P1.5 Trigonometry: triangles, radians and graphs

How do you use the sine rule?

Label the triangle with capital letters for the angles and the matching small letter for the side opposite each angle. Then, in any triangle,

• Use the first form to find a side and the second form to find an angle, so that the unknown sits on top.
• You need one complete "pair": a side and the angle opposite it.
• The sine rule is not in the formulae list, so learn it; you will use it constantly.

Why it works: drop a perpendicular from \(C\) to \(AB\). Its height is \(b\sin A\) from one side and \(a\sin B\) from the other, so \(b\sin A=a\sin B\). In a right-angled triangle the rule reduces to sin \(=\) opposite \(\div\) hypotenuse, because \(\sin90°=1\).

When does the sine rule give two triangles?

\(\sin\theta=\sin(180°-\theta)\), so an equation such as \(\sin R=0.82\) has an acute solution and an obtuse one. The calculator only shows the acute one, and you must decide whether the obtuse one also works.

• The obtuse angle is valid whenever it still leaves room in the triangle: (given angle) \(+\) (obtuse angle) \(<180°\).
• 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 the shorter of the two given sides.
• If the side opposite the given angle is the longer one, the obtuse option always fails the angle sum and there is one triangle.

How do you use the cosine rule and the area formula?

• First form: two sides and the angle between them give the third side.
• Second form: three sides give any angle. A negative cosine means an obtuse angle, and there is never any ambiguity, because \(\cos^{-1}\) gives a unique angle between 0° and 180°.
• The cosine rule is in the formulae list in its side form; be ready to rearrange it for \(\cos A\).

When \(A=90°\), \(\cos A=0\) and the rule becomes Pythagoras' theorem: the term \(-2bc\cos A\) is the correction for a non-right angle.

How do you solve a bearings problem?

• A bearing is measured clockwise from north and written with three figures: 072°, 195°.
• Draw a north line at every point you measure from.
• The bearing of \(A\) from \(B\) and the bearing of \(B\) from \(A\) differ by exactly 180° (the back bearing).
• North lines are parallel, so use alternate and co-interior angles to find the angles inside the triangle, then use the sine or cosine rule.

What is a radian?

• A full turn is an arc of \(2\pi r\), which is \(2\pi\) radii, so \(2\pi\text{ rad}=360°\) and \(\pi\text{ rad}=180°\).
• Degrees → radians: multiply by \(\dfrac{\pi}{180}\). Radians → degrees: multiply by \(\dfrac{180}{\pi}\).
• \(1\text{ rad}=\dfrac{180°}{\pi}\approx57.3°\).
• "rad" is often left out: an angle written without the ° symbol is in radians.
• Set your calculator to radian mode whenever the angles are in radians, including inside the sine and cosine rules.

How do you find arc lengths, sectors and segments?

With \(\theta\) in radians, \(r\) the radius, \(s\) the arc length and \(A\) the sector area:

Neither is in the formulae list, so learn both. They come from fractions of a circle: the sector is \(\dfrac{\theta}{2\pi}\) of the whole, so \(s=\dfrac{\theta}{2\pi}\times2\pi r\) and \(A=\dfrac{\theta}{2\pi}\times\pi r^2\). In degrees you would need extra factors of \(\dfrac{\pi}{180}\).

Segments

• A segment is the region between a chord and its arc.
• Segment \(=\) sector \(-\) triangle: \(\tfrac12r^2\theta-\tfrac12r^2\sin\theta=\tfrac12r^2(\theta-\sin\theta)\).
• The triangle is isosceles with two sides \(r\) and the included angle \(\theta\), so its area is \(\tfrac12r^2\sin\theta\) (with sin of an angle in radians).
• The perimeter of a segment is the chord plus the arc.

What do the graphs of sin, cos and tan look like?

Exact values to know

Using symmetry: from the graph, \(\sin\tfrac{5\pi}{6}=\sin\left(\pi-\tfrac{\pi}{6}\right)=\tfrac12\) and \(\cos\tfrac{4\pi}{3}=-\cos\tfrac{\pi}{3}=-\tfrac12\). The curves repeat, so \(\sin(x+2\pi)=\sin x\). The number of solutions of an equation like \(\sin x=k\) in an interval is the number of times the line \(y=k\) crosses the curve there.

How do simple transformations change a trig graph?

• Changes outside the function act on \(y\) as you expect; changes inside act on \(x\) the "opposite" way: \(+\tfrac{\pi}{6}\) moves left, \(\times2\) squashes.
• For \(y=\sin bx\) and \(y=\cos bx\) the period is \(\dfrac{2\pi}{b}\); for \(y=\tan bx\) it is \(\dfrac{\pi}{b}\).
• To find new key points, solve "inside \(=\) old key value": the maximum of \(\sin\left(x+\tfrac{\pi}{6}\right)\) is where \(x+\tfrac{\pi}{6}=\tfrac{\pi}{2}\), so \(x=\tfrac{\pi}{3}\).

Exam tips for P1.5

Common misconceptions

How this topic connects

P2 Pure Mathematics 2

The graphs and symmetries here are what you use to find every solution of a trigonometric equation in an interval, and the identities \(\tan x=\dfrac{\sin x}{\cos x}\) and \(\sin^2x+\cos^2x=1\) build on the same unit-circle ideas.

P1.3 Transformations, P3 and P4

The four transformation rules of P1.3 are exactly the ones used on trig graphs here. Radians are essential in calculus: the derivative of \(\sin x\) is \(\cos x\) only when \(x\) is in radians, and combined transformations such as \(R\sin(x+\alpha)\) extend the last section.

M1 Mechanics and IAL Physics

Resolving forces, finding resultants and velocity triangles use the sine and cosine rules and bearings-style directions. Angular displacement and angular speed in circular motion are measured in radians, and oscillations follow sine and cosine graphs.

Key formulae

Check your understanding

Specification checklist

Tick every box before the exam. The wording is ours; the numbers refer to the P1 specification sections.

• (P1 3.1) I can use the sine rule to find a side or an angle, and I can find both triangles when the data are ambiguous.
• (P1 3.1) I can use the cosine rule to find a side from two sides and the included angle, or an angle from three sides.
• (P1 3.1) I can find the area of any triangle using \(\tfrac12ab\sin C\).
• (P1 3.1) I can turn a bearings description into a triangle and solve it with the sine and cosine rules.
• (P1 3.2) I can convert between degrees and radians, and I know the radian values of common angles exactly.
• (P1 3.2) I can use arc length \(r\theta\) and sector area \(\tfrac12r^2\theta\), and find the areas and perimeters of segments and composite shapes.
• (P1 3.3) I can sketch \(y=\sin x\), \(y=\cos x\) and \(y=\tan x\), and use their periods and symmetries.
• (P1 3.3) I can sketch simple transformations such as \(y=3\sin x\), \(y=\sin\left(x+\tfrac{\pi}{6}\right)\) and \(y=\sin2x\), labelling intercepts, maxima and minima.