5 Trigonometry — 9MA0 spec coverage

5.1

Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form ½ab sin C; work with radian measure, including use for arc length and area of sector.

For a unit-circle angle $\theta$ , $\cos\theta$ and $\sin\theta$ are the point's horizontal and vertical coordinates, while $\tan\theta=\sin\theta/\cos\theta$ where $\cos\theta\neq0$ ; this extends the ratios beyond acute angles.
For a triangle, use $a/\sin A=b/\sin B=c/\sin C$ when an opposite side-angle pair is known, $a^2=b^2+c^2-2bc\cos A$ for three sides or an included angle, and $\tfrac12ab\sin C$ for area.
Radian measure makes arc and sector formulae direct: an angle $\theta$ radians in a circle of radius $r$ gives arc length $s=r\theta$ and sector area $A=\tfrac12r^2\theta$ .
Keep the calculator in the required angle mode and label sides opposite their matching angles; using degrees in $s=r\theta$ or pairing the wrong side and angle is a common error.

Tier 1 · Easy

2 marks

ORIGINAL

A circular arc has radius $7.5\,\text{cm}$ and subtends $1.2$ radians at the centre. Find its length.

Tier 2 · Standard

4 marks

ORIGINAL

Two sides of a triangular sail are $8\,\text{m}$ and $11\,\text{m}$ , with included angle $0.9$ radians. Calculate the third side and the area of the sail, giving each answer to $3$ significant figures.

Tier 3 · Hard

5 marks

ORIGINAL

A minor segment is cut from a circle of radius $6\,\text{cm}$ by a chord whose endpoints subtend $1.4$ radians at the centre. Determine the perimeter and area of the segment, giving both to $3$ significant figures.

5.2

Understand and use the standard small angle approximations of sine, cosine and tangent: sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ.

For $|\theta|$ close to zero and measured in radians, $\sin\theta\approx\theta$ , $\tan\theta\approx\theta$ and $\cos\theta\approx1-\theta^2/2$ .
Replace each trigonometric function by its stated approximation, simplify algebraically, and retain only a solution whose magnitude is small enough for the approximation to be credible.
For example, $1-\cos\theta\approx\theta^2/2$ , so $(1-\cos\theta)/\theta^2\approx1/2$ for a small non-zero $\theta$ .
The approximations are radian results, not degree results; another common error is to accept a large root created by the approximate polynomial.

Tier 1 · Easy

1 mark

ORIGINAL

Use a standard small-angle approximation to estimate $\sin(0.064)$ .

Tier 2 · Standard

2 marks

ORIGINAL

Without using a calculator's trigonometric keys, estimate $\dfrac{1-\cos(0.08)}{(0.08)^2}$ by a small-angle approximation.

Tier 3 · Hard

4 marks

ORIGINAL

A small positive angle $x$ satisfies $\sin x+\cos x=1.08$ . Use the standard small-angle approximations to estimate $x$ , giving $4$ decimal places, and explain which algebraic root is admissible.

5.3

Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity; know and use exact values of sin, cos and tan for standard angles and their multiples.

$\sin x$ and $\cos x$ have range $[-1,1]$ and period $2\pi$ , while $\tan x$ has range $\mathbb{R}$ , period $\pi$ and vertical asymptotes at $x=\pi/2+k\pi$ .
Use $\sin(-x)=-\sin x$ , $\cos(-x)=\cos x$ and $\tan(-x)=-\tan x$ , then reduce an angle by a whole period before using its reference angle and quadrant.
The exact first-quadrant values come from the $45^\circ$ and $30^\circ$ - $60^\circ$ triangles; quadrant signs then determine standard-angle values around the unit circle.
Do not read the period from the amplitude: in $a\cos(bx)+c$ , the amplitude is $|a|$ , the period is $2\pi/|b|$ , and $c$ moves the midline.

Tier 1 · Easy

2 marks

ORIGINAL

Find the exact value of $\tan(5\pi/6)$ .

Tier 2 · Standard

4 marks

ORIGINAL

For $y=3\cos(2x)-1$ , state the amplitude, period, maximum value and minimum value.

Tier 3 · Hard

4 marks

ORIGINAL

Evaluate exactly $\sin(-11\pi/6)$ , $\cos(13\pi/3)$ and $\tan(7\pi/4)$ .

