Trigonometric Ratios in Right Triangles
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Trigonometric Ratios in Right Triangles — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionDefining Sine Cosine and TangentAssessment Practice
2 marks~3 minCriterion D
A surveyor stands 40 m from the base of a telecommunications tower on level ground. Using a clinometer, she measures the angle of elevation to the top of the tower as 62°.
a
Explain how the tangent ratio is used to calculate the height of the tower, and state the height correct to one decimal place. [1]
b
Discuss one limitation of this method that affects the accuracy of the calculated height, and advise the surveyor whether she can confidently report that the tower meets a minimum height requirement of 75 m. [1]
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Solutions

2QuestionDefining Sine Cosine and TangentAssessment Practice
4 marks~6 minCriterion D
A surveyor estimates the height of a vertical cliff. She stands 50 m from the base along flat horizontal ground and measures the angle of elevation to the cliff top as 30°30°.
a
Calculate the height of the cliff using the tangent ratio. [1]
b
Explain why the tangent ratio is more practical than the sine ratio for this measurement. [1]
c
The angle measurement has an uncertainty of ±2°\pm 2°. Calculate the range of possible heights, then advise the surveyor whether her single estimate of 28.9 m is sufficiently reliable to report without qualification. [2]
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Solutions

3QuestionDefining Sine Cosine and TangentAssessment Practice
4 marks~6 minCriterion B
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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Solutions

4QuestionUsing SOH-CAH-TOA to Find Missing SidesAssessment Practice
6 marks~9 minCriterion A
A surveyor needs to find the height hh of a vertical cliff on the opposite bank of a river. From point A, the angle of elevation to the cliff top is 35°35°. She walks 50 m50\ \text{m} directly away from the cliff to point B, where the angle of elevation is 25°25°. Both points are at the same horizontal level as the base of the cliff.
a
Let xx be the horizontal distance from A to the base of the cliff. Construct two equations using tangent ratios and solve the system to find hh, rounded to the nearest metre. [3]
b
The surveyor's instrument has a precision of ±1°\pm 1°. Deduce the value of hh when a +1°+1° error is applied to the angle at A (using 36°36° instead of 35°35°), rounded to the nearest metre. [1]
c
The surveyor must report this height to a construction team planning to build a bridge. Advise the surveyor whether the trigonometric model alone is sufficient for this purpose, considering the assumptions made and the sensitivity of hh to measurement error shown in part (b). [2]
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Solutions

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