Trig Graphs, Knowledge Organiser

Three Graphs at a Glance

• sin x: starts (0, 0), period 360°, amp 1
• cos x: starts (0, 1), period 360°, amp 1
• tan x: period 180°, asymptotes at 90°, 270°
• cos x = sin(x + 90°): shifted left 90°

Transformations

• sin x + c → translation up c
• sin(x − a°) → translation right a°
• −sin x → reflection in the x-axis

Key Vocabulary

• Amplitude: distance from midline to max/min
• Period: x-length of one full cycle
• Translation: a shift of the whole graph left, right, up or down
• Asymptote: vertical line where tan is undefined

Common Errors

• Moving sin(x + 30°) to the right (it moves left)
• Only finding one solution to sin x = k
• Thinking cos x starts at 0 (it starts at 1)
• Translation direction: inside the bracket means the opposite way

Key Formulas

• sin x: period 360°, amplitude 1, starts at (0, 0)
• cos x: period 360°, amplitude 1, starts at (0, 1)
• tan x: period 180°, asymptotes at 90°, 270°, …
• sin x + c: up c; sin(x − a°): right a°; −sin x: reflected in the x-axis

Common Mistakes

Misconception 1: "y = sin(x + 30°) moves the graph to the right" A change inside the brackets works in the OPPOSITE direction to the sign. y = sin(x + 30°) moves the graph 30° to the LEFT, and y = sin(x − 30°) moves it 30° to the RIGHT. Check with the first peak: sin x peaks at 90°, so sin(x + 30°) peaks at 60°, which is further left. A change outside the brackets, such as sin x + 30, goes the way you expect (UP). Misconception 2: "sin x = 0.5 has only one solution between 0° and 360°" The sine graph crosses any horizontal line y = k (where −1 < k < 1) TWICE in every 360° period. For sin x = 0.5, the solutions are x = 30° AND x = 150°. The sine graph is symmetric about x = 90°, so the second solution is always 180° − (first solution). For cosine, the graph is symmetric about x = 180°, so the second solution is 360° − (first solution). Always check for two solutions in trig equations. Misconception 3: "y = cos x starts at zero like y = sin x" y = cos x starts at its MAXIMUM value of 1 when x = 0, not at zero. This is because cos 0° = 1 (the cosine of zero degrees is 1). The cosine graph looks like the sine graph shifted 90° to the left: it starts at a peak rather than at the middle. If you mistakenly sketch cos as starting at zero, all your key points will be wrong. Always mark (0°, 1) as the starting point for the cosine graph.

Worked Examples

• Example 1: Identifying Key Points on Trig Graphs Question: State the maximum and minimum values of y = sin x + 2, and the x-values where they occur for 0° ≤ x ≤ 360°. Show Solution Standard y = sin x: maximum = 1 at x = 90°, minimum = -1 at x = 270° y = sin x + 2: the whole graph moves up 2 units, so add 2 to every y-value Maximum = 1 + 2 = 3 (at x = 90°), minimum = -1 + 2 = 1 (at x = 270°) x-values: they do not change, because the graph only moves up Answer: Maximum = 3 at 90°, minimum = 1 at 270°. Same shape as sin x but translated 2 units up.
• Example 2: Solving a Trigonometric Equation Using the Graph Question: Find all solutions to sin x = 0.5 for 0° ≤ x ≤ 360°. Show Solution Step 1: First solution: x = sin⁻¹(0.5) = 30° Step 2: Use symmetry of the sine graph: the sine curve is symmetric about x = 90°, so the second solution = 180° - 30° = 150° Step 3: Check both solutions sin 30° = 0.5 ✓ sin 150° = sin(180° - 30°) = sin 30° = 0.5 ✓ Answer: x = 30° or x = 150°
• Example 3: Describing a Transformation of a Trig Graph Question: Describe the transformation that maps y = cos x onto y = cos(x + 60°). Show Solution Step 1: Identify the change: 60° has been added to x inside the brackets Step 2: Apply the rule: inside the brackets means the opposite direction, so +60° moves the graph LEFT by 60° Step 3: Check with a key point: the maximum of cos x is at x = 0°. On the new graph the maximum is at x = -60°, which is 60° to the left. Effect: The shape, the maximum (1) and the minimum (-1) are all unchanged. The graph only slides sideways. Answer: A translation of 60° to the left (vector (−60, 0)).