三角函數圖像與方程
Intermediatey=sin x rises and falls with period 360°; trig equations usually have two solutions.
- Sketch with five points: 0°,90°,180°,270°,360°.
- Period 360°; amplitude 1; range -1 to 1.
- Equations: sine's partner is 180°-x; cosine's is 360°-x.
文體小知識：三角函數圖像——對應 DSE 核心單元：y=sin x 與 y=cos x 的形狀、週期與解三角方程。
§Text Passage
1The graph of y=sin x (0°≤x≤360°) rises from (0,0) to a peak (90°,1), falls to (180°,0), dips to (270°,-1) and returns to (360°,0) — the wave repeats every 360°, its period. y=cos x has the same shape shifted left 90°, starting at (0,1). Both ranges run from -1 to 1.
2Graphs solve equations: sin x=0.5 has two solutions between 0° and 360° — 30° and 150°. The graph shows why: the horizontal line y=0.5 cuts the wave twice. Method: the calculator gives the basic angle (30°); then check quadrants — sine is positive in quadrants 1 and 2, so 180°-30°=150°. For cosine (quadrants 1 and 4): cos x=0.5 gives 60° and 300°.
3Transformations: y=sin 2x compresses the period to 180°; y=2sin x doubles the amplitude; y=sin x+1 lifts everything. Real applications — seasonal temperatures, tides, Ferris-wheel heights — all follow y=a sin(bx)+c, where a is amplitude, b controls period and c is the mean level.
1Summary 重點摘要
y=sin x rises and falls with period 360°; trig equations usually have two solutions.
2Vocabulary cards Vocabulary
The repeat length: 360° for sin and cos.
Maximum distance from the mean line.
Both peak at 1.
Both bottom out at -1.
The acute angle from the calculator.
Quadrants decide the sign.
Equations with two solutions per cycle.
Describing periodic phenomena mathematically.
3Step by step Steps
- Step 1: Five key points of y = sin xSketching y = sin x needs only five key points: (0°, 0), (90°, 1), (180°, 0), (270°, -1), (360°, 0), joined by a smooth curve — with these five the sketch already meets the basic standard. For y = cos x the same wave starts at (0°, 1): (0°, 1), (90°, 0), (180°, -1), (270°, 0), (360°, 1). The two curves are identical in shape, differing only by a horizontal shift of 90°. Use a smooth curve rather than straight segments, and remember cos 0° is 1, not 0 — miscopying that first point shifts the whole wave.
- Step 2: Three readings — maximum, minimum, periodDescribe any trigonometric graph with three numbers: maximum, minimum, and period. For y = sin x these are 1 (at x = 90°), -1 (at x = 270°), and 360°. Transformations act predictably: y = 2sin x doubles the amplitude to peak at 2 and dip to -2; y = sin 2x halves the period to 180°, because the coefficient of x compresses the wave; y = sin x + 1 lifts the whole curve by 1. Do not confuse the amplitude with the maximum value — a vertical shift moves both extremes together but leaves the amplitude unchanged.
- Step 3: Solving — the basic angle firstTo solve a trigonometric equation, find the basic angle with the calculator first: for sin x = 0.5, press sin⁻¹(0.5) to get 30°. The basic angle is only the starting point, not the full answer, because trigonometric functions are periodic and take the same value twice within 0° to 360°; the calculator always reports just one of them, and the other comes from quadrant analysis. Treat the first calculator output as the only solution and half the marks vanish. Confirm DEGREE mode before typing — RAD mode produces numbers that are wrong for this course.
- Step 4: Quadrants give the second solutionUse the sign pattern of each function to place the remaining solutions: sine is positive in quadrants 1 and 2, so sin x = 0.5 gives x = 30° and x = 180° - 30° = 150°; cosine is positive in quadrants 1 and 4, so cos x = 0.5 gives 60° and 360° - 60° = 300°; tangent is positive in quadrants 1 and 3, adding 180° to the basic angle. The chant: sin pairs with 180° - θ, cos with 360° - θ, tan with 180° + θ. When the function value is negative the quadrants switch sides — decide the sign first, then pick the quadrants, instead of applying the formula blindly.
- Step 5: Modelling — the Ferris wheel and periodic phenomenaTrigonometric functions model any periodic phenomenon; the classic example is a Ferris wheel of radius 10 m with its axle 12 m above the ground, turning once every 4 minutes. The height after t minutes is h = 12 + 10sin(90t). The formula has three parts: 10 is the amplitude, the largest departure from the mean position; 12 is the mean height, the height of the axle; the 90 inside the sine comes from 360 ÷ 4 = 90, the number of degrees turned per minute. The ride peaks at 22 m and dips to 2 m. Mixing up amplitude with mean height, or letting the unit of t disagree with the period, are the usual faults.
- Step 6: Answer format and scoring detailsA trigonometric equation usually has more than one solution in the given range, so list them all: write 'x = 30° or x = 150°' — a single solution forfeits half the marks. In sketches, label the coordinates of the five key points and the scales on both axes, and keep the curve smooth. In word problems, finish with a complete sentence that includes the unit and follows the accuracy demanded. Check the stated range too: answers outside 0° to 360° do not belong, and omitting the word 'or' between two solutions can make them read as one coordinate pair.
4Diagrams Diagrams
5Worked example Example
Example: solve the trigonometric equation 2sin x = 1 for 0° ≤ x ≤ 360°.
- Step 1: Divide both sides by 2 to get sin x = 0.5; applying the same operation to both sides keeps the equation balanced.
- Step 2: Find the basic angle with the calculator: x = sin⁻¹(0.5) = 30°; the calculator reports only the first-quadrant solution.
- Step 3: Analyse the quadrants: sine is positive in quadrants 1 and 2, so the second solution is x = 180° - 30° = 150°, by the identity sin(180° - θ) = sin θ.
- Step 4: Verify: sin 30° = 0.5 and sin 150° = 0.5, so both solutions satisfy the equation and lie inside the given range ✓
- Answer: The solutions are x = 30° and x = 150°. The routine has four steps: simplify the coefficient, find the basic angle, add the second solution using the quadrant signs, then verify each answer by substitution. Sine reaches 0.5 twice in every 360° cycle, so two solutions must appear, and answering only 30° forfeits half the marks. The second solution works because sin(180° - θ) = sin θ — the identity behind the quadrant analysis, and worth stating so the examiner can follow the reasoning.
6Common mistakes Mistakes
7Quick check Quick check
Q1Period and maximum of y=sin x?
Q2y=cos x at x=0?
Q3Both solutions of sin x=0.5?
Q4Period of y=sin 2x?
Q5Maximum of h=12+10sin(90t)?
8Recap Recap
- Sketch with five points: 0°,90°,180°,270°,360°.
- Period 360°; amplitude 1; range -1 to 1.
- Equations: sine's partner is 180°-x; cosine's is 360°-x.
- Transforms: b doubles → period halves; a doubles → amplitude doubles.
- Modelling kit: a + c + period.