Trigonometry

Trigonometry is the second most heavily examined focus area in Standard 2: 12 marks in NESA's sample HSC paper and about 12 marks a year in recent HSC papers. Expect at least one longer bearings question that needs a diagram and two steps.

• Right-angled triangles: sin, cos and tan to find sides and angles, with angles in degrees, or degrees and minutes.
• Angles of elevation and depression.
• Sine rule to find a side, or an angle (including an obtuse angle when you're told it is obtuse).
• Cosine rule to find a side, or an angle.
• Area of a triangle using $A=\frac{1}{2}ab\sin C$.
• True bearings and compass bearings, converting between them, and solving problems with them.

Check your calculator is in degree mode before every trig question. A quick test: $\sin 30^{\circ }$ should give 0.5.

Label the sides from the angle you are using: hypotenuse (opposite the right angle, the longest side), opposite and adjacent.

To find an angle, use the inverse key (${\sin }^{-1}$, ${\cos }^{-1}$ or ${\tan }^{-1}$). To give an angle in degrees and minutes, use the DMS key (often shown as ° ' ''). There are 60 minutes in a degree, so $36.1^{\circ }=36^{\circ }{6}^{\prime }$.

Angles of elevation and depression

The angle of elevation is measured up from the horizontal to an object; the angle of depression is measured down from the horizontal. Both are measured from the horizontal, never from a vertical wall or cliff.

Example. A kite string is pegged to the ground. The kite is 35 m above the ground and 48 m horizontally from the peg. The angle of elevation of the kite from the peg is:

Example. A 6.5 m ladder makes an angle of $72^{\circ }12^{\prime }$ with the ground. It reaches $6.5\times \sin 72^{\circ }12^{\prime }=6.19$ m up the wall (2 d.p.). Enter $72^{\circ }12^{\prime }$ with the DMS key, or as $72+\frac{12}{60}$.

For any triangle $ABC$, side $a$ is opposite angle $A$, and so on. The sine rule links each side with the angle opposite it.

Use the sine rule when you know a side and the angle opposite it, plus one more side or angle. Use the first form to find a side and the second form to find an angle (the unknown on top makes the algebra easier).

Finding a side. In a triangle, $A=48^{\circ }$, $B=67^{\circ }$ and $a=21$ cm.

Worked example: an obtuse angle

In triangle $ABC$, $a=8.5$ m, $b=14$ m and $A=31^{\circ }$. Angle $B$ is obtuse. Find $B$ to the nearest degree.

The calculator gives ${\sin }^{-1}(0.8483)=58^{\circ }$ which is acute. An obtuse angle has the same sine as its supplement, so

Only do this when the question tells you the angle is obtuse (or the diagram clearly shows it). The calculator always gives the acute angle.

Use the cosine rule when you know two sides and the angle between them (to find the third side), or all three sides (to find an angle).

In the angle form, $c$ is the side opposite the angle you want. The largest angle is opposite the longest side.

Example. A triangle has sides 8 m, 11 m and 15 m. Find the largest angle.

A negative cosine means the angle is obtuse, and the calculator gives the obtuse angle directly: there is no supplement step with the cosine rule.

$a$ and $b$ are two sides, and $C$ is the angle between them (the included angle). For a triangular paddock with sides 16 m and 21 m meeting at $58^{\circ }$:

If you don't have the included angle, find it first with the sine or cosine rule.

A true bearing is measured clockwise from north, written with three digits: $065^{\circ }$T, $210^{\circ }$T. A compass bearing starts from N or S and turns towards E or W: N$25^{\circ }$E, S$40^{\circ }$W.

To go back the way you came (a back bearing), add or subtract $180^{\circ }$. If B is on a bearing of $040^{\circ }$T from A, then A is on a bearing of $220^{\circ }$T from B.

Worked example: a two-leg trip

A boat sails 15 km from A to B on a bearing of $040^{\circ }$T, then 22 km from B to C on a bearing of $110^{\circ }$T. Find the distance from A to C, and the bearing of C from A.

Step 1: draw a diagram with a north line at A and at B. The back bearing from B to A is $040+180=220^{\circ }$T. The angle at B is the gap between the directions to A and to C:

Step 2: two sides and the included angle, so use the cosine rule.

Step 3: find $\angle BAC$ with the sine rule, keeping the full value of $AC$ in your calculator.

Step 4: the bearing of C from A is the bearing of B plus this angle: $40^{\circ }+42.5^{\circ }=082.5^{\circ }$ so C is on a bearing of about $083^{\circ }$T from A, 30.6 km away.

Radial surveys

In a radial survey, distances and bearings are measured from one central point to each corner of a block of land. Each pair of neighbouring corners makes a triangle with the centre: find the angle between them by subtracting bearings, then use the cosine rule for a boundary length and $\frac{1}{2}ab\sin C$ for the area.

• Calculator in radians. Check with $\sin 30^{\circ }=0.5$.
• Using the cosine rule with an angle that isn't between the two known sides.
• Mixing up which side is opposite which angle in the sine rule. Label the triangle first.
• Forgetting the $180^{\circ }$ supplement for an obtuse angle found with the sine rule.
• Rounding a length in Step 2 and using the rounded value in Step 3. Store the full value.
• Measuring a bearing anticlockwise, or from east. True bearings go clockwise from north.

• Draw and label a diagram for every bearings question, even if one is given. Mark north at each point.
• Write the rule you are using before substituting; a correct substitution usually earns a mark.
• Give angles to the stated accuracy (nearest degree or minute) and lengths with units.
• A show that question gives the answer: show every step so the marker can see you didn't work backwards.
• If a question has several parts, an answer from part (a) is often needed in part (b). Use your unrounded value.

• Applications of measurement (Year 11): Pythagoras' theorem, perimeter and area of composite shapes.
• Time and location (Year 11): directions and positions on the Earth.
• Ratios and rates: scale drawings and land areas from radial surveys and site plans.