9 Trigonometry
- Syllabus
- 2016
- Topic
- 9
- Level
- —
Learning objectives
9A Sine, cosine and tangentUse of sine, cosine and tangent of angles up to 180° Angles will be measured in degrees and decimals of a degree9B 2D and 3D trigonometrySolution of problems in two and three dimensions by calculation and by drawing Use of the sine and cosine rule Area of a triangle = ¹⁄₂ab sin C Questions on latitude and longitude will not be set Calculations of the angle between two planes, or of the angle between a straight line and a plane will not be set9C Elevation and depressionAngles of elevation and depression Angles will be given in degrees and decimals of a degree9D BearingsBearings The normal convention of bearings being measured clockwise will be adopted
Choose sine, cosine and tangent
In a right-angled triangle, name the sides relative to the marked angle: opposite is across from it, adjacent touches it but is not the hypotenuse, and the hypotenuse is opposite the right angle.
| Known or wanted sides | Ratio |
|---|---|
| opposite and hypotenuse | sinθ=HO |
| adjacent and hypotenuse | cosθ=HA |
| opposite and adjacent | tanθ=AO |
Mark the right angle and target angle, label O/A/H, choose the ratio containing the known and unknown quantities, substitute, solve, then round only at the end. Use inverse sine, cosine or tangent when the angle is unknown.
\theta=\sin^{-1}!\left(\frac{O}{H}\right),\quad\cos^{-1}!\left(\frac{A}{H}\right),\quad\tan^{-1}!\left(\frac{O}{A}\right)
Use degree mode and give angles in degrees or decimals of a degree. These right-triangle ratios choose sides relative to the target angle; do not reuse O and A labels unchanged when the target angle changes. The syllabus considers angles up to 180°.
Solve non-right triangles and 3D routes
| Information pattern | Tool |
|---|---|
| opposite side-angle pair plus another side or angle | sine rule |
| three sides, or two sides and included angle | cosine rule |
| two sides and included angle, area wanted | 21absinC |
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},\qquad c^2=a^2+b^2-2ab\cos C
\text{Area}=\frac12ab\sin C
For a 3D problem, redraw only the triangle that contains the requested length or angle. Establish its sides from earlier right-triangle, Pythagoras, sine-rule or cosine-rule calculations, keep unrounded values, then solve the final triangle.
Match each side with its opposite angle. For the sine rule, an unknown angle may have a second solution 180°−heta; test it against the diagram, side ordering and angle sum. In the cosine rule, isolate the cosine before applying cos−1.
Problems may be solved by calculation or accurate drawing. Latitude and longitude will not be set, nor will direct calculations of the angle between two planes or between a line and a plane. Do not treat a perspective sketch as a scale drawing.
Model elevation and depression
An angle of elevation is measured upward from an observer's horizontal line of sight. An angle of depression is measured downward from that horizontal.
Draw a horizontal through the observer, a vertical height, and the line of sight. Mark the angle at the observer—not at the object—and use the right angle between horizontal and vertical to expose a right triangle.
\text{horizontal sight lines are parallel};\Rightarrow;\text{alternate elevation/depression angles are equal}
Translate the context into a right triangle, include any observer height or different ground levels, choose sine/cosine/tangent from the labelled sides, and report the requested height or distance with units.
Angles are in degrees or decimals of a degree. Depression is not measured from the vertical, and a person's eye height cannot be ignored when the question distinguishes it from ground level.
Read and calculate three-figure bearings
A bearing is measured clockwise from north at the starting point and written with three figures, such as 047°, 120° or 305°.
Draw a north line at every relevant point. Put the protractor centre at the journey's start, begin at north, turn clockwise, then draw the route. The bearing of B from A starts at A; the bearing of A from B starts at B.
\text{reverse bearing}=\begin{cases}b+180°,&b<180°\b-180°,&b\ge180°\end{cases}
Use parallel north lines to transfer angles into the route triangle, then apply angle facts, the sine rule or cosine rule as appropriate. Convert the final direction back to a clockwise angle from north and pad it to three figures.
Do not measure anticlockwise, from east, or from the arrival point unless that is the named start. A triangle's interior angle is not automatically the bearing; relate it explicitly to a north line first.