## Monday, October 21, 2019

### The 31 Critical ACT Math Formulas You MUST Know

The 31 Critical ACT Math Formulas You MUST Know SAT / ACT Prep Online Guides and Tips The two biggest challenges of ACT Math are the time crunch- the math test has 60 questions in 60 minutes!- and the fact that the test doesnÃ¢â‚¬â„¢t provide you with any formulas. All the formulas and math knowledge for the ACT comes from what youÃ¢â‚¬â„¢ve learned and memorized. In this complete list of critical formulas you'll need on the ACT, I'll lay out every formula you must have memorized before test day, as well as explanations for how to use them and what they mean. I'll also show you which formulas you should prioritize memorizing (the ones that are needed for multiple questions) and which ones you should memorize only when you've got everything else nailed down tight. Already Feeling Overwhelmed? Does the prospect of memorizing a bunch of formulas make you want to run for the hills? We've all been there, but don't throw in the towel just yet! The good news about the ACT is that it is designed to give all test-takers a chance to succeed. Many of you will already be familiar with most of these formulas from your math classes. The formulas that show up on the test the most will also be most familiar to you. Formulas that are only needed for one or two questions on the test will be least familiar to you. For example, the equation of a circle and logarithm formulas only ever show up as one question on most ACT math tests. If youÃ¢â‚¬â„¢re going for every point, go ahead and memorize them. But if you feel overwhelmed with formula lists, donÃ¢â‚¬â„¢t worry about it- itÃ¢â‚¬â„¢s only one question. So letÃ¢â‚¬â„¢s look at all the formulas you absolutely must know before test day (as well as one or two that you can figure out yourself instead of memorizing yet another formula). Algebra Linear Equations Functions There will be at least five to six questions on linear equations and functions on every ACT test, so this is a very important section to know. Slope Slope is the measure of how a line changes. ItÃ¢â‚¬â„¢s expressed as: the change along the y-axis/the change along the x-axis, or $\rise/\run$. Given two points, $A(x_1,y_1)$, $B(x_2,y_2)$, find the slope of the line that connects them: $$(y_2 - y_1)/(x_2 - x_1)$$ Slope-Intercept Form A linear equation is written as $y=mx+b$ m is the slope and b is the y-intercept (the point of the line that crosses the y-axis) A line that passes through the origin (y-axis at 0), is written as $y=mx$ If you get an equation that is NOT written this way (i.e. $mxÃ¢Ë†â€™y=b$), re-write it into $y=mx+b$ Midpoint Formula Given two points, $A(x_1,y_1)$, $B(x_2,y_2)$, find the midpoint of the line that connects them: $$((x_1 + x_2)/2, (y_1 + y_2)/2)$$ Good to Know Distance Formula Find the distance between the two points $$Ã¢Ë†Å¡{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ You donÃ¢â‚¬â„¢t actually need this formula, as you can simply graph your points and then create a right triangle from them. The distance will be the hypotenuse, which you can find via the pythagorean theorem Logarithms There will usually only be one question on the test involving logarithms. If youÃ¢â‚¬â„¢re worried about having to memorize too many formulas, donÃ¢â‚¬â„¢t worry about logs unless youÃ¢â‚¬â„¢re trying for a perfect score. $log_bx$ asks Ã¢â‚¬Å"to what power does b have to be raised to result in x?Ã¢â‚¬  Most of the time on the ACT, youÃ¢â‚¬â„¢ll just need to know how to re-write logs $$log_bx=y = b^y=x$$ $$log_bxy=log_bx+log_by$$ $$log_b{x/y} = log_bx - log_by$$ Statistics and Probability Averages The average is the same thing as the mean Find the average/mean of a set of terms (numbers) $$\Mean = {\sum\of\the\terms}/{\the\number(\amount)\of\different\terms}$$ Find the average speed $$\Speed = {\total\distance}/{\total\time}$$ May the odds be ever in your favor. Probabilities Probability is a representation of the odds of something happening. A probability of 1 is guaranteed to happen. A probability of 0 will never happen. $${\ProbabilityÃ¢â‚¬Å'\ofÃ¢â‚¬Å'\anÃ¢â‚¬Å'\outcomeÃ¢â‚¬Å'\happening}={\numberÃ¢â‚¬Å'\ofÃ¢â‚¬Å'\desiredÃ¢â‚¬Å'\outcomes}/{\total\number\of\possible\outcomes}$$ Probability of two independent