° and 90°. A worked example appears step by step with a pause between steps. Then two challenges with a triangle drawn, whose values are calculated with the table and come out as whole numbers.
Student tasks
- Follow the worked example.
- With angle 30° and adjacent leg √27, calculate the hypotenuse (6).
- With angle 45° and hypotenuse 6√2, calculate the opposite leg (6).
Room 7 · The sign depends on where you are
A circle divided into four coloured quadrants, with a draggable point. Next to the point the pair (cos θ, sin θ) is shown and the active quadrant is highlighted. The student finds the combination of signs of each quadrant by moving the point.
Student tasks
- Drag the point through the four quadrants and watch the signs of cos θ and sin θ.
- Say the sign of (cos θ, sin θ) in quadrant III (−, −).
- Say the sign in quadrant II (−, +).
- Say the sign in quadrant IV (+, −).
Room 8 · The height IS the sine
A circle and, beside it, a graph that is drawn while the point turns: the height of the point at each instant is carried over to the graph with a dashed guide. The sine stops being an abstract number and becomes the height of the spinning point, “unrolled” in time.
Student tasks
- Watch how the height of the point produces the sine curve.
- Pause the animation at any instant and say whether the height (sin t) is positive, negative or close to zero.
Room 9 · The same idea with width
The same scene, now with the x-coordinate of the point: the segment on the x-axis is highlighted and carried over to the graph. The student compares the two animations (sine and cosine) and sees that the cosine curve is ahead of the sine curve.
Student tasks
- Watch how the x value of the point produces the cosine curve.
- Say whether cosine increases or decreases in the first quadrant, from 0° to 90° (it decreases).
Room 10 · Sine and cosine are the same wave, shifted
The sine and cosine curves drawn together, with a marker that moves along both at once. One is the other shifted by a quarter of a turn: cos θ = sin(θ + π/2).
Student tasks
- Follow the marker along the two curves.
- Choose which relation is true among cos θ = sin(θ + π), cos θ = sin(θ + π/2) and cos θ = sin(π − θ) (the second).
Room 11 · Half a turn, turning backwards
A point at angle θ and a second point obtained by turning half a turn (+π), a quarter turn (+π/2) or reflecting in the x-axis (−θ). With this the student sees how the coordinates change and works out the symmetries of sine and cosine. When a challenge is missed, the drawing shows both points and marks the relevant coordinate on each.
Student tasks
- Move θ and choose half turn, quarter turn or reflection to see the second point.
- If sin x = 0.6, calculate sin(x + π) (−0.6).
- If sin x = 0.6, calculate sin(−x) (−0.6).
- If cos x = 0.6, calculate cos(x + π) (−0.6).
- If sin x = 0.6, calculate cos(x + π/2) (−0.6).
Room 12 · Guess before you look
The circle appears without the point. The student must predict the value of sin θ for angles that reduce to a notable one (150°, 210°, 300°, 120°) by combining the reference angle and the sign of the quadrant. The six options are always the same (1/2, √2/2, √3/2 and their negatives), so the answer cannot be found by elimination. On answering (or on a mistake) the point and its reference angle are drawn.
Student tasks
- Choose sin(150°), sin(210°), sin(300°) and sin(120°) among the six options, without seeing the point.
Room 13 · Reading the graph without seeing the circle
Two drawings side by side: the graph of sin x with its axis marked in multiples of π, and the unit circle. Touching or dragging on the graph shows the matching point on both. Every crest, trough and zero of the graph is a specific position of the point on the circle.
Student tasks
- Explore the graph and see where the point falls on the circle.
- For x = π, 3π/2, 3π/4 and 7π/6, say in which part of the circle the point is (left, bottom, top left, bottom left). When the answer is right, the point is marked on the graph and on the circle.
Room 14 · How far does the function reach?
The graph of the sine inside a band marking the range [−1, 1]. The domain is all of ℝ because the point can keep turning forever, and the range is bounded because the point is always at distance 1 from the centre.
Student tasks
- Say whether any x has sin x = 1.5 (no, because the point is never farther than 1 from the centre).
- Give the maximum value of cos x (1).
- Give the minimum value of cos x (−1).
Room 15 · Sine and cosine are periodic functions
The graph of the sine with a shaded window of one full turn (2π) that a slider moves by 0, 1 or 2 turns. The values repeat every 2π: sin(x + 2π) = sin x and cos(x + 2π) = cos x.
Student tasks
- Shift the window by one or two turns and compare the graph.
- If sin x = 0.3, calculate sin(x + 2π) (0.3).
- If sin x = 0.3, calculate sin(x + 4π) (0.3).
- If sin x = 0.3, calculate sin(x − 2π) (0.3).
Room 16 · Spinning faster: sin(Nx)
An animated circle in which the point turns with angle N·t, and to its right two stacked graphs with a tracer point moving at the same horizontal speed in both: sin x (or cos x) above as a reference and sin(Nx) below. A slider changes N from 1 to 6 and a button switches between sine and cosine. Multiplying the angle by N makes the point go round N times in the time it used to take for one, and the graph is compressed N times.
