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Trigonometry

Trigonometry — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionWorking with Two Sides and Included Angle SASConcept Practice
2 marks~3 minCriterion A
A surveyor measures two boundary lines from a fixed point AA: one line AB=5 cmAB = 5 \text{ cm} and another AC=7 cmAC = 7 \text{ cm}, with the angle between them ∠BAC=40°\angle BAC = 40°.

Calculate the length of BCBC. [2]
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2QuestionMulti-Step Navigation ProblemsConcept Practice
2 marks~3 minCriterion A
A coast guard vessel is at point PP and a lighthouse is at point QQ. The diagram shows a north arrow at PP and the line PQPQ.

The angle measured clockwise from north to line PQPQ is 47°47°.
a
Deduce the three-figure bearing of QQ from PP. [1]
b
The coast guard must sail on a bearing between 040°040° and 055°055° to stay within a safe navigation corridor. Justify whether the vessel is heading within the safe corridor. [1]
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3QuestionSketching Graphs of y = sin x cos x tan xConcept Practice
2 marks~3 minCriterion B
The diagram shows the graph of y=tan⁡xy = \tan x for −π<x<π-\pi < x < \pi, with asymptotes at x=−π2x = -\frac{\pi}{2} and x=π2x = \frac{\pi}{2} clearly marked. Below it is an incomplete sketch of y=tan⁡(2x)y = \tan(2x) for the same domain, with only one asymptote drawn at x=−π4x = -\frac{\pi}{4}. Outline the pattern in the spacing of the asymptotes as the coefficient inside the tangent function changes. Use this pattern to predict the x-coordinate of the next asymptote for y=tan⁡(2x)y = \tan(2x) to the right of the origin. State a general rule linking the coefficient kk in y=tan⁡(kx)y = \tan(kx) to the distance between consecutive asymptotes.
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4QuestionSketching Graphs of y = sin x cos x tan xConcept Practice
2 marks~3 minCriterion A
A buoy bobs in the ocean. Its vertical displacement, dd metres, from its rest position is modelled by d=sin⁡xd = \sin x, where xx is measured in degrees and 0°≤x≤360°0° \leq x \leq 360° represents one tidal cycle.
a
Construct a sketch of y=sin⁡xy = \sin x for −360°≤x≤360°-360° \leq x \leq 360°, plotting key points at multiples of 90°90° and drawing a smooth curve through them. [1]
b
The buoy is considered "active" when d>0d > 0. Interpret how many degrees of each 360°360° cycle the buoy spends above its rest position, and explain what this reveals about the symmetry of the model. [1]
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5QuestionUsing SOH-CAH-TOA to Find Missing SidesConcept Practice
4 marks~6 minCriterion A
A solar panel installation company assesses whether a roof section is suitable for fitting panels. The slant length of the roof is 15 m and the angle between the slant and the horizontal is 40°.
a
Write down the trigonometric ratio that links the adjacent side, the hypotenuse, and the angle in a right triangle. [1]
b
Calculate the horizontal length of the roof section. [2]
c
The installer states: "A horizontal span greater than 11 m confirms the roof is wide enough for the panel." Advise the installer whether the roof section should be approved for panel installation, justifying your answer with reference to your result from part (b). [1]
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6QuestionFinding the Area of Any Triangle Using TrigonometryConcept Practice
2 marks~3 minCriterion B
Triangle ABCABC has sides a=BCa = BC and b=ACb = AC, with a perpendicular height hh drawn from vertex BB to side ACAC. The angle at vertex CC is denoted CC.

Show that the area of triangle ABCABC can be written as

Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C

by starting from Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} and substituting h=asin⁡Ch = a\sin C. [2]
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7QuestionFinding the Area of Any Triangle Using TrigonometryConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a triangular garden plot. Two boundary fences meet at a corner, with lengths AB=10AB = 10 m and AC=7AC = 7 m enclosing an angle of 42°42° at vertex AA.
a
Calculate the area of the triangular plot. [1]
b
Turf costs 18 dollars per m². The architect has a budget of 2 900 dollars. Advise the architect whether the budget is sufficient to turf the entire plot, justifying your answer with a cost calculation. [1]
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8QuestionSin and Cos rules based problemsConcept Practice
2 marks~3 minCriterion A
A surveyor measures a triangular plot of land, ABCABC. Angle A=35°A = 35°, angle B=72°B = 72°, and side a=8.2a = 8.2 cm (opposite angle AA).

