AC9M10M03 · YEAR 10 · MEASUREMENT

Pythagoras and Trigonometry

ACARA v9 CONTENT DESCRIPTION “solve practical problems applying Pythagoras’ theorem and trigonometry of right-angled triangles, including problems involving direction and angles of elevation and depression”
Builds on: Pythagoras and Trigonometry in Right-Angled Triangles. Measuring lengths and areas of figures prepares the way for this unit, which turns to right-angled triangles: first the theorem of Pythagoras relating their sides, then trigonometry, which links the sides to the angles and lets us find a missing length or angle.

Pythagoras and the right-angled triangle

The right-angled triangle is one of the most useful shapes in all of measurement, and its first great property is the theorem of Pythagoras. It says that in any right-angled triangle, the square built on the longest side, the hypotenuse opposite the right angle, has an area equal to the two squares built on the other sides added together. Written with letters, a squared plus b squared equals c squared, where c is the hypotenuse. With legs of 3 and 4, the squares are 9 and 16, and their sum, 25, is the square on the hypotenuse, which is therefore 5. This relationship lets you find the third side of a right-angled triangle whenever the other two are known, a tool used constantly in building, navigation and design.

Pythagoras: squares on the sides
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
A right-angled triangle has shorter sides 3 and 4. Reveal the squares on each side to see the relationship Pythagoras discovered.

Calculating a side with Pythagoras

Using the theorem is direct. To find the hypotenuse, square each of the two shorter sides, add the results, and take the square root. For legs of 6 and 8, the hypotenuse squared is 36 plus 64, which is 100, so the hypotenuse is the square root of 100, exactly 10. The method also works in reverse to find a shorter side: if the hypotenuse and one leg are known, you subtract the square of that leg from the square of the hypotenuse before taking the root. Because the formula involves squares and a square root, an answer is often an exact surd rather than a whole number, and knowing when to leave it exact and when to round is part of using appropriate units and accuracy.

Finding the hypotenuse
Square the legs, add, and take the square root to find the hypotenuse of a right-angled triangle.
To find the hypotenuse, square the two legs, add them, and take the square root. For legs 6 and 8, c squared is 36 plus 64, which is 100, so c is the square root of 100, exactly 10. The same method finds a shorter side by subtracting instead of adding.

Trigonometry: linking sides to angles

Pythagoras connects the three sides, but it says nothing about the angles. Trigonometry fills that gap through three ratios of the sides, each tied to one of the acute angles. Labelling the sides relative to an angle as opposite, adjacent, and hypotenuse, the sine of the angle is opposite over hypotenuse, the cosine is adjacent over hypotenuse, and the tangent is opposite over adjacent. The memory phrase SOH CAH TOA captures all three. The remarkable fact that makes trigonometry work is that these ratios depend only on the angle, not on the size of the triangle: every right-angled triangle with a given angle has the same sine, cosine and tangent for it, so the ratios can be tabulated and used for any such triangle.

The three trigonometric ratios
Sine, cosine and tangent relate an angle to ratios of the sides: SOH CAH TOA.
Sine of an angle is the opposite side over the hypotenuse. The ratio depends only on the angle, not the triangle size, which is what makes trigonometry work.

Finding an unknown side

The trigonometric ratios let you find a missing side when an angle and one side are known. The trick is to choose the ratio that connects what you know to what you want. Suppose a right-angled triangle has a 30 degree angle and a hypotenuse of 10, and you need the side opposite the angle. Sine links opposite and hypotenuse, so sine 30 degrees equals the opposite over 10, which rearranges to opposite equals 10 times sine 30 degrees. Since sine 30 degrees is exactly 0.5, the opposite side is 5. Choosing sine, cosine or tangent according to which sides are involved, then rearranging, solves a vast range of practical problems, from the height of a tree to the slope of a ramp.

Solving for an unknown side
Choose the ratio that links the known angle and side to the unknown side, then solve.
To find a side, pick the ratio linking the known angle and side to the unknown. With a 30 degree angle and hypotenuse 10, sine connects them: the opposite side is 10 times sine 30 degrees, and since sine 30 degrees is exactly 0.5, the opposite side is 5.

Finding an unknown angle

Trigonometry also runs the other way: when two sides are known, it can recover the angle between or beside them. Here the inverse ratios are needed, written as inverse sine, inverse cosine and inverse tangent. If the side opposite an angle is 3 and the adjacent side is 4, then the tangent of the angle is 3 over 4, or 0.75, and the angle itself is the inverse tangent of 0.75, which is about 36.9 degrees. The inverse ratio simply asks the reverse question: not what is the ratio for this angle, but what angle gives this ratio. Together, Pythagoras for sides and trigonometry for the link between sides and angles make the right-angled triangle a complete and powerful measuring instrument.

Solving for an unknown angle
When two sides are known, an inverse trigonometric ratio recovers the angle.
This triangle has opposite 3 and adjacent 4, but the angle is unknown. Reveal how the inverse tangent recovers the angle from the ratio of the sides.

Angles of elevation and depression

Most real uses of trigonometry start from a horizontal line of sight. The angle of elevation is the angle you look up from the horizontal to see something higher, such as the top of a tree or a tower; the angle of depression is the angle you look down from the horizontal to something lower, such as a boat seen from a clifftop. Both are measured from the horizontal, and because the horizontal at your eye and the horizontal at the object are parallel, an angle of elevation up to an object equals the angle of depression back down from it.

These problems become right-angled triangles. To find the height of a tree when you stand 20 metres away and the angle of elevation to its top is 35 degrees, the height is the opposite side and the 20 metres is the adjacent side, so tan 35 degrees equals height over 20, giving a height of 20 times tan 35 degrees, about 14 metres. The same reasoning with an angle of depression finds how far away a boat is from the foot of a cliff of known height, and the same right-triangle thinking handles direction and navigation problems. Choosing the ratio that links the side you know to the side you want is the whole skill.

Quick self-check
1. Pythagoras’ theorem for a right-angled triangle states that:
2. A right-angled triangle has legs 6 and 8. Its hypotenuse is:
3. In a right-angled triangle, sine of an angle equals:
4. With a 30° angle and hypotenuse 10, the side opposite the angle is:
5. If two sides of a right-angled triangle are known, the unknown angle is found using:
6. You stand 20 m from a tree and the angle of elevation to its top is 35 degrees. Its height is...
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