Ratios and Graphs of Trigonometric Functions · 6.2.3
Problems using trig function graphs
Students use a trigonometric graph, or an equation representing one, to solve related problems, such as finding where the graph meets a horizontal line, determining the number of solutions to an equation within 0° to 360°, or reading specific values directly from a sketched or given graph.
The official learning standard (6.2.3)
“Solve problems involving graphs of sine, cosine and tangent functions.”
What it means
Students use a trigonometric graph, or an equation representing one, to solve related problems, such as finding where the graph meets a horizontal line, determining the number of solutions to an equation within 0° to 360°, or reading specific values directly from a sketched or given graph.
How it is examined
This standard is examined mainly in Paper 2, where a graph (drawn or given) is used to find solutions of an equation such as a sin bx + c = k, often by drawing a line on the same axes. Paper 1 may ask how many solutions an equation has within a given range.
Worked example
The graph of y = 2 cos x, 0° ≤ x ≤ 360°, is drawn together with the line y = 1. Find the values of x at the points of intersection.
- At the points of intersection, 2 cos x = 1, so cos x = 0.5.
- Reference angle: cos⁻¹(0.5) = 60°.
- cos x is positive, so x lies in Quadrant I or Quadrant IV.
- Quadrant I: x = 60°. Quadrant IV: x = 360° - 60° = 300°.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How many times can a horizontal line cross a trig graph in one cycle?
It depends on the line's height relative to the graph's maximum and minimum. A line strictly between the minimum and maximum usually crosses a sine or cosine graph twice per cycle; a line exactly at the maximum or minimum touches it only once.
Can I solve this type of problem without drawing the graph?
Yes, the graph is a visual aid, but the underlying maths is the same as solving the trigonometric equation algebraically using reference angles and the quadrant rule. Drawing the graph simply helps you see how many solutions to expect and check your answers make sense.
What if the line does not touch the graph at all?
This happens when the line's y-value lies outside the graph's range, for example, a line above the maximum value or below the minimum value of the function. In that case, the equation has no solution within 0° to 360°, so state clearly that there is no intersection.