Trigonometry and Measure of Angles

Trigonometry and measure of angles is examined most years, usually as a 2–4 mark triangle or sector problem in Section II and a multiple-choice question on exact values or radians. Radian measure is used for the rest of the course, so this topic underpins all of Year 12 trigonometric calculus.

• Acute angles: exact ratios for $30^{\circ }$, $45^{\circ }$ and $60^{\circ }$; right-angled triangle problems, including angles of elevation and depression and bearings.
• Angles of any size: the definitions of sin, cos and tan using a point on a circle, related angles, the ASTC signs, and $\sin (-\theta )=-\sin \theta$ and similar results.
• Non-right-angled triangles: the sine rule (including the ambiguous case), the cosine rule and the area formula.
• Radians: converting to and from degrees, exact values for multiples of $\frac{\pi }{6}$ and $\frac{\pi }{4}$.
• Arcs and sectors: arc length, sector area, and segment area.
• Graphs of $y=\sin x$, $y=\cos x$ and $y=\tan x$, with their key features.

SOH CAH TOA gives the ratios for an acute angle. The exact values come from two triangles: half an equilateral triangle with sides $1,\sqrt{3},2$ (for $30^{\circ }$ and $60^{\circ }$) and an isosceles right triangle with sides $1,1,\sqrt{2}$ (for $45^{\circ }$).

• Angle of elevation: measured up from the horizontal. Angle of depression: measured down from the horizontal. They are equal as alternate angles.
• True bearings are measured clockwise from north and written with three digits, such as $075^{\circ }$. The reverse bearing differs by $180^{\circ }$.
• Compass bearings such as S$40^{\circ }$W start from north or south and turn towards east or west.

Always draw a diagram with north lines at every point where a bearing is given.

Take a point $(x,y)$ on a circle of radius $r$ centred at the origin, where $\theta$ is measured anticlockwise from the positive $x$-axis. Then

These work for any angle, including negative angles (measured clockwise) and angles beyond $360^{\circ }$. At multiples of $90^{\circ }$ the point lies on an axis, so for example $\sin 90^{\circ }=1$ and $\cos 180^{\circ }=-1$, while $\tan 90^{\circ }$ is undefined because $x=0$.

The signs follow from the signs of $x$ and $y$ in each quadrant (ASTC: All, Sin, Tan, Cos are positive in quadrants 1, 2, 3, 4).

The related angle is the acute angle between the ray and the $x$-axis. To find a ratio of any angle, find the ratio of the related angle and attach the sign for that quadrant.

For negative angles: $\cos (-\theta )=\cos \theta$, $\sin (-\theta )=-\sin \theta$ and $\tan (-\theta )=-\tan \theta$.

Example. Find the exact value of $\sin \left(-\frac{3\pi }{4}\right)$. The angle $-\frac{3\pi }{4}$ ends in the third quadrant, where sine is negative, and its related angle is $\frac{\pi }{4}$. So the value is $-\frac{1}{\sqrt{2}}$.

For any triangle $ABC$, with side $a$ opposite angle $A$ and so on:

• Use the sine rule when you know a side and its opposite angle, plus one more side or angle.
• Use the cosine rule when you know two sides and the included angle (find the third side), or all three sides (find an angle).
• The area formula needs two sides and the angle between them.

The syllabus also expects you to follow the proofs of these rules for a given triangle.

The ambiguous case

When you use the sine rule to find an angle, $\sin B=k$ has two solutions between $0^{\circ }$ and $180^{\circ }$: $B$ and $180^{\circ }-B$. Both are valid if the second angle still fits in the triangle (the angles sum to less than $180^{\circ }$). This can only happen when you are given two sides and an angle that is not between them, and the side opposite the given angle is shorter than the other given side. If you are asked for the larger angle in a triangle, it is opposite the longest side, so use the cosine rule for it to avoid the ambiguity.

