Demonstration 1 of 4
The angle decides the ratio
If you make the three-four-five triangle bigger or smaller, do sine, cosine and tangent change?
Scaling multiplies every side by the same number, so each ratio's top and bottom change together. Choosing the other acute angle swaps which leg is opposite.
\[\sin\theta\div\cos\theta=\frac35\div\frac45=\frac34\]
θ is the chosen acute angle. The opposite leg faces it, the adjacent leg touches it, and the hypotenuse sits across from the right angle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
Predict first. Keep the same angle and shrink the triangle to one fifth of its size. What is the new hypotenuse, and does the sine change?
Choose an example
Constructed example: the chapter's three-four-five triangle, scaled by 2 and by 1/5.
Calculated values
- Opposite / hypotenuse (sine)
- 3/5
- Adjacent / hypotenuse (cosine)
- 4/5
- Opposite / adjacent (tangent)
- 3/4
- Pythagorean check
- 3^2 + 4^2 = 9 + 16 = 25 = 5^2
sin = 3/5, cos = 4/5, tan = 3/4. Check: (3/5) / (4/5) = 3/4, the same as tangent. This is the book's 3-4-5 triangle at its original size; choose another scale to see the lengths change while the three ratios stay put.
Use the idea
Before writing a ratio, point at the angle and say which leg faces it. That one habit settles opposite and adjacent.
Where the conclusion applies
A right triangle with an acute chosen angle and all sides in the same unit, so the units cancel. If the triangle has no right angle, these three ratios do not apply.
Check your understanding: In a triangle with legs 6 and 8 and hypotenuse 10, what is the sine of the angle opposite the 6?
Chapter 20 source: section "A triangle supplies three ratios". Demonstration C20-D01.
Demonstration 2 of 4
Where the turning point lands
At each quarter turn, what are the point's two coordinates, and what happens to tangent?
The radius carries the point around. Its across position is cosine, its up position is sine, and the quadrant decides each sign.
\[(\cos\theta, \sin\theta)\]
\[\tan\theta=\sin\theta/\cos\theta\]
The unit circle has radius 1 and center (0, 0). The angle θ starts on the positive horizontal axis and turns counterclockwise, in radians. Cosine is the across coordinate and sine is the up coordinate.
Predict first. Turn to π. Which coordinate becomes negative, and is tangent defined there?
Choose an example
Constructed example: the chapter's table of the cardinal angles on the unit circle.
Calculated values
- Angle (radians)
- π/2
- Point
- (0, 1)
- Cosine
- 0
- Sine
- 1
- Tangent
- undefined (cosine is 0, so sin/cos would divide by zero)
- Radius check
- (0)^2 + (1)^2 = 1
At π/2 the point is (0, 1), so cos = 0 and sin = 1. Radius check: (0) x (0) + (1) x (1) = 1. tan = 1/0 is undefined. The signs come from where the point sits: left is negative across, below is negative up.
Use the idea
When a sine or cosine value surprises you, sketch the circle and ask whether the point is above or below, left or right.
Where the conclusion applies
Angles start on the positive horizontal axis and turn counterclockwise. Change that convention and every label must change with it. Tangent is undefined wherever cosine is 0.
Check your understanding: At 2π, what are the cosine and sine, and why do they match the values at 0?
Chapter 20 source: section "One circle, two coordinate stories". Demonstration C20-D02.
Demonstration 3 of 4
Unroll one coordinate into a wave
If you record only one coordinate as the point turns, what shape does the record make?
Each quarter turn adds one marked value to the trace. Sine starts at mid height; cosine starts at the right edge, so it starts at 1.
\[(π/2, 1)\]
\[1, 0, -1, 0, 1\]
Across the graph is the angle θ in radians, not a position on the circle. Up the graph is the recorded coordinate: sine records height, cosine records across.
Predict first. Stop at a half turn. Is the sine trace back at 0? Is the cosine trace?
Choose an example
Constructed example: the chapter's five sine points and five cosine values.
Calculated values
- Recorded coordinate
- vertical (sine)
- Turned so far
- 4/4 of a turn = 2π
- Values at each quarter turn
- 0, 1, 0, -1, 0
- Starting value
- 0
Recording the vertical coordinate from 0 to 2π: 0, 1, 0, -1, 0. The turn so far is 4/4 x 2π = 2π. The last marked point on the circle is (1, 0): (1) x (1) + (0) x (0) = 1, so it is still one unit from the center. One full turn is complete, so the value is back where it began and the pattern repeats.
Use the idea
When you read a wave-shaped graph, first ask which coordinate it records and what the horizontal axis measures.
Where the conclusion applies
The input is an angle in radians and the circle has radius 1. Equal values at 0 and π do not make π a period: the full cycle needs 2π.
Check your understanding: What is the sine value at 5π/2, one full turn past π/2?
Chapter 20 source: section "Unroll the vertical coordinate". Demonstration C20-D03.
Demonstration 4 of 4
Midline first, then amplitude
In the height model, how high and how low can the modeled height go?
Sine runs from -1 to 1. Multiplying by the amplitude stretches that range, and adding the midline lifts it. The top is midline plus amplitude, the bottom is midline minus amplitude.
\[h(\theta) = 2\ \text{m} + (1.5\ \text{m}) \sin\theta\]
\[2\ \text{m} - 1.5\ \text{m} = 0.5\ \text{m}\]
h is the modeled height in meters and θ is the angle in radians. The 2 m is the midline. The 1.5 m is the amplitude: the distance from the midline up to the top or down to the bottom.
Predict first. Raise the midline to 3 m and keep the amplitude at 1.5 m. What are the new top and bottom?
Choose an example
Constructed example: the chapter's height model, with other midlines and amplitudes.
Calculated values
- Midline
- h = 2 m
- Amplitude
- 1.5 m
- Maximum
- 2 + 1.5 = 3.5 m
- Minimum
- 2 - 1.5 = 0.5 m
- Heights at 0, π/2, π, 3π/2, 2π
- 2, 3.5, 2, 0.5, 2 m
Top: 2 + 1.5 = 3.5 m. Bottom: 2 - 1.5 = 0.5 m. At π/2: 2 + 1.5 x 1 = 3.5 m. Any calculated height above 3.5 m or below 0.5 m does not belong to this model.
Use the idea
After substituting into a sine model, check the answer against the top and bottom. A height outside them came from a different expression.
Where the conclusion applies
A constructed model with the angle in radians. No time, speed or physical motion is supplied. With amplitude 0 the model is a flat line at the midline and has no least positive period.
Check your understanding: For h = 2 m + (1.5 m) sin θ, what is the height at 3π/2?
Chapter 20 source: section "Amplitude measures distance from a midline". Demonstration C20-D04.