5.4

Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains.

The reciprocal functions are $\sec x=1/\cos x$ , $\cosec x=1/\sin x$ and $\cot x=\cos x/\sin x$ ; secant and cosecant have range $(-\infty,-1]\cup[1,\infty)$ and period $2\pi$ , while cotangent has range $\mathbb{R}$ and period $\pi$ .
The principal ranges are $-\pi/2\leq\arcsin x\leq\pi/2$ , $0\leq\arccos x\leq\pi$ and $-\pi/2<\arctan x<\pi/2$ ; $\arcsin x$ and $\arccos x$ require $-1\leq x\leq1$ .
To determine a composite inverse-trigonometric domain, first restrict its input to the inverse function's domain; for example, $\arccos(2x-1)$ requires $-1\leq2x-1\leq1$ .
The notation $\sin^{-1}x$ means the inverse function $\arcsin x$ , not the reciprocal $1/\sin x$ ; the reciprocal is $\cosec x$ .

Tier 1 · Easy

1 mark

ORIGINAL

Given $\cos\theta=-4/5$ , write down $\sec\theta$ .

Tier 2 · Standard

2 marks

ORIGINAL

Give the principal values, in radians, of $\arcsin(-1/2)$ and $\arctan(-1)$ .

Tier 3 · Hard

5 marks

ORIGINAL

For $y=2\sec x-1$ , state the period and range, then give all vertical asymptotes in $-\pi\leq x\leq\pi$ .

5.5

Understand and use tan θ = sin θ / cos θ; understand and use sin²θ + cos²θ = 1, sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.

The quotient identity is $\tan\theta=\sin\theta/\cos\theta$ , and the three Pythagorean identities are $\sin^2\theta+\cos^2\theta=1$ , $\sec^2\theta=1+\tan^2\theta$ and $\cosec^2\theta=1+\cot^2\theta$ .
Choose an identity containing the known and required functions, rearrange it, and use the stated quadrant to select the correct sign after taking a square root.
If $\tan\theta=-7/24$ in quadrant II, a reference triangle has side magnitudes $7$ , $24$ and $25$ , so $\sin\theta=7/25$ and $\cos\theta=-24/25$ .
From $\sin^2\theta+\cos^2\theta=1$ one obtains two possible signs; ignoring the quadrant or silently choosing the positive root can change every reciprocal ratio that follows.

Tier 1 · Easy

3 marks

ORIGINAL

An acute angle $\theta$ satisfies $\sin\theta=5/13$ . Find $\tan\theta$ and $\sec\theta$ exactly.

Tier 2 · Standard

4 marks

ORIGINAL

Given that $\tan\theta=-7/24$ and $\pi/2<\theta<\pi$ , determine $\sin\theta$ , $\cos\theta$ and $\cosec\theta$ .

Tier 3 · Hard

5 marks

ORIGINAL

An angle $\theta$ lies in the first quadrant and satisfies $\sec\theta+\tan\theta=5$ . Find $\sin\theta$ exactly.

5.6

Understand and use double angle formulae; formulae for sin(A ± B), cos(A ± B), tan(A ± B) with geometrical proofs; express a cos θ + b sin θ in the form r cos(θ ± α) or r sin(θ ± α).

The compound-angle formulae are $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$ , $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$ and $\tan(A\pm B)=(\tan A\pm\tan B)/(1\mp\tan A\tan B)$ ; a geometrical proof can calculate the same unit-circle chord by coordinates and by the cosine rule.
Setting $A=B=\theta$ gives $\sin2\theta=2\sin\theta\cos\theta$ , $\cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1$ and $\tan2\theta=2\tan\theta/(1-\tan^2\theta)$ .
To write $a\cos\theta+b\sin\theta=R\cos(\theta-\alpha)$ , compare coefficients to obtain $R\cos\alpha=a$ , $R\sin\alpha=b$ , hence $R=\sqrt{a^2+b^2}$ with the quadrant of $\alpha$ set by the signs.
For a compound-angle expression, the sign in the cosine formula reverses; for an $R$ -form, expanding the proposed form before choosing $\alpha$ prevents a wrong sign.

Tier 1 · Easy

3 marks

ORIGINAL

Use a compound-angle formula to find the exact value of $\sin(75^\circ)$ .

