Pythagoras and right-angled trigonometry

Explanation

Read the right triangle before choosing a method

A right-angled triangle contains one 90° angle. The side opposite that angle is the hypotenuse; it is always the longest side. The other two sides are perpendicular legs. In triangle ABC with angle B = 90°, AC is the hypotenuse, while AB and BC are the legs. Naming the right angle first prevents the most common labelling error.

When one acute angle is selected, the two legs receive relative names. The leg touching the selected angle is adjacent, unless it is the hypotenuse; the leg across from the angle is opposite. These labels can change when the selected angle changes, but the hypotenuse never changes. Write the labels beside the stated lengths before selecting Pythagoras or a trigonometric ratio.

Explanation

Exact values, rounding and right-angle checks

If the squared length is not a perfect square, an exact answer can be left as a square root, such as √74 cm. Give a decimal only when requested or when a practical context needs one. Keep the full calculator value during later steps and round only the final result; early rounding can move a final angle or length outside the required accuracy.

To test three positive side lengths, first identify the longest as c. Compare c² with the sum a² + b² for the other two sides. Equality confirms a right angle opposite c; inequality means the three lengths are not the sides of a right-angled triangle. This check does not identify every property of a non-right triangle, so do not extend it into non-right-angled trigonometry.

Explanation

Sine, cosine and tangent compare side lengths

For an acute angle θ in a right-angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. A compact memory aid is SOH-CAH-TOA, but the words matter more than the initials. Label the sides relative to θ before writing a ratio.

Choose the ratio containing the known side and the required side. If the opposite side and hypotenuse are involved, use sine; if adjacent and hypotenuse are involved, use cosine; if opposite and adjacent are involved, use tangent. Calculator angle mode must be degrees for this syllabus context.

Explanation

Inverse trigonometry finds an acute angle

When two sides are known and an acute angle is required, first form the appropriate side ratio and then use the matching inverse function. For example, if opposite/adjacent = 3/4, then θ = tan⁻¹(3/4). The superscript -1 here means inverse function, not reciprocal: tan⁻¹ is not 1/tan.

Use degree mode and normally give an angle answer to one decimal place, following the syllabus instruction. A right triangle's two non-right angles add to 90°, so an acute answer must lie strictly between 0° and 90°. Keep the side ratio unrounded when entering the inverse function.

Explanation

Turn a two-dimensional situation into a right triangle

In an applied problem, identify a horizontal line and a vertical or explicitly perpendicular line. Their intersection supplies the right angle. State what every side represents and keep all lengths in compatible units. The shortest distance from a point to a straight line is measured along the perpendicular segment, not along a sloping route.

An angle of elevation is measured upward from an observer's horizontal line; an angle of depression is measured downward from a horizontal line at the observer. Horizontal lines at different heights are parallel, so the matching acute angle inside the right triangle can be identified. Eye height or instrument height must be added or subtracted when the question distinguishes it from ground level.

Pythagoras decision check

Use Pythagoras when a right angle is known and the information concerns side lengths only. Mark the hypotenuse first. To find it, add the squares of the legs; to find a leg, subtract the square of the known leg from the square of the hypotenuse. Take the positive square root for a geometric length.

Before accepting an answer, ask whether the hypotenuse is the longest side, whether the unit is a length rather than a squared unit, and whether the precision instruction has been followed. For coordinate points, state both coordinate changes. For a right-angle test, square the longest side and compare it with the sum of the other two squares.

Trigonometric ratio and calculator check

Label opposite, adjacent and hypotenuse relative to the named acute angle. Select the one ratio containing both the known side and the target. When finding a side, rearrange the ratio algebraically; when finding an angle, apply the matching inverse function to a ratio of two sides.

Check degree mode before evaluating. Leg lengths must be positive and shorter than the hypotenuse. Acute-angle answers must lie between 0° and 90° and should normally be written to one decimal place. Preserve unrounded calculator values through a multi-step solution and attach the requested unit only to lengths, not to ratios.

Final method, scope and common-error review

For every problem: identify the right angle, name the target, label the sides from the relevant acute angle, choose Pythagoras or one trigonometric ratio, substitute unrounded values, and check units and size. Elevation and depression begin at a horizontal line. Include eye or instrument height only when the description requires it.

This pack covers the mapped two-dimensional content of sections 6.1 and 6.2. Coordinate distance is included only as an explicit Pythagoras application. Bearings may appear as supporting direction information in a 6.2 problem, but this pack does not teach bearings as a standalone technique. It does not teach sine rule, cosine rule, area using 1/2 ab sin C, ambiguous cases, three-dimensional trigonometry, or angles between a line and a plane.

Every example and question supplies coordinates or a complete text description of the right triangle; no answer depends on a missing diagram. Consequently, this text-only pack practises calculation and interpretation but does not claim to cover skills that require constructing, measuring or reading an unstated graphical figure.