Student tasks
- Change N and compare the two graphs.
- Say how many periods sin(3x) completes in [0, 2π] (3).
- Say how many periods cos(5x) completes in [0, 2π] (5).
Room 17 · The range does not depend on frequency
The graph of y = A · sin(Nx), with the formula updated live and the range band. Two sliders: amplitude A (0.5 to 4) and N (1 to 5). Changing N does not change the height of the wave, and changing A scales the range to [−A, A].
Student tasks
- Move A and N and see what changes in the graph and the formula.
- Give the range of 3 sin(5x) ([−3, 3]).
- Give the range of 0.5 cos(2x) ([−0.5, 0.5]).
Room 18 · A pure sound is a sine wave
A pure tone is A · sin(2π f t): the frequency f sets the pitch and the amplitude A the volume. Two sliders (f between 200 and 800 Hz, A between 0.2 and 1.4) and a button that plays the tone with the browser’s audio; the drawn wave reflects both values in real time. Two buttons, tone A and tone B, play two tones of 300 and 600 Hz in random order and show their wave with the real frequency.
Student tasks
- Move frequency and amplitude, listen to the tone and watch the wave.
- Listen to tone A and tone B and say which has the higher frequency (sounds higher-pitched).
Room 19 · How a computer draws a circle
A circle is drawn by computing many points (cx + r·cos t, cy + r·sin t) and joining them with segments: it is a loop that calls sine and cosine. A slider changes the number of steps (from 3 to 64) and the polygon gets closer to the circle; the code snippet that draws it is also shown.
Student tasks
- Increase the number of steps and watch the polygon approach the circle.
- Choose how many steps (6, 12 or 60) it takes for the polygon to be indistinguishable from a smooth circle (60).
Room 20 · When the circle stretches: the ellipse
The curve (a·cos t, b·sin t) is a circle if a = b and an ellipse if a ≠ b. The student adjusts two sliders, a and b (0.5 to 3), which start at a = 0.5 and b = 1.4, and compares their curve with a dotted target ellipse.
Student tasks
- Adjust a and b until the curve matches the target ellipse (a = 2.5, b = 1, within 0.15). The challenge closes when it does.
Room 21 · The sign of sin(2x)
A circle with two points, the angle x and its double 2x, and the formula sin(2x) = 2 · sin x · cos x. Doubling the angle can move it to another quadrant and, with it, change the sign. A slider explores x from 0° to 170°. In all three challenges the double 2x crosses into another quadrant, so the sign cannot be deduced by looking at x alone. On a mistake, the drawing is set to the angle in the question.
Student tasks
- Move x and watch which quadrant 2x falls in.
- Say the sign of sin(2x) for x = 100° (negative), x = 200° (positive) and x = 300° (negative).
Room 22 · Calculate sin(2x)
With the formula sin(2x) = 2 · sin x · cos x, the student must calculate sin(2x) from sin x, which is not a notable value. Each case has two steps: first find cos x with sin²x + cos²x = 1 (numerical answer) and then choose sin(2x) among five similar options (the same radical with another numerator or sign, or the result of wrongly applying the cosine formula).
Student tasks
- With sin x = 2√6/5, find cos x (1/5) and choose sin(2x) (4√6/25).
- With sin x = √5/3, find cos x (2/3) and choose sin(2x) (4√5/9).
Room 23 · The sign of cos(2x)
Like room 21, with the formula cos(2x) = cos²x − sin²x: reduce 2x to its quadrant and remember where cosine is positive or negative. There is a circle with x and 2x and a slider to explore.
Student tasks
- Say the sign of cos(2x) for x = 80° (negative), x = 200° (positive) and x = 280° (negative).
Room 24 · Calculate cos(2x)
The same mechanics as room 22, now with cos(2x) = cos²x − sin²x. cos x is given, sin x is found with the Pythagorean identity and cos(2x) is chosen among five options. The result is rational, with no radical, because squaring removes the root.
Student tasks
- With cos x = 2√6/5, find sin x (1/5) and choose cos(2x) (23/25).
- With cos x = √5/3, find sin x (2/3) and choose cos(2x) (1/9).
Room 25 · Another way to give a point: polar coordinates
A point given by (r, θ), with two sliders (r between 0.5 and 6, θ from 0° to 350°), concentric guide circles and a live readout (r, θ) → (x, y). The relations are x = r·cos θ and y = r·sin θ: the unit circle was the case r = 1. It closes the lab by tying together what came before.
Student tasks
- Move r and θ and see how x and y change.
- Choose the expression for the x-coordinate (r cos θ) and for the y-coordinate (r sin θ).
- For the Cartesian point (3√3, −3), find r (6).
- For that same point, find the angle θ (330°).
Rooms to project
The most striking ones to show and discuss in class.