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

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In a sports equipment storage facility in Perth, a right-angled triangular shelf is designed to hold basketballs. The shelf has a base of 84 cm and a height of 27 cm, forming a right angle at the base.

Using the constancy of the tangent ratio for the angle between the base and the hypotenuse in similar right-angled triangles, determine the exact value of tan(θ) for this angle. Express your answer as a simplified fraction.

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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

Worked solution and answer (click to reveal)

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Worked solution

  1. Identify the given elements of the right-angled triangle: base = 84 cm, height = 27 cm, and the right angle is between them.
  2. The angle θ is defined as the angle between the base and the hypotenuse. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side.
  3. For angle θ, the side opposite to θ is the height (27 cm), and the side adjacent to θ is the base (84 cm).
  4. Apply the definition of tangent: tan(θ) = opposite / adjacent = 27 / 84.
  5. Simplify the fraction 27/84 by dividing both numerator and denominator by their greatest common divisor (GCD). The GCD of 27 and 84 is 3.
  6. Divide 27 by 3 to get 9, and divide 84 by 3 to get 28. So, 27/84 = 9/28.
  7. The simplified fraction 9/28 is in lowest terms because 9 and 28 share no common factors other than 1.
  8. The constancy of the tangent ratio in similar right-angled triangles means that for any triangle with the same angle θ, the ratio of opposite to adjacent will always be 9/28.
  9. Therefore, tan(θ) = 9/28.
  10. Verify the result by checking that 9/28 cannot be simplified further: 9 = 3², 28 = 4×7, no common prime factors.
  11. The value of tan(θ) for the angle between the base and hypotenuse is exactly 9/28.
  12. The answer is confirmed as a simplified fraction, consistent with the constancy of the tangent ratio in similar right-angled triangles.

Reference answer: 9/28

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