SPM Mathematics: How to Master Trigonometry (Ratios and Graphs)

Beyond 90 degrees, sine, cosine and tangent can be negative, and which are positive depends on the quadrant; the familiar All, Sine, Tangent, Cosine pattern around the four quadrants keeps this straight. Find the basic acute angle first, then attach the correct sign from the quadrant.

Getting the sign wrong turns a correct value into a lost mark.

The sine and cosine graphs are waves between minus 1 and 1; the cosine curve is the sine curve shifted, and the tangent graph repeats more sharply with breaks. For a sketch, mark the key points where each curve crosses zero and reaches its maximum and minimum before joining them smoothly.

Memorising the basic shapes means you can draw them even when nerves hit.

An equation like sin 2x = 0.5, solved for x between 0° and 360°, needs one extra step compared to a plain sin x equation, and skipping it is a common way to lose solutions. Because the equation is written in terms of 2x rather than x, first work out the range for 2x itself: if x runs from 0° to 360°, then 2x runs from 0° to 720°, doubling the range you need to search.

Solve the equation for 2x across this doubled range first, finding every solution using the basic angle and the sign pattern in each quadrant as usual, and only divide each solution by 2 at the very last step to get the values of x. Missing this range-doubling step is the single most common reason students find only two solutions for a multiple-angle equation when the question actually has three or four within the required range for x.

Writing out the working range for the multiple angle as an explicit first line, "for 2x, 0° ≤ 2x ≤ 720°" before solving anything makes this step visible and much harder to forget.

When a sine or cosine graph is written as y = a sin bx or y = a cos bx, the two numbers a and b tell you exactly how the basic graph has been stretched, without needing to plot a single point first. The number a is the amplitude, it sets the maximum value of y to a and the minimum to −a, stretching the graph vertically compared to the basic graph, which normally has a maximum of 1 and a minimum of −1.

The number b controls how quickly the graph repeats: a larger b squeezes the graph horizontally, so it completes a full cycle in a shorter distance along the x-axis, while a smaller b stretches the cycle out over a wider range. Before sketching, it helps to write both numbers down separately and state in words what each one changes, "a stretches the height, b squeezes the width" since mixing up which number controls which direction is an easy way to sketch a graph that has the right general shape but the wrong maximum value or the wrong number of complete waves across the given range.