outcomes both happening is $$\ProbabilityÃ¢â‚¬Å'\ofÃ¢â‚¬Å'\eventÃ¢â‚¬Å'\A*\probabilityÃ¢â‚¬Å'\ofÃ¢â‚¬Å'\event\B$$ e.g., Event A has a probability of $1/4$ and event B has a probability of $1/8$. The probability of both events happening is: $1/4 * 1/8 = 1/32$. There is a 1 in 32 chance of both events A and event B happening. Combinations The possible amount of different combinations of a number of different elements A Ã¢â‚¬Å"combinationÃ¢â‚¬  means the order of the elements doesnÃ¢â‚¬â„¢t matter (i.e. a fish entree and a diet soda is the same thing as a diet soda and a fish entree) Possible combinations = number of element A * number of element B * number of element CÃ¢â‚¬ ¦. e.g. In a cafeteria, there are 3 different dessert options, 2 different entree options, and 4 drink options. How many different lunch combinations are possible, using one drink, one, dessert, and one entree? The total combinations possible = 3 * 2 * 4 = 24 Percentages Find x percent of a given number n $$n(x/100)$$ Find out what percent a number n is of another number m $$(100n)/m$$ Find out what number n is x percent of $$(100n)/x$$ The ACT is a marathon. Remember to take a break sometimes and enjoy the good things in life. Puppies make everything better. Geometry Rectangles Area $$\Area=lw$$ l is the length of the rectangle w is the width of the rectangle Perimeter $$\Perimeter=2l+2w$$ Rectangular Solid Volume $$\Volume = lwh$$ h is the height of the figure Parallelogram An easy way to get the area of a parallelogram is to drop down two right angles for heights and transform it into a rectangle. Then solve for h using the pythagorean theorem Area $$\Area=lh$$ (This is the same as a rectangleÃ¢â‚¬â„¢s lw. In this case the height is the equivalent of the width) Triangles Area $$\Area = {1/2}bh$$ b is the length of the base of triangle (the edge of one side) h is the height of the triangle The height is the same as a side of the 90 degree angle in a right triangle. For non-right triangles, the height will drop down through the interior of the triangle, as shown in the diagram. Pythagorean Theorem $$a^2 + b^2 = c^2$$ In a right triangle, the two smaller sides (a and b) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle) Properties of Special Right Triangle: Isosceles Triangle An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides. An isosceles right triangle always has a 90 degree angle and two 45 degree angles. The side lengths are determined by the formula: x, x, xÃ¢Ë†Å¡2, with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides * Ã¢Ë†Å¡2. E.g., An isosceles right triangle may have side lengths of 12, 12, and 12Ã¢Ë†Å¡2. Properties of Special Right Triangle: 30, 60, 90 Degree Triangle A 30, 60, 90 triangle describes the degree measures of its three angles. The side lengths are determined by the formula: x, xÃ¢Ë†Å¡3, and 2x. The side opposite 30 degrees is the smallest, with a measurement of x. The side opposite 60 degrees is the middle length, with a measurement of xÃ¢Ë†Å¡3. The side opposite 90 degree is the hypotenuse, with a length of 2x. For example, a 30-60-90 triangle may have side lengths of 5, 5Ã¢Ë†Å¡3, and 10. Trapezoids Area Take the average of the length of the parallel sides and multiply that by the height. $$\Area = [(\parallel\side\a + \parallel\side\b)/2]h$$ Often, you are given enough information to drop down two 90 angles to make a rectangle and two right triangles. YouÃ¢â‚¬â„¢ll need this for the height anyway, so you can simply find the areas of each triangle and add it to the area of the rectangle, if you would rather not memorize the trapezoid formula. Trapezoids and the need for a trapezoid formula will be at most one question on the test. Keep this as a minimum priority if you're feeling overwhelmed. Circles Area $$\Area=Ãâ‚¬r^2$$ Ãâ‚¬ is a constant that can, for the purposes of the ACT, be written as 3.14 (or 3.14159) Especially useful to know if you donÃ¢â‚¬â„¢t have a calculator that has a $Ãâ‚¬$ feature or if you're not using a calculator on the test. r is the radius of the circle (any line drawn from the center point straight to the edge of the circle). Area of a Sector Given a radius and a degree measure of an arc from the center, find the area of that sector of the circle. Use the formula for