Laws of Sines and Cosines The Laws of Sines and Cosines are useful for geometry questions involving non-right triangles. Law of Sines The Law of Sines states that aSin(A)=bSin(B)=cSin(C) where a, b, and c are the side lengths of the triangle, and A, B, and C are the internal angles opposite those respective sides, respectively. Questions involving the Law of…
Arithmetic sequence Arithmetic sequences are similar to function, in that they indicate a way to work with numbers. However, instead of working with numbers continuously (including terms with decimals), sequences work only with whole numbers. Questions will normally provide a sequence, which students will then have to determine a specific term in (for example, the…
Difference of two squares Difference of two squares is a pattern that can be used to factor certain quadratic equations on the ACT math section. The default formula is that: a2 − b2 = (a+b)(a−b). The terms in this formula can be either variables or constants. There is no center term because the two binomials are conjugates of each other. When…
Circle equations The default equation of a circle looks like this: (x−h)2 + (y−k)2 = r2 where r is the radius of the circle, and the x and y binomials indicate where the center of the circle is. Circle questions on the ACT math section will normally provide either a graph or an equation. Whichever one is provided must be used…
c = product of roots, b = sum of roots, a = scaling When factoring an exponential equation (ax2+bx+c into (x-d)(x-e)), the signs and magnitudesof b and c can provide a lot of information that is useful for factoring. Example Equation: x2 + 3x − 10 C C should always be the starting place any time a student factors. It gives direct and specific information…
Volume of a prism Simple Shapes The volume of a simple 3d object is equal to the area of the base object multiplied by the height of the object Complex Shapes The shapes above all have right vertices, meaning that their heights are consistently perpendicular. Not all 3D shapes have perpendicular vertices, which means that…
Permutation Permutation is similar to the counting principles discussed earlier. However, permutation questions involve options which are mutually exclusive. Here is a sample ACT question involving Permutation: Unlike with the security lock question covered above, once a plant is used in one display spot, it cannot be used in a different display spot. So instead…
Vectors Vectors are a way of expressing magnitude (size) and direction. The magnitude of a vector will often be described using variables, most commonly the variables i and j. The direction of the vector will be indicated by an arrowhead at the end of the vector line. Vector Skills Adding To add together any two…
Ellipses Ellipses are a rare but important shape to know about for the ACT math section. Here are the main properties of an ellipse that students will be expected to know: Key Skills for Ellipses What is an Ellipse? An ellipse is an oval shape. It is similar to a circle, but where one of its…
Scientific notation Scientific Notation is a way to express very large or very small numbers by the number’s most significant digits combined with an exponential expression of base 10. This notation usually includes a whole number with one or two decimals after it: 2.38. This will then be followed by an exponential expression of: × 10a , where…