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IMAT Mathematics · Formula sheet

Trigonometry formulas

6 formulas, 1 reference table and 8 common traps for IMAT Mathematics Trigonometry, grouped by subtopic.

Full notes with worked examples

Angles, Ratios and Exact Values

Learn this subtopic in the notes

Measuring angles in radians and degrees

Converting angles

θrad=θdeg×π180s=rθ\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \qquad s = r\theta
  • θ\thetathe angle
  • ssarc length, with θ\theta in radians
  • rrradius

Sine, cosine and tangent in a right triangle

SOH CAH TOA

sin⁡α=opphypcos⁡α=adjhyptan⁡α=oppadj\sin\alpha = \frac{\text{opp}}{\text{hyp}} \qquad \cos\alpha = \frac{\text{adj}}{\text{hyp}} \qquad \tan\alpha = \frac{\text{opp}}{\text{adj}}

The unit circle: any angle, signs in four quadrants and sin² + cos² = 1

The two basic identities

sin⁡2θ+cos⁡2θ=1tan⁡θ=sin⁡θcos⁡θ\sin^2\theta + \cos^2\theta = 1 \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Exact values of sine, cosine and tangent at the standard angles

Anglesincostan
0∘0^\circ (0)010
30∘30^\circ (π/6)(\pi/6)12\dfrac{1}{2}32\dfrac{\sqrt{3}}{2}33\dfrac{\sqrt{3}}{3}
45∘45^\circ (π/4)(\pi/4)22\dfrac{\sqrt{2}}{2}22\dfrac{\sqrt{2}}{2}1
60∘60^\circ (π/3)(\pi/3)32\dfrac{\sqrt{3}}{2}12\dfrac{1}{2}3\sqrt{3}
90∘90^\circ (π/2)(\pi/2)10Not defined
tan⁡90∘\tan 90^\circ does not exist, because cos⁡90∘=0\cos 90^\circ = 0.
From the triangles with sides 1,3,21, \sqrt{3}, 2 and 1,1,21, 1, \sqrt{2}.

Common traps

Pi radians is 180 degrees, not 360

A full turn is 2π2\pi, so π\pi alone is half a turn, 180∘180^\circ. Using 360∘360^\circ for π\pi doubles every converted angle.

Opposite goes with sine, adjacent with cosine

The leg facing the angle is hypotenuse times sine; the leg touching the angle is hypotenuse times cosine. Swapping them is the most common wrong option, so label the triangle from the given angle before writing anything.

sin 30° is one half; cos 30° is root 3 over 2

The two values swap between 30° and 60°. Remember that sine grows with the angle: the smaller angle has the smaller sine, so sin⁡30∘=12\sin 30^\circ = \tfrac{1}{2} and sin⁡60∘=32\sin 60^\circ = \tfrac{\sqrt{3}}{2}.

The identity gives cos only up to a sign

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 gives cos⁡θ=±1−sin⁡2θ\cos\theta = \pm\sqrt{1 - \sin^2\theta}. The quadrant decides the sign, and the option with the wrong sign is always offered.

Graphs, Equations and the Sine and Cosine Rules

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Graphs of sine, cosine and tangent: period and amplitude

Period and amplitude

y=asin⁡(bx)+c:amplitude ∣a∣,period 360∘by = a\sin(bx) + c: \quad \text{amplitude } |a|, \quad \text{period } \frac{360^\circ}{b}

Solving trigonometric equations in a given range

General solutions

sin⁡x=k: x=α, 180∘−αcos⁡x=k: x=±αtan⁡x=k: x=α(+whole periods)\sin x = k: \ x = \alpha,\ 180^\circ - \alpha \qquad \cos x = k: \ x = \pm\alpha \qquad \tan x = k: \ x = \alpha \quad (+\text{whole periods})

The sine rule and the cosine rule for any triangle

Sine rule, cosine rule and area

asin⁡A=bsin⁡B=csin⁡Cc2=a2+b2−2abcos⁡CArea=12absin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \qquad c^2 = a^2 + b^2 - 2ab\cos C \qquad \text{Area} = \tfrac{1}{2}ab\sin C

Common traps

The period of sin(bx) is 360° divided by b

A larger bb means faster repetition and a shorter period: sin⁡(3x)\sin(3x) has period 120∘120^\circ, not 1080∘1080^\circ. The amplitude is ∣a∣|a|, a positive number, even when aa is negative.

Solve for the whole angle first

In cos⁡(2x)=k\cos(2x) = k the angle 2x2x runs over twice the range of xx. Solving for xx directly over one turn misses half the solutions; solve for 2x2x over the doubled range, then halve.

Throw away values of sin x outside minus 1 to 1

A quadratic in sin⁡x\sin x can have a root such as 2 or −3-3. It gives no angles at all, so do not count it when you count solutions.

The cosine rule uses the angle between the two sides

In c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C, the angle CC is the one between sides aa and bb, facing side cc. Using any other angle gives a wrong side, and forgetting the minus sign turns an obtuse angle into an acute one.

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