Mathematics 1 — breadth, trigonometry and no formula sheet

Fourteen questions across the range the ACT actually covers — which is wider than the SAT's and shallower, and reaches into territory a lot of students have simply never prepared.

Trigonometry is on this test. Three questions here are trigonometry, and that proportion is deliberate: it is the topic most often missing from a student's preparation altogether rather than weak in it, because the SAT barely touches it and most preparation is written for the SAT. Sine and cosine as ratios and the identity connecting them; amplitude and period read off an equation; the law of cosines on a triangle with no right angle in it.

There is no formula sheet. The SAT gives you one and the ACT does not, and the explanations here name the formula each question needed so you can see exactly what you are expected to arrive with. The law of cosines, the volume of a cylinder, the relationship between a period and the coefficient inside the sine — all of it comes from you.

Four choices, not five, and no penalty for a wrong answer. The Math section was cut from five options to four, so a blind guess is now worth a quarter and eliminating one option makes it a third. With roughly a minute per question, anything you cannot start within fifteen seconds should be guessed, marked and revisited — never left blank.

Two of the questions can be half-solved before any arithmetic, and the explanations say so, because that instinct is worth more than speed. In the law-of-cosines question, one option is simply a + b, and the third side of a triangle is always shorter than the sum of the other two — so it is gone before you start. In the probability question, one option is greater than 1, and no probability is.

The rest is the ACT's real spread: unit conversion where the conversion is the question, a percentage increase and decrease that do not cancel, averages worked backwards from a total, a function evaluated at x + 1, logarithms, a line through two points, and what happens to a median and a mean when every value in a set shifts by the same amount.

That last category — ACT calls it Integrating Essential Skills and it is about a fifth of the section — is the arithmetic of the middle-school years applied to problems that are not phrased simply. It is where confident students lose marks, because nothing in it is difficult and everything in it is easy to read too fast.

What this cannot give you is the clock. Forty-five questions in fifty minutes, and pace is this section's real difficulty. Take a timed full-length section from ACT's own free practice material to find out where you actually are.

Every question here is written by Zestly, and nothing is reproduced from any ACT publication.

• Use the law of cosines on a triangle with no right angle, and recognise when Pythagoras does not apply
• Read amplitude and period off a sine or cosine equation
• Connect sine and cosine through the Pythagorean identity
• Convert between units where the conversion is the substance of the question
• See why a percentage rise and an equal percentage fall do not cancel
• Work an average backwards from a total, and from a total within a total
• Predict what happens to a mean and a median when every value shifts by the same amount
• Discard an impossible option before computing — a probability above 1, a triangle side longer than the other two combined
• Arrive with the formulas, since the ACT provides no formula sheet

In triangle ABC, side a = 5, side b = 7, and the angle C between those two sides measures 60 degrees. What is the length of side c? — The law of cosines is c² = a² + b² − 2ab·cos C, and it is one of the formulas you must bring with you, since the ACT provides no formula sheet. With cos 60° = ½: c² = 74 − 35 = 39. One of the wrong options is simply a + b, the length you would get if the triangle were flattened into a straight line — so it can be discarded without computing anything.