- 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.
- 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.
- 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.
- 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 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.
- 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.
- 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.
- 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.