Calculate the length of side bb, opposite angle BB, using the sine rule:

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

Give your answer in centimetres, correct to 3 significant figures. [2]
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9QuestionWorking with Two Sides and Included Angle SASAssessment Practice
5 marks~8 minCriterion B
In triangle ABCABC, let BC=aBC = a, AC=bAC = b, AB=cAB = c, and let ∠ACB=C\angle ACB = C.
a
Construct a perpendicular from vertex AA to side BCBC, meeting BCBC at point DD. Label CD=xCD = x, DB=a−xDB = a - x, and height AD=hAD = h. Using right triangle ACDACD, write expressions for xx and hh in terms of bb and CC. [2]
b
Apply the Pythagorean theorem to right triangle ABDABD and substitute your expressions from part (a). Expand and simplify fully. [2]
c
Hence prove that c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C, justifying the final algebraic step that reduces the expression. [1]
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10QuestionWorking with Two Sides and Included Angle SASAssessment Practice
4 marks~6 minCriterion D
A surveyor is planning a triangular plot of land with two fixed boundary sides a=8a = 8 m and b=11b = 11 m and an adjustable included angle CC. The cosine rule states:

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
a
Calculate the length of side cc when C=60°C = 60° and when C=120°C = 120°. [2]
b
Show that when C=90°C = 90°, the cosine rule reduces to Pythagoras' theorem, and calculate cc. [1]
c
The surveyor needs the third boundary cc to be at least 15 m to meet a planning regulation. Advise the surveyor whether C=90°C = 90° is sufficient, and justify the minimum whole-number angle (in degrees) that should be used. [1]
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11QuestionWorking with Two Sides and Included Angle SASAssessment Practice
6 marks~9 minCriterion C
A surveyor measures two sides of a triangular plot as a=85a = 85 m and b=62b = 62 m, with an included angle of C=47°C = 47°. The cosine rule states:

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
a
Calculate the length of the third side, cc, correct to one decimal place. [2]
b
The surveyor's measurements have possible errors of ±0.5\pm 0.5 m in each side length and ±1°\pm 1° in the angle. Analyse which of these three measurement errors has the greatest impact on the calculated value of cc, and justify your reasoning mathematically. [3]
c
The surveyor's client requires the third side to be shorter than 65 m to meet a planning regulation. Advise the surveyor whether the measurement errors identified in part (b) affect the reliability of the conclusion drawn from part (a). [1]
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12QuestionMulti-Step Navigation ProblemsAssessment Practice
12 marks~18 minCriterion B
A small boat completes two-leg voyages. The captain records the bearing and distance of each leg.

Voyage 1: Leg 1: 045°045°, 1010 km; Leg 2: 135°135°, 1010 km; Final displacement: 14.114.1 km at 090°090°
Voyage 2: Leg 1: 060°060°, 1515 km; Leg 2: 120°120°, 1515 km; Final displacement: 26.026.0 km at 090°090°
Voyage 3: Leg 1: 030°030°, 88 km; Leg 2: 150°150°, 88 km; Final displacement: 8.08.0 km at 090°090°
Voyage 4: Leg 1: 000°000°, 55 km; Leg 2: 180°180°, 55 km; Final displacement: 00 km
a
Calculate the sum of the two bearings for each voyage. Deduce what the four sums have in common. [2]
b
For Voyages 1–3, the two leg distances are equal and the bearing sums equal 180°180°. Analyse the final displacement directions and formulate a general rule predicting the direction of the final displacement whenever two legs of equal distance have bearings summing to 180°180°. [4]
c
A navigator plans Voyage 6: Leg 1: 225°225°, 1212 km; Leg 2: 135°135°, 1212 km. Using vector components, calculate the actual final displacement (magnitude and direction). Advise the navigator whether the rule from part (b) can be used to plan this voyage, justifying your advice with reference to both the bearing sum and the calculated result. [6]

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13QuestionSolving Bearings Questions Using TrigonometryAssessment Practice
6 marks~9 minCriterion D
A search-and-rescue helicopter uses bearings from two coastal stations, A and B, which are 12 km apart along a straight coastline. Station A reports a distress signal at a bearing of 055°055°, and station B reports it at a bearing of 320°320°. The helicopter crew triangulates the signal position, assuming a flat Earth and that bearings are accurate to within ±1°\pm 1°.
a
Calculate the distance from station A to the distress signal. Show all working. [2]
b
Explain how the bearing uncertainty of ±1°\pm 1° affects the size of the search zone and how the flat-Earth assumption may limit the accuracy of this model. [2]
c
Justify whether bearing-based triangulation provides a sufficiently reliable method for directing the helicopter to the distress signal in this scenario. [2]
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14QuestionUnderstanding Bearings as Directional AnglesAssessment Practice
6 marks~9 minCriterion C
A port is located at the origin (0,0)(0, 0) and a lighthouse is at coordinates (3,4)(3, 4) km, where the positive xx-axis points east and the positive yy-axis points north. A student proposes the following model to calculate the bearing of the lighthouse from the port:

