Angles in standard position, degrees and radians, and why the circle prefers radians; the unit circle with its special points — sin 30° = 1/2, cos 60° = 1/2, sin 45° = √2/2 — and the signs and reference angles in every quadrant. Then sine and cosine as waves: amplitude, period, midline and phase shift read off y = A sin(B(x − C)) + D; a tide and a Ferris wheel modeled and checked; the Pythagorean identity, inverse trig, and equations solved on [0, 2π). Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
One circle, radius 1, centered at the origin. Every angle puts you at one point on it.
The point on the unit circle is (cos θ, sin θ). Cosine is how far across, sine is how far up — so a LOW angle like 30° has the small sine, and tangent is just the slope, sin over cos.
What people get wrong
⚠️People often think…
sin 30° = √3/2.
30° is a LOW angle, so its height is the small number: sin 30° = ½ and cos 30° = √3/2. At 60° they trade places. Picture the point before you answer.
Low angle, small sine. Picture the point.
⚠️People often think…
If the max is 5 and the min is 1, the amplitude is 4.
Amplitude is half the swing, measured from the middle. Midline (5 + 1)/2 = 3, amplitude (5 − 1)/2 = 2. Max − min is the whole swing, top to bottom.
Halve the swing. Amplitude starts at the middle.
⚠️People often think…
y = sin(2x) has period 2.
B is how many cycles fit, not the length. Period = 2π/B, so 2π/2 = π. A bigger B packs more waves in, which makes each one SHORTER.
Period is 2π/B. Bigger B, shorter wave.
⚠️People often think…
On [0, 2π), sin x = ½ has one solution, π/6.
Going all the way round, you pass every height twice — once climbing, once coming down. sin x = ½ at π/6 AND 5π/6. The calculator only ever hands you the first one.
One turn, two crossings. Find the second yourself.
Middle, height, length, start
1Watch one
A wave rises to 5 and falls to 1. Find the midline, amplitude and period of y = 3 + 2 sin(2x).
Midline: average the max and the min. (5 + 1)/2 = 3.
Amplitude: half the swing. (5 − 1)/2 = 2.
Period: 2π/B with B = 2, so 2π/2 = π.
Start: no shift here, so the wave begins at the midline heading up.
2Do one with me
Read y = 3 + 2 sin(2x) and the special angles.
The amplitude is
The period is
sin 30° as a fraction is
💬One sentence, then you move on
Why does a bigger B make the wave shorter, not longer?
3Try one
Give the SECOND solution of sin x = ½ on [0, 2π).
I want a hint first
Mirror π/6 across π/2: subtract it from π. π − π/6 = 6π/6 − π/6.
💬Last one — then you're done here
Why does sin x = ½ have two answers in one turn?
Where this goes
Where this lives
Anything that comes back around: daylight over a year, a tide, a heartbeat on a monitor, a sound wave. Midline, amplitude and period describe all of them.
What this feeds
Next unit is sequences and series, where a pattern is listed instead of drawn.
Name one thing in your week that repeats on a fixed cycle.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
〰️ Reading a sinusoid — four numbers in order
MIDLINE D = (max + min)/2 → AMPLITUDE A = (max − min)/2 → PERIOD P = the distance from one max to the next, then B = 2π/P → SHIFT C = where the wave crosses the midline going up (for sine) or sits at its max (for cosine). Write y = A sin(B(x − C)) + D. Example: max 5, min 1, period 8, midline crossing up at x = 2 → D = 3, A = 2, B = π/4, C = 2: y = 2 sin(π(x − 2)/4) + 3.
Middle, height, length, start.
⭕ The whole topic in one idea
Walk around a circle of radius 1. Your height is the sine of the angle, your horizontal position is the cosine, and the slope of the line back to the center is the tangent. Everything else follows: the signs by quadrant, the special values from two triangles, the wave when you plot height against angle, and its period 2π because that is one lap. Radians measure the lap in radii, so the circle's own arithmetic (s = rθ) works without conversion.
