Pearson Edexcel International Advanced Level Mathematics: Triangle trig, radians and trig graphs (YMA01) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover Unit P1 topics 3.1 to 3.3 (sine and cosine rules, radian measure, trigonometric graphs) of the Pearson Edexcel International Advanced Subsidiary/Advanced Level Mathematics specification, Issue 3, April 2019. P1 is compulsory for IAS Mathematics and IAL Mathematics. Learn the content first in the study guide and the revision notes. Course hub: Edexcel A Level Mathematics. See also the P1 practice questions.

Give angles in degrees to 1 decimal place and lengths to 3 significant figures unless told otherwise. Use radians where an angle is given in radians.

Questions

1. (a) Write 135° in radians, in terms of π. [1] (b) Write 5π/12 radians in degrees. [1] (c) Write down the exact value of sin(5π/6). [1]

2. In triangle PQR, PQ = 6.4 cm, PR = 9.1 cm and angle QPR = 63°.

(a) Find the length of QR. [3] (b) Find the size of angle PQR. [3]

3. In triangle ABC, angle A = 28°, BC = 7.5 cm and AC = 10 cm. Angle B can take two values.

(a) Find both possible values of angle B. [3] (b) Find the two possible lengths of AB. [3]

4. A triangle has two sides of length 12 cm and 15 cm enclosing an angle θ. Its area is 72 cm².

(a) Find the two possible values of θ. [3] (b) For the obtuse value of θ, find the length of the third side. [3]

5. A sector of a circle has radius 7.5 cm and angle 0.8 radians. Find

(a) the arc length, [1] (b) the area of the sector, [2] (c) the perimeter of the sector. [1]

6. A sector of a circle has radius 8 cm and area 40 cm².

(a) Find the angle of the sector, in radians. [2] (b) Find the perimeter of the sector. [2]

7. The points A and B lie on a circle with centre O and radius 10 cm. Angle AOB = 1.4 radians.

(a) Find the area of triangle OAB. [2] (b) Find the area of the minor segment cut off by chord AB. [3] (c) Find the perimeter of this segment. [3]

8. Sketch the graph of y = sin(x − π/3) for 0 ≤ x ≤ 2π. Give the exact coordinates of the points where the curve crosses the axes, and of its maximum and minimum points. [5]

9. The angle x is acute and cos x = p. Write each of the following in terms of p.

(a) cos(−x) [1] (b) cos(π − x) [1] (c) cos(x + 2π) [1] (d) cos(x + π) [1] (e) sin(π/2 − x) [1]

10. ABCD is a convex quadrilateral with AB = 5 cm, BC = 8 cm, CD = 7 cm, DA = 6 cm and angle ABC = 70°.

(a) Find the length of AC. [3] (b) Find the size of angle ADC. [3] (c) Find the area of ABCD. [3]

11. A sector of a circle has radius r cm and angle θ radians. Its perimeter is 20 cm and its area is 24 cm².

(a) Show that r² − 10r + 24 = 0. [3] (b) Hence find the two possible pairs of values of r and θ. [4]

12. State (a) the period of y = cos 3x, [1] (b) the period of y = tan(x/2), [1] (c) the greatest and least values of y = 4 sin x, [1] (d) the equations of the asymptotes of y = tan 2x for 0 ≤ x ≤ π. [1]

Answers

1. (a) 135 × π/180 = 3π/4 [1] (b) 5π/12 × 180/π = 75° [1] (c) sin(5π/6) = sin(π − π/6) = sin(π/6) = 1/2 [1] Examiner insight: write the conversion factor you used. For an exact value, show the symmetry step, such as sin(π − x) = sin x, instead of stating the answer alone. [3]

2. (a) QR² = 6.4² + 9.1² − 2(6.4)(9.1)cos 63° [1] = 40.96 + 82.81 − 52.88 = 70.89 [1] QR = 8.42 cm [1] (b) cos Q = (6.4² + QR² − 9.1²) / (2 × 6.4 × QR) [1] = (40.96 + 70.89 − 82.81) / 107.77 = 0.2695 [1] Angle PQR = 74.4° [1] Examiner insight: PR is the longest side, so angle PQR could be obtuse. The cosine rule settles this, whereas the sine rule would offer 74.4° and 105.6° with no way to choose. Keep the unrounded value of QR for later parts. [6]

3. (a) sin B / 10 = sin 28° / 7.5, so sin B = 10 sin 28° / 7.5 = 0.6260 [1] B = 38.8° [1] or B = 180° − 38.8° = 141.2°, which is allowed because 28° + 141.2° < 180° [1] (b) B = 38.8°: C = 113.2° and AB = 7.5 sin 113.2° / sin 28° = 14.7 cm [1][1] B = 141.2°: C = 10.8° and AB = 7.5 sin 10.8° / sin 28° = 2.98 cm [1] Examiner insight: the question states that B takes two values, so a single answer is incomplete. State why the second value is accepted by checking the angle sum, then give each length with its angle. [6]

4. (a) ½ × 12 × 15 × sin θ = 72 [1] sin θ = 72/90 = 0.8 [1] θ = 53.1° or 126.9° [1] (b) x² = 12² + 15² − 2(12)(15)cos 126.9° [1] = 369 + 216 = 585 (since cos 126.9° = −0.6) [1] x = 24.2 cm [1] Examiner insight: sin θ = 0.8 always gives a second angle, 180° − θ. Use the stated angle in the cosine rule and expect a larger side than for the acute case; a smaller value signals a sign slip. [6]

