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ACT Math Tips Types of Math Questions:
§ Multiple Choice (A, B, C, D, E and F, G, H, J, K) o You are not penalized for wrong answers. Therefore, try every
problem.
ACT Mathematics: 60 minutes – 60 items Area Number of Items Pre-Algebra 14 Elementary Algebra 10 Intermediate Algebra 9 Coordinate Geometry 9 Plane Geometry 14 Trigonometry 4 Scores Reported: Pre-Algebra and Elementary Algebra Intermediate Algebra and Coordinate Geometry Plane Geometry and Trigonometry Total Number Correct Note: You only have a minute per math question.
Test-Taking Strategies
1. Relax. 2. You’re going to make mistakes. So be careful with calculations and using the
calculator. 3. All that matters is which circle you fill in. 4. You can know the answer to an item but be marked wrong. If you get the
right answer but fill in the wrong circle, the machine will mark it wrong. 5. Save the hard items for last. 6. They try to trick you. Test writers often include distracters. 7. Watch out for except, not, or least. 8. Still don’t have the answer? Eliminate and guess. 9. Do your work in the test booklet. 10. Write the letter for the answer choice in your test booklet.
ACT Math Tips
2
Things to Know c 2x 60° 45° s 2 r b x s a 30° 45°
c a b2 2 2= + x 3 s A r= π 2
Special Right Triangles C r= 2π l r w h h
w h b l
A lw= A bh=12 V lwh= V r h= π 2
§ The measure of a straight angle is 180° . § The sum of the measures of the interior angles in a triangle is 180° . § The total number of degrees in a circle is 360° .
• Common Squares and Square Roots
x 1 2 3 4 5 6 7 8 9 10 11 12 15 20 25 30 x2 1 4 9 16 25 36 49 64 81 100 121 144 225 400 625 900
• Common Cubes and Cube Roots
x 1 2 3 4 5 x3 1 8 27 64 125
• Algebraic Relationships
o Distributive Law: a(x + y) = ax + ay a(x – y) = ax – ay
o Exponents: a a am n m n= + aa
am
nm n= − a am n mnc h= a
an
n− =
1
abcb g0 1= a an n1= a a
mn n
m=d i
o Radicals: a b ab× = xy
xy
=
• Solve systems of equations by graphing (calculate the intersection), substitution, elimination, or using matrices (matrix equation or augmented matrix (RREF))
ACT Math Tips
3
• You can solve equations using a graph by typing in the left side as Y1 and the right side as Y2 , then calculate the intersection.
• Solve and graph linear and quadratic equations and inequalities • Factoring Polynomials
o
a b a b a b a ab b
a b a b a b a ab b
a b a b a b
+ = + + = + +
− = − − = − +
+ − = −
b g b gb gb g b gb gb gb g
2 2 2
2 2 2
2 2
2
2
o Quadratic Formula x b b aca
=− ± −2 4
2
o Discriminant, b ac2 4− , tells you the nature and number of roots
Exponential and Logarithmic Functions
1. If yx aa = , then yx = .
2. yxa =log if and only if yax = 3. log10 x = log x 4. loge x = ln x
5. Special Logarithms:
xa
xa
a
x
xa
a
a
a =
=
=
=
log
log
01log1log
or
ln e = 1ln1= 0
ln ex = x
eln x = x
6. logamn = logam + logan
7. nmnm
aaa logloglog −=
8. )(loglog mpm ap
a =
9. ax
axxa ln
lnlogloglog == , a ≠ 1
10. If yx aa loglog = , then yx = .
ACT Math Tips
4
Trigonometry
Unit Circle
For an angle θ on the unit circle, let P(x, y) be the point on the terminal side of θ .
sin θ = y cos θ = x tan θ = xy
csc θ = y1
sec θ = x1 cot θ =
yx
For an angle θ in standard position, let P(x, y) be the point on the terminal side of
θ and 22 yxr += . sin θ =
ry cos θ =
rx tan θ =
xy
csc θ = yr
sec θ = xr cot θ =
yx
ACT Math Tips
5
Trigonometric Graphs
Amplitude = |a| Periodb
=2π or π
b Phase Shift = c
b
Note: The tangent and cotangent functions do not have amplitude.
Trigonometric Formulas, Properties, and Identities • Pythagorean Triples you should know are
o 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17
• Right Triangle Trigonometry (SOHCAHTOA)
o Sin OppHyp
= Cos AdjHyp
= Tan OppAdj
=
o opphyp
=csc adjhyp
=sec oppadj
=cot
o sin cos∠ = ∠A B cos sin∠ = ∠A B tan sincos
∠ =∠
∠A A
A tan sin
cos∠ =
∠
∠B B
B
• Reciprocal Identities o sin
cscθ
θ=
1 cos
secθ
θ=
1 tan
cotθ
θ=
1
o cscsin
θθ
=1
seccos
θθ
=1
cottan
θθ
=1
ACT Math Tips
6
• Quotient Identities
o tansincos
θθθ
= cot cossin
θθθ
=
• Pythagorean Identities o sin cos2 2 1θ θ+ = o tan sec2 21θ θ+ = o cot csc2 21θ θ+ =
• Double Angle Formulas
o uuu cossin22sin = o uuuuu 2222 sin211cos2sincos2cos −=−=−=
o uuu 2tan1
tan22tan−
=
• Sum and Difference Formulas
o sin sin cos cos sinu v u v u v± = ±b g
o cos cos cos sin sinu v u v u v± =b g
o tantan tantan tan
u v u vu v
± =±b g
1
• Review verifying trigonometric identities. • Review solving trigonometric equations.