ACT Mathematics — right-triangle trigonometry, functions and sequences

The last stretch of the ACT mathematics section is where the questions stop being arithmetic and start being notation: the three trigonometric ratios in a right triangle, function notation, what restricts a domain, and sequences. This material covers that ground. Arithmetic, algebra, coordinate geometry and plane geometry are handled by the other materials in this category.

Twelve questions. Three on the ratios themselves, where the whole difficulty is deciding which side is opposite the named angle and which is beside it — and where the missing side often has to be found with the Pythagorean theorem before any ratio can be written down. Two on the special right triangles, including one that requires working out which angle each side faces before the 1 : the square root of 3 : 2 pattern can be applied at all; getting that backwards is the commonest way to turn a solvable question into a wrong answer. Two on function notation, one evaluating at a negative number and one working backwards from an output to its input. Two on domains, one where a denominator may not be zero and one where a square root may not take a negative — and in that second one the boundary value itself is allowed, which is precisely where the two closest options part company. Two on sequences, one geometric and one arithmetic asked far enough along that the answer turns on counting steps rather than terms. And one on reading a rule stated in words.

Every triangle is described completely in words, naming the points and saying which angle is the right angle, and each of the five triangle questions also shows a labeled drawing. Every wrong option is a real quantity in the same problem: asked for the sine of X, the option 8/17 is its cosine and 15/8 is its tangent; asked for the leg of an isosceles right triangle, the option with the square root of 2 in it is that triangle's hypotenuse. The explanations name which quantity each one is.

Two habits run through the material and are worth more than any formula in it. First, check the answer against what the figure forbids: no sine or cosine of an acute angle can exceed 1, because both put a leg over the hypotenuse and the hypotenuse is the longest side — so an answer of 17/15 is wrong before the triangle is even looked at. Second, count steps rather than terms: reaching the 20th term of a sequence takes nineteen moves, the same off-by-one as expecting 20 gaps in a fence of 20 posts, when there are only 19.

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• Write the sine, cosine and tangent of a named angle from a triangle described in words
• Find a missing side with the Pythagorean theorem before writing a ratio that needs it
• Reject an impossible answer on sight: no sine or cosine of an acute angle exceeds 1
• Work out which angle each side faces before applying the 30-60-90 or 45-45-90 ratios
• Evaluate a function at a negative input, squaring and subtracting with the signs intact
• Find the input that produces a stated output by undoing the operations in reverse order
• State a domain, including whether the boundary value belongs in it
• Extend a sequence by counting the steps between terms rather than the terms themselves

Written for the functions and trigonometry strands of the ACT mathematics section, following the categories the test's own published description names: right-triangle trigonometry and the values at special angles, function notation, domain, and arithmetic and geometric sequences. Every triangle is described in full inside its own question, naming the points and the right angle, and each one also comes with a labeled drawing. Nothing is taken from any ACT publication or released test.