ACT Mathematics — radians, the unit circle, trig graphs and oblique triangles
ACT describes its mathematics section as covering the skills students have typically acquired in courses taken up to the beginning of grade 12. Its Geometry subcategory includes trigonometric ratios and missing values in triangles and circles, and its Functions subcategory the manipulation of functions and the features of their graphs. On the test this reaches past right-triangle ratios to radian measure, the unit circle, the graphs of sine and cosine and triangles that have no right angle. This material covers that trigonometry beyond right triangles.
It begins with radian measure in both directions and exact values on the unit circle: the cosine of 2π/3 and the tangent of 3π/4, where the reference angle gives the size and the quadrant gives the sign. Two questions practice that sign logic directly — naming the quadrant from the signs of sine and tangent, and finding a tangent from a given cosine in Quadrant II with the Pythagorean identity.
The graph questions read the range of y = 4 sin(2x) + 1 from its amplitude and midline, identify a phase shift, and find a period when the coefficient of x contains π. For oblique triangles, the law of sines and the law of cosines are printed in the question, and you use the first to find a side and the second to find an angle. The last question solves 2 sin x − 1 = 0 on one full turn, where the second solution in Quadrant II is the one most often missed.
The material offers two formats. The quiz has twelve multiple-choice questions with four answer choices each, as on the enhanced ACT, and every explanation names the slip behind each wrong choice, so a miss tells you what to fix. The flashcards collect the rules and formulas worth knowing by heart.
The content is based on ACT's published description of the mathematics section (reporting categories and the topics listed for each). It is independent practice and is not produced or endorsed by ACT.
- Convert between degrees and radians
- Find exact trigonometric values and signs on the unit circle
- Read the amplitude, period, midline and phase shift of sine and cosine graphs
- Solve oblique triangles with the laws of sines and cosines
- Solve a simple trigonometric equation over one full turn
Practice material written by Zestly, based on ACT's published description of the ACT mathematics section (act.org, Mathematics Test Description: Preparing for Higher Math — Geometry and Functions). Formulas beyond the basics are given in the questions, as ACT states that recall of complex formulas is not required.
Sample question
The angle $\theta$ is in Quadrant II and $\cos\theta = -\frac{5}{13}$. What is the value of $\tan\theta$?
- $-\frac{12}{5}$
- $\frac{12}{5}$
- $-\frac{5}{12}$
- $-\frac{12}{13}$
See the answer
$-\frac{12}{5}$
From $\sin^2\theta + \cos^2\theta = 1$: $\sin^2\theta = 1 - \frac{25}{169} = \frac{144}{169}$, and $\sin\theta = \frac{12}{13}$ because sine is positive in Quadrant II. Then $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{12/13}{-5/13} = -\frac{12}{5}$. The value $\frac{12}{5}$ ignores the quadrant, $-\frac{5}{12}$ is the reciprocal (the cotangent), and $-\frac{12}{13}$ is the sine with the wrong sign, not the tangent.
- ACT Mathematics — statistics, counting and probability
- ACT Mathematics — factoring, rational, radical, exponential and absolute-value equations
- ACT Mathematics — multi-step percent, rate, mixture and measurement problems
- ACT Mathematics — function families, composition and transformations
- ACT Mathematics — complex numbers, matrices, vectors and radicals
- ACT Mathematics — right-triangle trigonometry, functions and sequences
ACT® is a registered trademark of ACT Education Corp., which is not affiliated with and does not endorse Zestly. All questions here are original and reproduce no official ACT material.