θ=tan⁡−1 ⁣(xy),B=90°−θ\theta = \tan^{-1}\!\left(\frac{x}{y}\right), \qquad B = 90° - \theta
a
Calculate the bearing BB predicted by the student's model. [2]
b
Construct the correct bearing using a clearly labelled diagram and trigonometric reasoning. [2]
c
The lighthouse is the destination of a vessel that must follow a bearing accurate to within ±2°\pm 2° to avoid shallow waters. Advise the navigator whether the student's model should be used for this voyage, identifying the source of any error in the model. [2]
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15QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice
7 marks~11 minCriterion D
A Ferris wheel has a diameter of 20 m. Its lowest point is 2 m above the ground. The height hh (in metres) of a passenger above the ground after tt seconds is modelled by

h(t)=10sin⁡(0.5t)+12,h(t) = 10\sin(0.5t) + 12,

where the wheel rotates at constant speed.
a
Calculate the height of the passenger after 3 seconds. Give your answer to one decimal place. [2]
b
Deduce the first time, to one decimal place, at which the passenger reaches a height of 17 m. [2]
c
The Ferris wheel operator uses this model to plan safe stopping positions. Advise the operator whether this model alone is sufficient for that purpose, referring to its assumptions and to real-world conditions. [3]
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16QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice
3 marks~5 minCriterion C
A sound engineer models the pressure wave of a pure musical tone using y=sin⁡xy = \sin x, where xx represents time in degrees over one complete cycle (0°≤x≤360°0° \leq x \leq 360°).

The function passes through (0°,0)(0°, 0), (90°,1)(90°, 1), (180°,0)(180°, 0), (270°,−1)(270°, -1), and (360°,0)(360°, 0).
a
Calculate sin⁡45°\sin 45°, giving your answer in exact form. [1]
b
Deduce the values of sin⁡135°\sin 135° and sin⁡225°\sin 225°, using symmetry properties of the sine function. [1]
c
The engineer states: "The pressure wave is perfectly balanced — positive and negative peaks are equal in magnitude throughout the cycle." Using your results from (a) and (b), and the value sin⁡315°\sin 315°, justify whether this statement is correct. [1]
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17QuestionUsing Trig Ratios to Find Missing AnglesAssessment Practice
12 marks~18 minCriterion B
A surveyor uses a drone to photograph a hillside. The drone's camera records the horizontal distance (adjacent side) and vertical rise (opposite side) for three sight lines from a fixed base point.

Sight line 1opposite = 1 madjacent = 2 m
Sight line 2opposite = 2 madjacent = 3 m
Sight line 3opposite = 3 madjacent = 4 m


The angle of elevation θ\theta is measured from the horizontal at the base point.
a
Calculate θ1\theta_1, θ2\theta_2, and θ3\theta_3, the angles of elevation for sight lines 1, 2, and 3. [3]
b
Deduce the angle of elevation θ4\theta_4 for a fourth sight line where opposite = 4 m and adjacent = 5 m, and justify whether the angles are increasing at a constant rate. [4]
c
The surveyor needs an angle of elevation of exactly 45° to calibrate the camera correctly. Advise the surveyor whether this calibration angle can ever be achieved using the pattern in this sequence, and recommend what alternative measurement setup would be required. [5]
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18QuestionDefining Sine Cosine and TangentAssessment Practice
4 marks~6 minCriterion C
A surveyor uses complementary angles when measuring elevation from two positions. The graph of y=sin⁡θy = \sin \theta is shown for 0∘≤θ≤90∘0^\circ \leq \theta \leq 90^\circ.
a
State the value of sin⁡30∘\sin 30^\circ and the value of cos⁡60∘\cos 60^\circ. [1]
b
Explain why cos⁡(90∘−θ)=sin⁡θ\cos(90^\circ - \theta) = \sin \theta for all θ\theta in a right triangle, referring to the sides of the triangle. [2]
c
Advise the surveyor whether the sine of an elevation angle and the cosine of its complementary angle can always be used interchangeably in the field, justifying your answer. [1]
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19QuestionSolving Word Problems with Right-Angled TrianglesAssessment Practice
8 marks~12 minCriterion D
A surveyor measures the angle of elevation to five structures from a point 50 m away horizontally.

