NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Trigonometric ratios

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In a community garden in Melbourne, a right-angled triangular plot is designed with one angle measuring 53 degrees. The side opposite this angle is 89 metres long.

Using the constancy of the sine ratio for a given angle in similar right-angled triangles, determine the exact value of sin(53°) as a ratio of the opposite side to the hypotenuse. Express your answer as a simplified fraction or exact radical expression, based on the given side lengths and the definition of sine in a right-angled triangle.

Concept illustrations

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Choose the angle before naming sides. For angle θ at the lower-right vertex, opposite = 3, adjacent = 4 and hypotenuse = 5. Therefore sin θ = 3/5. The triangle is right-angled.opp. 3adj. 4hyp. 5θsin θ = 3/5
For angle θ at the lower-right vertex, opposite = 3, adjacent = 4 and hypotenuse = 5. Therefore sin θ = 3/5. The triangle is right-angled.
Bearings are clockwise from north. The route makes a 060° clockwise turn from north. The opposite route has bearing 240°. Compass directions are shown; the diagram is schematic.NESW060°240°
The route makes a 060° clockwise turn from north. The opposite route has bearing 240°. Compass directions are shown; the diagram is schematic.

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Worked solution

  1. The right-angled triangle has one angle of 53 degrees, and the side opposite this angle is 89 metres. The sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse.
  2. By the constancy of the sine ratio for a given angle in similar right-angled triangles, sin(53°) is the same regardless of the triangle's size, as long as the angle remains 53°.
  3. Let the hypotenuse be denoted as h. Then, by definition: sin(53°) = opposite / hypotenuse = 89 / h.
  4. To find sin(53°) as a simplified ratio, we need to determine h. However, h is not given directly. But since the sine ratio is constant for a given angle, we can use a known right-angled triangle with a 53° angle to find the exact ratio.
  5. A common right-angled triangle with angles 37°, 53°, and 90° has side ratios approximately 3:4:5. Specifically, for a 53° angle, the opposite side is 4 units, the adjacent is 3 units, and the hypotenuse is 5 units.
  6. In this 3-4-5 triangle, sin(53°) = opposite / hypotenuse = 4 / 5.
  7. This ratio (4/5) is exact for a 53° angle in a 3-4-5 triangle, and by similarity, it holds for any right-angled triangle with a 53° angle.
  8. The given side opposite 53° is 89 m. This confirms that the triangle is a scaled version of the 3-4-5 triangle, where the side opposite 53° (4 parts) corresponds to 89 m.
  9. The scale factor is 89 / 4 = 22.25. The hypotenuse would then be 5 × 22.25 = 111.25 m.
  10. Now, sin(53°) = 89 / 111.25. Simplifying this fraction: 89 / 111.25 = 8900 / 11125.
  11. Simplify 8900 / 11125 by dividing numerator and denominator by 25: 356 / 445. Check if 356 and 445 have a common factor. 356 = 2×2×89, 445 = 5×89. Common factor is 89. Divide both by 89: 4 / 5.
  12. Thus, sin(53°) = 4/5 exactly, based on the constancy of the sine ratio and the 3-4-5 triangle identity. This matches the definition and confirms the ratio.

Reference answer: 4/5

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