5. (a) s = 7.5 × 0.8 = 6 cm [1] (b) A = ½ × 7.5² × 0.8 [1] = 22.5 cm² [1] (c) perimeter = 2 × 7.5 + 6 = 21 cm [1] Examiner insight: state the formula before substituting. The perimeter includes both straight edges, so adding only the arc to one radius is a typical slip. [4]

6. (a) ½ × 8² × θ = 40 [1] 32θ = 40, so θ = 1.25 radians [1] (b) arc = 8 × 1.25 = 10 [1] perimeter = 16 + 10 = 26 cm [1] Examiner insight: the area formula only works with θ in radians, so a degree answer here would not fit the next part. Show the arc length as a separate step before adding the radii. [4]

7. (a) Area = ½ × 10 × 10 × sin 1.4 [1] = 49.3 cm² [1] (b) Sector = ½ × 100 × 1.4 = 70 [1] Segment = 70 − 49.27 [1] = 20.7 cm² [1] (c) arc AB = 10 × 1.4 = 14 [1] chord AB = 2 × 10 × sin 0.7 = 12.88 [1] (or by the cosine rule) perimeter = 14 + 12.88 = 26.9 cm [1] Examiner insight: a segment is a sector minus a triangle. Evaluate sin 1.4 with the angle in radians (about 0.985), not degrees. The chord is a straight line, not the arc, so do not stop at 14. [8]

8. Sine curve shifted right by π/3, period 2π, range −1 to 1 [1] y-intercept: x = 0 gives sin(−π/3) = (0, −√3/2) [1] Zeros: x − π/3 = 0 and π give (π/3, 0) and (4π/3, 0) [1] Maximum when x − π/3 = π/2: (5π/6, 1); minimum when x − π/3 = 3π/2: (11π/6, −1) [1] Right end: x = 2π gives sin(5π/3) = (2π, −√3/2), with a smooth curve through all labelled points [1] Examiner insight: a right shift moves each key point of sin x to the right by π/3; check one point, such as the zero at π/3, to catch a shift in the wrong direction. Use exact values, not decimals. [5]

9. (a) cos(−x) = p [1] (b) cos(π − x) = −p [1] (c) cos(x + 2π) = p [1] (d) cos(x + π) = −p [1] (e) sin(π/2 − x) = cos x = p [1] Examiner insight: sketch the cosine curve and read off each relationship, rather than trusting memory. Cosine is even, so the sign changes only for a shift by π or a reflection through π − x. [5]

10. (a) AC² = 5² + 8² − 2(5)(8)cos 70° [1] = 89 − 27.36 = 61.64 [1] AC = 7.85 cm [1] (b) cos D = (6² + 7² − 61.64) / (2 × 6 × 7) [1] = 23.36 / 84 = 0.2781 [1] Angle ADC = 73.9° [1] (c) Area of ABC = ½ × 5 × 8 × sin 70° = 18.79 [1] Area of ACD = ½ × 6 × 7 × sin 73.85° = 20.17 [1] Area = 39.0 cm² [1] Examiner insight: split a quadrilateral along a diagonal and keep the unrounded diagonal for later parts. Make sure each area uses the angle between the two sides you multiply. [9]

11. (a) Perimeter: 2r + rθ = 20 [1] Area: ½ r² θ = 24, so rθ = 48/r [1] 2r + 48/r = 20, so 2r² − 20r + 48 = 0, giving r² − 10r + 24 = 0 [1] (b) (r − 4)(r − 6) = 0 [1] r = 4: rθ = 12, so θ = 3 [1] r = 6: rθ = 8, so θ = 4/3 [1] Both angles are below 2π, so both sectors exist [1] Examiner insight: in a “show that” question, every line must lead to the printed equation. Do not skip the substitution of rθ. “Hence” asks you to use part (a), and the two answers should be checked against the original conditions. [7]

12. (a) Period = 2π/3 [1] (b) Period = π ÷ ½ = 2π [1] (c) Greatest value 4, least value −4 [1] (d) 2x = π/2 and 3π/2, so asymptotes x = π/4 and x = 3π/4 [1] Examiner insight: for sin or cos, the period is 2π divided by the coefficient of x; for tan, it is π divided by it. Give the equations of asymptotes as x = value, not just the numbers. [4]

Where marks are usually lost

• Using the sine rule for the largest angle without considering that it may be obtuse.
• Giving one answer for an ambiguous-case or area-to-angle question that has two.
• Putting the angle in the area formula when it is not between the two sides used.
• Calculating with the angle in degrees when the formula needs radians, or the reverse.
• Forgetting the radii in a sector perimeter, or the chord in a segment perimeter.
• Writing a segment area as ½r²θ, which is the sector.
• Rounding intermediate lengths, which can shift the last digit of the final answer.
• Sketching a shifted curve in the wrong direction, or not labelling exact values.
• Mixing up the period of sin(ax) with that of tan(ax).

Next steps

• Revise with the revision notes.
• Re-read the study guide.
• Return to the Edexcel A Level Mathematics hub or the printable checklist.
• Try the free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and Pure Mathematics, Specification, Issue 3, April 2019 (Pearson Education Limited), Unit P1 topics 3.1 to 3.3.