4.8 Trigonometry and Pythagoras’ theorem

Learning objectives

4.8A Pythagoras' theorem in two dimensionsknow, understand and use Pythagoras’ theorem in two dimensions 4.8B Trigonometry in right-angled trianglesknow, understand and use sine, cosine and tangent of acute angles to determine lengths and angles of a right-angled triangle 4.8C Two-dimensional trigonometry and bearingsApply trigonometric methods to solve two-dimensional problems, including bearings.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: $c^2=a^2+b^2$ . The hypotenuse $c$ is always opposite the right angle.

Unknown	Rearrangement
hypotenuse	$c=\sqrt{a^2+b^2}$
shorter side	$a=\sqrt{c^2-b^2}$

Mark the right angle, identify the hypotenuse, substitute lengths with units, then take the positive square root. In a compound shape, form one right triangle at a time and carry the unrounded result forward.

Do not add the squares when the unknown is a shorter side, and do not use Pythagoras unless the triangle is right-angled.

Relative to an acute angle $\theta$ , $\sin\theta=\frac{O}{H}$ , $\cos\theta=\frac{A}{H}$ and $\tan\theta=\frac{O}{A}$ . Label opposite, adjacent and hypotenuse before choosing a ratio.

Known and wanted sides	Ratio
opposite and hypotenuse	sine
adjacent and hypotenuse	cosine
opposite and adjacent	tangent

For a length, rearrange the chosen ratio. For an angle, use the matching inverse function, such as $\theta=\tan^{-1}(O/A)$ . Keep the calculator in degree mode and round only the final answer.

Adjacent means the non-hypotenuse side beside the chosen angle; its identity changes when the reference angle changes.

Translate the context into a labelled 2D diagram. Bearings are measured clockwise from north and written with three figures, so first convert the bearing information into the interior angle needed by the right triangle.

Step	Decision
1	draw north lines and known distances
2	use parallel north lines, right angles or angle sums to find the working angle
3	select Pythagoras or SOHCAHTOA
4	convert the result back to a clockwise three-figure bearing

A bearing such as $142^\circ$ describes a direction from north; an interior triangle angle such as $38^\circ$ is not automatically the final bearing.

This Foundation objective uses right-triangle decomposition. The sine rule and cosine rule for non-right triangles are taught separately in 4.8.HC.

For $90^\circ<\theta<180^\circ$ , the reference angle is $180^\circ-\theta$ . Sine stays positive, while cosine and tangent are negative.

Ratio	Obtuse-angle relationship
sine	$\sin\theta=\sin(180^\circ-\theta)$
cosine	$\cos\theta=-\cos(180^\circ-\theta)$
tangent	$\tan\theta=-\tan(180^\circ-\theta)$

Sketch the angle in the second quadrant, find its acute reference angle, apply the correct sign, and check the calculator is in degree mode.

An inverse-sine display gives a principal acute value; contextual or stated obtuse conditions may require the supplementary angle $180^\circ-\theta$ .

Angles of elevation and depression are measured from a horizontal line. Parallel horizontals make the angle of depression equal to the corresponding angle of elevation.

Information	Triangle quantity
two object heights	often subtract to obtain vertical separation
horizontal ground distance	adjacent side
line of sight	hypotenuse

Draw a horizontal through the observer, label the vertical difference and horizontal distance, then use the right-triangle ratio that connects the known sides to the required angle or length.

Do not measure the angle from the vertical, and do not use a full height when the line of sight joins points already above the ground.

For any triangle, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ and $a^2=b^2+c^2-2bc\cos A$ , where each side is paired with its opposite angle.

Given	Usually choose
an opposite side-angle pair	sine rule
three sides, or two sides and included angle	cosine rule

Label opposite pairs, choose a form with one unknown, substitute without premature rounding, and test the result against the largest-side/largest-angle relationship. For the sine-rule ambiguous case, check whether the supplementary angle also satisfies the data and angle sum.

The cosine rule uses the angle included between the two named sides. An inverse-sine answer alone can miss a valid obtuse solution.

A 3D distance can be built from right triangles on perpendicular planes. In a cuboid, the space diagonal satisfies $d^2=l^2+w^2+h^2$ .

Stage	Construction
face	find a diagonal from two perpendicular edges
space	combine that diagonal with the perpendicular third direction

Identify the two endpoints, draw or name a helpful face projection, prove the relevant angle is $90^\circ$ , then apply Pythagoras once or twice. Keep surds exact when requested.

Do not combine three lengths unless they represent mutually perpendicular directions; a sloping edge may already include more than one direction.

The area of a triangle with sides $a$ and $b$ enclosing angle $C$ is $A=\frac12ab\sin C$ . The angle must be between the two chosen sides.

Required quantity	Rearrangement
area	$A=\frac12ab\sin C$
side $a$	$a=\frac{2A}{b\sin C}$
included angle	$C=\sin^{-1}(2A/ab)$ , then check alternatives

Split compound shapes into triangles, calculate each contribution, and add or subtract as the geometry requires. Preserve full precision until the final stated accuracy.

Do not use a non-included angle with the two selected sides, and remember to double only when symmetry actually creates two congruent triangles.

The angle between a line and a plane is the angle between the line and its perpendicular projection onto that plane. This creates a right triangle containing the line, its projection and the perpendicular height.

Step	Action
1	identify where the line meets the plane
2	find the line's projection in the plane, often with Pythagoras
3	use the projection and perpendicular height in a right triangle
4	state the required line-plane angle, not a different 3D angle

A useful check is that the projection is shorter than the sloping line and the chosen angle lies between $0^\circ$ and $90^\circ$ .

The angle between a line and a plane is not the angle between two planes or between the line and an arbitrary edge in the plane.

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—4.9 Mensuration of 2D shapesRead

4.8 Trigonometry and Pythagoras’ theorem

Learning objectives

Use Pythagoras' theorem in two dimensions

Use trigonometry in right-angled triangles

Model two-dimensional trigonometry and bearings

Use trigonometric ratios for obtuse angles

Solve angles of elevation and depression

Use the sine rule and cosine rule

Use Pythagoras' theorem in three dimensions

Find triangle area using sine

Use trigonometry in three dimensions

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