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

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In a walking route in Brisbane, two paths form a right-angled triangle with a 62-degree angle at the starting point. The path opposite the 62-degree angle is 24 metres long.

Using the constancy of the sine ratio for a given angle in similar right-angled triangles, determine the exact value of sin(62°) 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.

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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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Explanation

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Worked solution

  1. A right-angled triangle has a 62-degree angle at the starting point. The side opposite this angle is 24 metres long. 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. The sine ratio is constant for a given angle across all similar right-angled triangles, regardless of size. This constancy allows us to use the given side lengths to determine sin(62°) exactly.
  3. Let the hypotenuse be denoted as h. By definition, sin(62°) = opposite / hypotenuse = 24 / h.
  4. To find sin(62°) as a simplified ratio, we need to determine h. However, since no other side length is given, we cannot compute h numerically from the given data alone.
  5. The question asks for sin(62°) as a ratio of opposite to hypotenuse, based on the given side lengths. Since only the opposite side (24 m) is known, and the hypotenuse is unknown, we cannot form a numerical fraction without additional information.
  6. The sine ratio for 62° cannot be expressed as a simplified fraction or exact radical expression using only the given side length (24 m) and the definition of sine, because the hypotenuse is not provided.
  7. Therefore, it is mathematically impossible to determine the exact value of sin(62°) as a simplified fraction or radical expression from the given information alone.
  8. The question requests an exact value based on given side lengths, but without the hypotenuse or another side, no exact ratio can be formed.
  9. Since the hypotenuse is not given, and no other side or angle is provided, the sine ratio cannot be evaluated to a simplified fraction or radical expression.
  10. The problem cannot be solved as stated because the hypotenuse is not known, and no other side or angle is given to calculate it.
  11. Thus, no exact simplified fraction or radical expression for sin(62°) can be determined from the given data.
  12. The correct response is that the exact value cannot be determined from the given information.

Reference answer: Cannot be determined from given information

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