Cosine across, sine up, tangent is the slope.
⚠️ Traps the test loves
Swapping the 30° and 60° points. 30° is LOW, so its sine is the small number: sin 30° = ½, cos 30° = √3/2.
Calling max − min the amplitude. Max 5, min 1 → amplitude 2, midline 3. Halve the swing.
Reading B as the period. y = sin(2x) has period 2π/2 = π; a bigger B means a SHORTER period.
Taking the shift of sin(2x − π) as π. Factor first: sin(2(x − π/2)) — the shift is π/2.
Giving one answer to sin x = ½ on [0, 2π). Sine hits each height twice per turn: π/6 AND 5π/6.
Using degrees in s = rθ. Radius 6, 60°: 6 × 60 = 360 is nonsense; 6 × π/3 = 2π ≈ 6.28 is the arc.
📐 The special angles — the chart
Degrees
Radians
Point (cos, sin)
sin
cos
tan
0°
0
(1, 0)
0
1
0
30°
π/6
(√3/2, ½)
½ = 0.5
√3/2 ≈ 0.87
√3/3 ≈ 0.58
45°
π/4
(√2/2, √2/2)
√2/2 ≈ 0.71
√2/2 ≈ 0.71
1
60°
π/3
(½, √3/2)
√3/2 ≈ 0.87
½ = 0.5
√3 ≈ 1.73
90°
π/2
(0, 1)
1
0
undefined
180°
π
(−1, 0)
0
−1
0
270°
3π/2
(0, −1)
−1
0
undefined
360°
2π
(1, 0)
0
1
0
〰️ y = A sin(B(x − C)) + D — the chart
Letter
Name
How to read it
Example: y = 2 sin(x − π/2) + 1
A
amplitude
(max − min)/2; a negative A flips the wave
2
B
frequency factor
period = 2π/B; to build a period P use B = 2π/P
1 → period 2π
C
phase shift
right by C for (x − C); factor B out first
π/2 to the right
D
midline
(max + min)/2; the vertical shift
y = 1
max / min
D + A and D − A
3 and −1
🎯 How the test will ask
“Convert” 135° to radians or 5π/6 to degrees — multiply by π/180 or 180/π.
“Find the exact value” of sin 150°, cos(4π/3), tan(−π/4) — reference angle, then the sign from the quadrant.
A graph of a wave — “write the equation,” or an equation — “give amplitude, period, midline, shift.”
A tide, a Ferris wheel, a temperature — “write a model and evaluate it at t = ….” Midline, amplitude, B = 2π/P, then where the max or the rising midline crossing sits.
“sin θ = 3/5 in Quadrant II; find cos θ and tan θ” — the Pythagorean identity for the size, the quadrant for the sign.
“Solve on [0, 2π)” — reference angle, the right quadrants, and every cycle that fits in the window.
Arc length or sector area — convert to radians, then s = rθ or A = ½r²θ.
✅ Before the test, can you…
Draw the unit circle from memory with the points at 0°, 30°, 45°, 60°, 90°, 180°, 270°, in degrees and radians?
Say the sign of sine, cosine and tangent in each quadrant, and find any reference angle?
Read amplitude, period, midline and phase shift off y = 2 sin(x − π/2) + 1 and off its graph?
Write h(t) = 3 + 3 cos(π(t − 4)/6) for a tide from its high and low, and check it?
Use sin²θ + cos²θ = 1 to get cos θ from sin θ, with the right sign?
Solve sin x = ½ and 3 sin(2x) = 1.5 on [0, 2π) and explain how many answers there are?
Find an arc length and a sector area with the angle converted to radians first?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: label the unit circle in degrees and radians, read amplitude, period, midline and phase shift off a sine wave, graph a day of tides and find the model behind it, then sort twelve angles into their quadrants by the signs of sine and cosine. Every activity checks itself, and hints are free.
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