ObjectTreeBuildingFlagpoleCliffTower
Height (m)15.028.035.042.050.0
Angle of elevation (θ\theta)16.7°29.2°35.0°40.0°45.0°
a
Calculate tan⁡θ\tan\theta for each object. Present your results as labelled rows matching the format above, rounding to 3 decimal places. [2]
b
Analyse the relationship between an object's height and its corresponding value of tan⁡θ\tan\theta, using evidence from your table. [2]
c
Deduce a general formula for the height hh of an object in terms of tan⁡θ\tan\theta and horizontal distance dd, then apply it to find the height of a lighthouse observed at an angle of elevation of 53.1° from the same point. [2]
d
A safety regulation states that any structure taller than 65 m within 50 m of a public path requires a warning beacon. Advise the safety inspector whether the lighthouse requires a warning beacon, justifying your decision with reference to the precision of your calculation. [2]
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20QuestionProblem Solving with Non-Right TrianglesAssessment Practice
7 marks~11 minCriterion D
A surveyor records the following measurements for a triangular plot of land: side a=120 ma = 120\text{ m} (opposite angle AA), side b=150 mb = 150\text{ m} (opposite angle BB), and angle A=35°A = 35°. The measurements carry known tolerances: distances ±2%\pm 2\%, angles ±1°\pm 1°.
a
Deduce how many distinct triangles can be formed with these measurements, justifying your reasoning using the sine rule. [2]
b
Calculate the area of each possible triangle using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C. [2]
c
Advise the surveyor whether the area estimate is sufficient for legal land registration, taking into account the effect of measurement tolerances on both the number of possible triangles and the range of possible areas. [3]
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21QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
4 marks~6 minCriterion C
A landscape architect is designing a triangular garden bed with two fixed side lengths a=9.2a = 9.2 cm and b=12.5b = 12.5 cm (on a scale drawing). The graph below shows how the area of the triangle varies with the included angle CC, using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C.
a
Show that the area formula simplifies to Area=57.5sin⁡C\text{Area} = 57.5\sin C. [1]
b
Explain how the shape of the graph shows that the maximum area occurs at C=90°C = 90°. [2]
c
The architect claims a right-angled triangular bed gives the most efficient use of the fixed side lengths. Calculate the maximum area correct to 3 significant figures, and advise the architect whether this right-angle design should be adopted to maximise the garden bed area. [1]
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22QuestionSin and Cos rules based problemsAssessment Practice
2 marks~3 minCriterion D
A surveying team needs to find the straight-line distance across a river to a landmark CC on the opposite bank. Direct measurement is impossible.

The team stands at two points, AA and BB, on the same bank, measuring a baseline AB=84 mAB = 84\ \text{m}. They record the angle to the landmark from each point: ∠CAB=71°\angle CAB = 71° and ∠CBA=63°\angle CBA = 63°.

Apply the Sine Rule to find the distance ACAC, then advise the team on whether a rope of length 110 m110\ \text{m} is sufficient to reach the landmark from point AA. [2]
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23QuestionSin and Cos rules based problemsAssessment Practice
5 marks~8 minCriterion B
A surveying team is mapping a triangular plot of land, ABCABC. Their instruments record side a=8a = 8 cm, side b=10b = 10 cm, and angle A=35°A = 35°.
a
Deduce how many distinct triangles can be formed with these conditions, showing your use of the sine rule. [2]
b
For each possible triangle, calculate angle BB and side cc. [2]
c
The surveying team must fence only the plot with the larger perimeter. Advise the team which triangle to select, justifying your answer with a comparison of both perimeters. [1]

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24QuestionSin and Cos rules based problemsAssessment Practice
8 marks~12 minCriterion D
Two search-and-rescue stations, A and B, are positioned 50 km apart along a straight coastline. Station A records the bearing of a distress signal as 30°30° from the coastline; station B records it as 50°50° from the coastline. Each bearing has an uncertainty of ±1°\pm 1° due to equipment limitations.
a
Show that the distance from station A to the signal is approximately 38.9 km, using the Law of Sines with exact bearings of 30°30° and 50°50°. [2]
b
Deduce the maximum possible range of distances from station A to the signal when the worst-case bearing errors of ±1°\pm 1° are applied simultaneously at both stations. [2]
c
A rescue coordinator must decide whether two-station triangulation is reliable enough to direct a rescue vessel in rough seas. Advise the coordinator whether this method should be used alone or combined with supplementary positioning, using your results from parts (a) and (b) and at least two real-world limitations of the model. [4]
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