Tier 2 · Standard

5 marks

ORIGINAL

Express $5\cos\theta-12\sin\theta$ as $R\cos(\theta+\alpha)$ , where $R>0$ and $0<\alpha<\pi/2$ . Hence state its maximum and minimum values.

Tier 3 · Hard

5 marks

ORIGINAL

Use two points on the unit circle and the cosine rule to prove geometrically that $\cos(A-B)=\cos A\cos B+\sin A\sin B$ .

5.7

Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.

Solve first for the trigonometric ratio, use a reference angle and the signs in each quadrant, then list only solutions in the stated interval.
For a quadratic in one trigonometric function, substitute a temporary variable, factorise or use the quadratic formula, and reject any ratio outside its possible range before solving each remaining branch.
When the equation involves $kx$ , transform the given interval for $x$ into the corresponding interval for $kx$ , find every solution there, and divide only at the end.
Inverse-trigonometric buttons return a principal value rather than the full solution set; endpoints, excluded endpoints and degree-versus-radian mode must all be checked explicitly.

Tier 1 · Easy

3 marks

ORIGINAL

Solve $\sin\theta=0.4$ for $0\leq\theta\leq2\pi$ , giving solutions to $3$ decimal places.

Tier 2 · Standard

4 marks

ORIGINAL

Determine all $x$ satisfying $2\cos^2x-3\cos x+1=0$ for $0\leq x<2\pi$ .

Tier 3 · Hard

5 marks

ORIGINAL

Find every solution of $\tan(2x)=-\sqrt3$ in the interval $-\pi/2\leq x\leq\pi$ .

5.8

Construct proofs involving trigonometric functions and identities.

A trigonometric identity is true for every value in its domain, so a proof transforms one side into the other using exact algebra and established identities rather than testing selected angles.
Usually begin with the more complicated side, replace secant, cosecant, cotangent or tangent by sine and cosine when helpful, and factor or take a common denominator before cancelling.
Multiplying numerator and denominator by a conjugate can expose $1-\sin^2x=\cos^2x$ or $1-\cos^2x=\sin^2x$ and complete the proof cleanly.
Never cancel terms across addition, and record domain restrictions: algebra such as division by $\sin x$ is valid only where that denominator is non-zero.

Tier 1 · Easy

2 marks

ORIGINAL

Prove that $\sin x\cot x=\cos x$ wherever the left-hand side is defined.

Tier 2 · Standard

3 marks

ORIGINAL

Prove that $\dfrac{1-\cos(2x)}{\sin(2x)}=\tan x$ for values at which both sides are defined.

Tier 3 · Hard

5 marks

ORIGINAL

Prove that $\dfrac{1}{1-\sin x}-\dfrac{1}{1+\sin x}=2\tan x\sec x$ wherever the expressions exist.

5.9

Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

Resolve a vector of magnitude $V$ at angle $\theta$ to the positive horizontal into components $V\cos\theta$ and $V\sin\theta$ , changing signs to match its actual direction.
For resultant or equilibrium problems, form separate equations in two perpendicular directions; a zero resultant requires both component sums to equal zero.
In a kinematics model, trigonometric functions can describe direction or periodic motion, and the mathematical solution must be interpreted using the stated time interval, units and physical constraints.
A calculator angle without a quadrant check can point in the opposite direction; draw and label a diagram, then state bearings or directions in the form the context requests.

Tier 1 · Easy

3 marks

ORIGINAL

A drone travels at $12\,\text{m s}^{-1}$ on a path $35^\circ$ above the horizontal. Calculate its horizontal and vertical velocity components to $3$ significant figures.

Tier 2 · Standard

5 marks

ORIGINAL

Two horizontal forces act on a crate: $8\,\text{N}$ due east and $11\,\text{N}$ at $60^\circ$ north of east. Find the magnitude and direction of their resultant, to $3$ significant figures and the nearest degree respectively.

Tier 3 · Hard

5 marks

ORIGINAL

A ring is in equilibrium under its weight of $18\,\text{N}$ , a tension $P$ directed $25^\circ$ above the horizontal, and a horizontal tension $Q$ acting oppositely. Calculate $P$ and $Q$ to $3$ significant figures.