the area multiplied by the angle of the arc divided by the total angle measure of the circle. $$Area\of\an\arc = (Ãâ‚¬r^2)(\degree\measure\of\center\of\arc/360)$$ Circumference $$\Circumference=2Ãâ‚¬r$$ or $$\Circumference=Ãâ‚¬d$$ d is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius. Length of an Arc Given a radius and a degree measure of an arc from the center, find the length of the arc. Use the formula for the circumference multiplied by the angle of the arc divided by the total angle measure of the circle (360). $$\Circumference\of\an\arc = (2Ãâ‚¬r)(\degree\measure\center\of\arc/360)$$ Example: A 60 degree arc has $1/6$ of the total circle's circumference because $60/360 = 1/6$ An alternative to memorizing the Ã¢â‚¬Å"formulasÃ¢â‚¬  for arcs is to just stop and think about arc circumferences and arc areas logically. If you know the formulas for the area/circumference of a circle and you know how many degrees are in a circle, put the two together. If the arc spans 90 degrees of the circle, it must be $1/4$th the total area/circumference of the circle, because $360/90 = 4$. If the arc is at a 45 degree angle, then it is $1/8$th the circle, because $360/45 = 8$. The concept is exactly the same as the formula, but it may help you to think of it this way instead of as a Ã¢â‚¬Å"formulaÃ¢â‚¬  to memorize. Equation of a Circle Useful to get a quick point on the ACT, but donÃ¢â‚¬â„¢t worry about memorizing it if you feel overwhelmed; it will only ever be worth one point. Given a radius and a center point of a circle $(h, k)$ $$(x - h)^2 + (y - k)^2 = r^2$$ Cylinder $$\Volume=Ãâ‚¬r^2h$$ Trigonometry Almost all the trigonometry on the ACT can be boiled down to a few basic concepts SOH, CAH, TOA Sine, cosine, and tangent are graph functions The sine, cosine, or tangent of an angle (theta, written as ÃŽËœ) is found using the sides of a triangle according to the mnemonic device SOH, CAH, TOA. Sine - SOH $$\SineÃ¢â‚¬Å' ÃŽËœ = \opposite/\hypotenuse$$ Opposite = the side of the triangle directly opposite the angle ÃŽËœ Hypotenuse = the longest side of the triangle Sometimes the ACT will make you manipulate this equation by giving you the sine and the hypotenuse, but not the measure of the opposite side. Manipulate it as you would any algebraic equation: $Sine ÃŽËœ = \opposite/\hypotenuse$ = $\hypotenuse * \sine ÃŽËœ = \opposite$ Cosine - CAH $$\Cosine ÃŽËœ = \adjacent/\hypotenuse$$ Adjacent = the side of the triangle nearest the angle ÃŽËœ (that creates the angle) that is not the hypotenuse Hypotenuse = the longest side of the triangle Tangent - TOA $$\TangentÃ¢â‚¬Å' ÃŽËœ = \opposite/\adjacent$$ Opposite = the side of the triangle directly opposite the angle ÃŽËœ Adjacent = the side of the triangle nearest the angle ÃŽËœ (that creates the angle) that is not the hypotenuse Cosecant, Secant, Cotangent Cosecant is the reciprocal of sine $\CosecantÃ¢â‚¬Å' ÃŽËœ = \hypotenuse/\opposite$ Secant is the reciprocal of cosine $\SecantÃ¢â‚¬Å' ÃŽËœ = \hypotenuse/\adjacent$ Cotangent is the reciprocal of tangent $\CotangentÃ¢â‚¬Å' ÃŽËœ = \adjacent/\opposite$ Useful Formulas to Know$$\Sin^2ÃŽËœ + \Cos^2ÃŽËœ = 1$$ $${\Sin ÃŽËœ}/{\Cos ÃŽËœ} = \Tan ÃŽËœ$$ Hurray! You've memorized your formulas. Now treat yo' self. But Keep in Mind Though these are all the formulas you should memorize to do well on the ACT math section, this list by no means covers all aspects of the mathematical knowledge youÃ¢â‚¬â„¢ll need on the exam. For example, youÃ¢â‚¬â„¢ll also need to know your exponent rules, how to FOIL, and how to solve for absolute values. To learn more about the general mathematical topics covered by the test, see our article on what's actually tested on the ACT math section. What's Next? Now that you know the critical formulas for the ACT, it might be time to check out our article on How to an Perfect Score on the ACT Math by a 36 ACT-Scorer. Don't know where to start?Look no further than our article onwhat is considered a good, bad, or excellent ACT score. Want to improve your score by 4+ points? Our completely online and customized prep program adapts to your strengths, weaknesses, and needs. And we guarantee your money back if you don't improve your score by 4points or more. 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