Worked example: two possible triangles

In triangle $ABC$, $A=35^{\circ }$, $a=7$ cm and $b=11$ cm. Find the possible values of $B$, to the nearest degree.

$B=64.3^{\circ }$ or $B=180^{\circ }-64.3^{\circ }=115.7^{\circ }$.

Check the second: $35^{\circ }+115.7^{\circ }=150.7^{\circ }<180^{\circ }$, so it fits. Both triangles exist, so $B\approx 64^{\circ }$ or $B\approx 116^{\circ }$. (The third side is then 12.0 cm or 6.0 cm.)

Worked example: bearings with the cosine rule

A hiker walks 6 km from $P$ on a bearing of $040^{\circ }$ to $Q$, then 9 km on a bearing of $110^{\circ }$ to $R$. How far is $R$ from $P$?

At $Q$, the bearing back to $P$ is $220^{\circ }$. The angle between the paths at $Q$ is $220^{\circ }-110^{\circ }=110^{\circ }$.

Using the sine rule, $\angle QPR\approx 43^{\circ }$, so the bearing of $R$ from $P$ is about $040^{\circ }+43^{\circ }=083^{\circ }$.

One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. So the angle in radians is arc length divided by radius, and a full revolution is $\frac{2\pi r}{r}=2\pi$.

To convert degrees to radians, multiply by $\frac{\pi }{180}$; to convert radians to degrees, multiply by $\frac{180}{\pi }$. Set your calculator to radians whenever the angle has no degree sign.

With $\theta$ in radians:

The segment formula is sector area minus the triangle area $\frac{1}{2}{r}^2\sin \theta$. A major sector or segment uses the reflex angle $2\pi -\theta$ (or subtract the minor one from the whole circle).

Worked example: a sector and its segment

A sector has radius 8 cm and angle 1.2 radians. Find its perimeter, its area, and the area of the segment cut off by the chord.

Remember the perimeter of a sector includes both radii.

• $y=\sin x$ has zeros at $0,\pm \pi ,\pm 2\pi ,\dots$ and a maximum at $x=\frac{\pi }{2}$.
• $y=\cos x$ starts at its maximum $(0,1)$ and has zeros at $\pm \frac{\pi }{2},\pm \frac{3\pi }{2},\dots$
• $y=\tan x$ has vertical asymptotes where $\cos x=0$, at $x=\frac{\pi }{2}+k\pi$.

Transformations of these graphs (such as $y=3\sin 2x$) are Year 12 content.

• Calculator in the wrong mode: degrees for a radian question, or the reverse.
• Using $l=r\theta$ or $A=\frac{1}{2}{r}^2\theta$ with $\theta$ in degrees.
• Missing the second triangle in the ambiguous case, or including an obtuse angle that makes the angle sum exceed $180^{\circ }$.
• Using the area formula with an angle that is not between the two given sides.
• Measuring bearings anticlockwise or from east, or forgetting the three-digit format.
• Forgetting both radii in the perimeter of a sector.
• Rounding in the middle of a calculation. Keep full calculator values until the final answer.

• Find or calculate questions: write the rule, substitute, then give the answer to the requested accuracy with units. The substitution line usually earns the first mark.
• For exact value questions, show the related angle and the quadrant sign; no decimals.
• In bearings questions, a labelled diagram with north lines often earns marks on its own and prevents most errors.
• If a question says "show that" the triangle has two possible shapes, state $\sin B=k$, give both angles, and check the angle sum for the obtuse one.
• Read whether the answer is wanted in degrees and minutes, degrees to 1 d.p., or radians.

• Trigonometric identities and equations (Year 11): related angles and ASTC are the method for solving trig equations.
• Further graph transformations and modelling (Year 12): transforms the graphs of sin, cos and tan for periodic models.
• Differential and integral calculus (Year 12): the derivatives and integrals of trig functions only work in radians.
• Applications of calculus: sector areas and arc lengths turn up in optimisation and rates problems.