Math 30-1 · Trigonometry
Angles, circles and waves
Trig in Math 30-1 is one picture — a circle with radius 1 — looked at four different ways. Get comfortable with the circle and everything else comes from it.
Radians
The idea: a radian measures an angle by arc length. Half a turn is π radians, so π = 180°. Everything else is a conversion from that.
To convert degrees to radians, multiply by π/180. To go back, multiply by 180/π — or just replace π with 180°.
- 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6
- 3π/4 = 3 × 180° ÷ 4 = 135°
Standard position, coterminal and reference angles
An angle in standard position starts on the positive x-axis and turns counterclockwise (clockwise if negative). Coterminal angles end on the same arm — they differ by full turns, so 45°, 405° and −315° are all coterminal. The reference angle is the acute angle between the terminal arm and the x-axis; 150°, 210° and 330° all have a reference angle of 30°.
Arc length
With θ in radians, arc length is a = rθ. A radius of 6 cm and an angle of 2π/3 gives a = 6 × 2π/3 = 4π cm.
Watch out: a = rθ only works in radians. Plugging in 120° gives 720 cm for a 6 cm circle, which is obviously too big.
The unit circle
The idea: on a circle of radius 1, the point at angle θ is (cos θ, sin θ). Cosine is the x-coordinate, sine is the y-coordinate, and tan θ = y/x.
Exact values worth knowing
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° = π/6 | 1/2 | √3/2 | 1/√3 |
| 45° = π/4 | √2/2 | √2/2 | 1 |
| 60° = π/3 | √3/2 | 1/2 | √3 |
| 90° = π/2 | 1 | 0 | undefined |
Any angle in three steps
Find cos(4π/3):
- Quadrant: 4π/3 is past π, so it is in Quadrant III.
- Reference angle: 4π/3 − π = π/3, and cos(π/3) = 1/2.
- Sign: only tangent is positive in III, so cosine is negative: −1/2.
A point that is not on the unit circle
If P(−3, 4) is on the terminal arm, first find r = √(x² + y²) = 5. Then sin θ = y/r = 4/5, cos θ = x/r = −3/5, and tan θ = y/x = −4/3.
Watch out: getting the number right but the sign wrong. Always decide the quadrant before you write the answer.
Sinusoidal graphs
The idea: y = a sin b(x − c) + d. Each letter does exactly one job: a stretches up and down, b squeezes sideways, c slides sideways, d slides up and down.
| Parameter | What it changes | How to read it |
|---|---|---|
| a | Amplitude | |a|, the distance from midline to peak |
| b | Period | 2π ÷ |b| (or 360° ÷ |b|) |
| c | Horizontal shift | Right by c if written as (x − c) |
| d | Midline | y = d; max is d + |a|, min is d − |a| |
For y = 3 sin[2(x − π/4)] − 1: amplitude 3, period π, shifted π/4 right, midline y = −1. So the range is −4 ≤ y ≤ 2.
Watch out: y = sin(2x − π/2) is not shifted π/2. Factor first: sin 2(x − π/4), so the shift is π/4.
Solving equations and using identities
The idea: isolate the trig function, find the reference angle, then use CAST to find every angle in the domain — there is almost always more than one.
Worked example
Solve sin θ = 1/2 for 0 ≤ θ < 2π.
- Reference angle: sin(π/6) = 1/2, so it is π/6.
- Sine is positive in Quadrants I and II.
- Quadrant I: π/6. Quadrant II: π − π/6 = 5π/6.
When it looks like a quadratic
2cos²θ − cos θ − 1 = 0 factors exactly like 2x² − x − 1: (2cos θ + 1)(cos θ − 1) = 0. So cos θ = −1/2, giving 2π/3 and 4π/3, or cos θ = 1, giving 0. Three solutions.
Double angles mean double the solutions
sin 2θ = 1/2 has a period of π, so it runs through two full cycles on [0, 2π) and hits 1/2 four times. Solve for 2θ over [0, 4π), then divide by 2.
Identities to know
- Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
- Quotient: tan θ = sin θ / cos θ
- Pythagorean: sin²θ + cos²θ = 1
- Sum and difference: sin(A ± B) = sin A cos B ± cos A sin B, and cos(A ± B) = cos A cos B ∓ sin A sin B
- Double angle: sin 2A = 2 sin A cos A, and cos 2A = cos²A − sin²A
To simplify (1 − cos²x) ÷ (sin x cos x): the Pythagorean identity turns the top into sin²x, and sin²x ÷ (sin x cos x) = sin x ÷ cos x = tan x.
Watch out: dividing both sides by a trig function, like cos θ, to "cancel" it. You lose the solutions where cos θ = 0. Factor instead.