Revision notes

Every Trigonometry topic condensed to one page: the facts to know, the mistakes to avoid, and the formulae. Use it as a last-minute check, or print it.

Formulae for this unit

sin 30° = cos 60° (Higher) $\tfrac{1}{2}$
sin 45° = cos 45° (Higher) $\dfrac{1}{\sqrt{2}}$
sin 60° = cos 30° (Higher) $\dfrac{\sqrt{3}}{2}$
tan 30° (Higher) $\dfrac{1}{\sqrt{3}}$
tan 45° (Higher) $1$
tan 60° (Higher) $\sqrt{3}$

Right-Angled Trigonometry (SOHCAHTOA)

SOHCAHTOA works only in right-angled triangles.
sin is opposite over hypotenuse, cos is adjacent over hypotenuse, tan is opposite over adjacent.
The hypotenuse is always opposite the right angle; label opp and adj from the angle you are using.
To find an angle, use the inverse function: sin, cos or tan to the power minus 1.

Watch out: Mislabelling which side is opposite and which is adjacent relative to the given angle, especially when the triangle is drawn in an unfamiliar orientation.

Exact Trig Values

You must learn sin, cos and tan of 0, 30, 45, 60 and 90 degrees by heart, with no calculator.
sin 30 is one half, cos 60 is one half, sin 45 equals cos 45 equals 1 over root 2, and tan 45 is 1.
sin 60 and cos 30 are both root 3 over 2; tan 60 is root 3 and tan 30 is 1 over root 3.
tan 90 is undefined.

Watch out: Reaching for a calculator or a rounded decimal instead of recalling the exact fraction or surd value.

The Sine Rule

Use the sine rule when you have a matching side and opposite angle, plus one more piece of information.
a over sin A equals b over sin B equals c over sin C, with each side opposite its angle.
Flip it to sin A over a when you are finding an angle.
Finding an obtuse angle can give two possible answers (the ambiguous case).

Watch out: Mislabelling which side is opposite which angle, using a side that is adjacent to the angle rather than directly across from it.

The Cosine Rule

Use the cosine rule for three sides, or for two sides and the angle between them.
a squared equals b squared plus c squared minus 2 b c cos A to find a side.
Rearranged, cos A equals (b squared plus c squared minus a squared) over 2 b c to find an angle.
If the angle is 90 degrees, cos A is 0 and the rule reduces to Pythagoras.

Watch out: Confusing which side is being found and which two sides go into the formula, substituting the wrong side as the subject.

Area of a Triangle

Area equals one half a b sin C, where C is the angle between sides a and b.
The angle must sit between the two sides you use.
This works for any triangle, not just right-angled ones.
If you only know three sides, find an angle with the cosine rule first.

Watch out: Using two sides that do not enclose the given angle, instead of the two sides that meet at that angle.

3D Trigonometry & Pythagoras

Pick a right-angled triangle inside the 3D shape and redraw it flat as a 2D diagram.
You often need Pythagoras or trig on one triangle to get a length for the next.
The angle between a line and a plane is measured to the line shadow on that plane.
Work in stages and keep full accuracy until the final answer.

Watch out: Trying to find a cuboid diagonal in a single step without applying Pythagoras twice (or the combined 3D formula), missing one of the three dimensions.

Trigonometric Graphs

y = sin x and y = cos x wave between -1 and 1 and repeat every 360 degrees.
The cos graph is the sin graph shifted 90 degrees to the left.
y = tan x repeats every 180 degrees and has asymptotes at 90, 270 degrees and so on.
Use the symmetry of the graph to find every solution in a given range.

Watch out: Forgetting that the period of both sin x° and cos x° is 360°, and misreading how often a graph completes one full cycle.

Trigonometric Equations

A trig equation usually has more than one solution in a given interval.
Find the first solution with the inverse, then use the graph or symmetry for the rest.
sin is positive in the first and second quadrants, cos in the first and fourth, tan in the first and third.
Check that every answer lies in the range the question asks for.

Watch out: Only giving one solution within 0° to 360° when the equation actually has two valid solutions.

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