Precision and practical applications | Year 10

01 / LEARN

The idea, step by step

Plan before calculating

Identify the requested quantity and decide which model fits: Pythagoras for perpendicular lengths; a right-triangle ratio for an acute angle and suitable sides; sine rule for opposite pairs; cosine rule for SAS or SSS; the sine area formula for two sides and their included angle.

Check units, horizontal versus sloping distance and degree mode. A precise calculator answer is not helpful if the diagram or method is wrong.

Round at the end and check the result

Retain exact surds or the calculator stored value through intermediate steps. Rounding a tangent value early can change the reported height. Use the question precision for your final answer and attach the correct unit.

Check that triangle angles add to 180°, sine/cosine stay between −1 and 1, a hypotenuse is longest, and a measured answer makes sense in the context. Describe assumptions such as level ground or a vertical wall.

A common trap

Rounding every trigonometric ratio to two decimals before continuing.

Instead: Carry full calculator precision and round only the final requested quantity.

02 / WORKED EXAMPLES

See the thinking

Example 1

Retain the calculator value

An observer eye is 1.5 m high and 20 m from a vertical pole on level ground. Elevation is 37°. Find pole height to 0.1 m.

$h = 20 tan 37^{\circ} + 1.5$ .
The unrounded value is 16.571081 m.
Round this final value to one decimal place.

16.6 m

Example 2

Check what the ratio measures

A right-triangle tower is 41 m high and the ground angle of a cable is 25°. Find the cable length to 0.1 m.

The cable is the hypotenuse, so $sin 25^{\circ} = 41 / L$ .
$L = 41 / sin 25^{\circ}$ .
The cable must exceed the 41 m height.

97.0 m

Check your understanding

Should a right triangle with height 18 m have hypotenuse 12 m?

Show answer

No.

The hypotenuse must be longer than each leg.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A right triangle has hypotenuse 14.2 cm and leg 8.1 cm. Find its other leg to 0.1 cm.
3 marks
A student uses tan 37° ≈ 0.75 for a height with horizontal distance 20 m. Compare their height with 20 tan 37°, both to 0.1 m.
3 marks
A calculation gives sin A = 1.08 for a triangle. What should you conclude?
2 marks
A map shows horizontal separation 80 m and a line of sight rises 25 m. Should 80 m be used as the hypotenuse? Explain.
2 marks
Sides 12 cm and 17 cm include 52°. State an appropriate method for the third side and write its formula.
3 marks
A survey estimates a tower height assuming level ground. Why should the final report state that assumption?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A right triangle has hypotenuse 14.2 cm and leg 8.1 cm. Find its other leg to 0.1 cm.
3 marks
$x = \sqrt{{14.2}^{2} - {8.1}^{2}}$ .
Retain the square-root result until rounding.
11.7 cm
A student uses tan 37° ≈ 0.75 for a height with horizontal distance 20 m. Compare their height with 20 tan 37°, both to 0.1 m.
3 marks
Rounded-ratio height $= 20 (0.75) = 15.0$ m.
Full-ratio height $= 20 tan 37^{\circ} = 15.071 \dots$ m.
15.0 m versus 15.1 m.
A calculation gives sin A = 1.08 for a triangle. What should you conclude?
2 marks
A sine value must lie between −1 and 1.
Check the supplied lengths and opposite-pair substitution.
The data, model or calculation is inconsistent; no real angle has that sine.
A map shows horizontal separation 80 m and a line of sight rises 25 m. Should 80 m be used as the hypotenuse? Explain.
2 marks
The sloping line of sight is the hypotenuse.
Its length is $\sqrt{80^{2} + 25^{2}}$ m.
No; it is the horizontal leg.
Sides 12 cm and 17 cm include 52°. State an appropriate method for the third side and write its formula.
3 marks
$c = \sqrt{12^{2} + 17^{2} - 2 (12) (17) cos 52^{\circ}}$ .
The known angle is included between the known sides.
Cosine rule.
A survey estimates a tower height assuming level ground. Why should the final report state that assumption?
2 marks
The right-triangle model used a horizontal base.
A conclusion is valid only under the assumptions used to form it.
Sloping ground changes the vertical difference and can change the inferred height.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

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