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.

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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))

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• 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 = .

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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

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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

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• 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

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• Review verifying trigonometric identities. • Review solving trigonometric equations.