Trigonometry questions in Additional Math include solving equations in a given range, using identities, sketching graphs and their periods, and triangle problems. The reports show that students find one answer easily and then lose marks on the others. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Missing solutions in the range. The most repeated problem. Students gave only the base angle, found one further solution by subtracting the base angle from π (and not the full set), gave three solutions where four were needed (−2.03 was commonly omitted), used only the positive square root, or did not consider negative values. Reports advise candidates to learn techniques for finding all the solutions in a range.
- Order of operations. Not dealing correctly with the order of operations when solving an equation such as sin(2x − π/3) = ½, and mixing degrees with radians. In another question, students found the angle for φ/3 and forgot to multiply by 3 at the end.
- The compound angle. Omitting the angle −4.59, or miscalculating a compound angle (−π + 0.2526... as −4.09, for example).
- Rounding. Rounding errors were common: harshly rounded working values for 3x + 1, ignoring the instruction on the front page to round to 3 significant figures, and incorrectly rounding a correct answer (0.853 was the most common).
- Periods and graphs. Not knowing that tan θ has a period of π (answers of 4π for the period of 3 tan 2θ rather than 2π), taking the period of the tangent function as 360° instead of 180°, and errors relating the period of sin(bx + c) to b. In another question, a wavelength of 180° was used for the sine function in place of the required factor of 2, which gave incorrect stretching.
- Quadratics in sin or cos. Stating the correct quadratic and making no attempt to solve it, or being unable to solve it because it does not factorise. In one question, students wrote 2 sin θ + sin² θ = 1, then sin θ (sin θ + 2) = 1, and then sin θ = 1 or sin θ = −1, which is not valid. A small number used an incorrect form of the quadratic formula.
- "Hence" and the previous part. Working with the unsimplified form of the given equation, not realising that part (a) is to be used in part (b), and not noticing that a result such as 2 sec² 3φ = −8 shows an error has been made.
- Rearranging. Losing one of the sines or using poor bracketing, which led to incorrect cancelling.
- Triangles. Omitting brackets in the sine rule that were not recovered later, using the wrong angle (BAC instead of ACB), using the wrong form for the area, and arithmetic errors when expanding in the cosine rule.
- Wrong coefficients. In one question, the coefficient of sin 3x was given as 6.
- Equations that should have alerted you. Candidates who used a value of k that made cos 2x negative should have gone back to their part (a) work to find errors.
What examiners advise
- Find all the solutions by working out the base angle, the other angles in a cycle, and then adding or subtracting a period to cover the range, including negative angles.
- Follow the order of operations when you solve an equation with a compound angle: solve for the whole bracket first, then rearrange.
- Keep intermediate steps to at least 4 significant figures, and round only at the end.
- Check the calculator mode at the start of each question involving angles.
- Look for the link between parts, and use "hence" as an instruction.
- Know the periods of sin, cos and tan, and how a change in the coefficient of x changes them.
Habits that help
- For an equation in a given range, write the range first, then change it for the compound angle.
- List all solutions in order, from the smallest to the largest, then count them.
- Check each final answer by substituting it back into the equation.
Questions students ask
Why do I keep missing solutions to trigonometric equations?
It is the most repeated problem. Students gave only the base angle, or subtracted the base angle from π and found no further solutions. Work out the base angle, the other angles in a cycle, then add or subtract a period to cover the range, including negative angles. List solutions in order and count them.
How do I solve sin(2x − π/3) = ½?
Solve for the whole bracket first, find all the values in the adjusted range, then rearrange for x. Reports record order-of-operations errors and mixing degrees with radians.
What is the period of tan θ, and how does the coefficient of x change a period?
tan θ has a period of π (180°). Students took it as 360°, and made errors relating the period of sin(bx + c) to b. Know the periods of sin, cos and tan and how a change in the coefficient of x changes them.
What do I do when an equation becomes a quadratic in sin or cos?
Solve it. Some students stated the correct quadratic and made no attempt to solve it, or could not solve it because it does not factorise.
Why must I check my calculator mode?
Because answers differ in degrees and radians. It is one of the most repeated reminders in the reports. Check at the start of each question involving angles.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.