ALEKS Math Master Placement Formula Guide

ALEKS Placement Roadmap & Strategy

• ALEKS placement is adaptive, meaning the difficulty of the next question depends on the performance of previous answers.

• Higher placement can be blocked by small errors in foundations such as fractions, signs, exponents, units, or restrictions, even if the student is familiar with advanced concepts.

• The strategy for a clean setup involves checking domain, units, signs, denominator restrictions, and graph behavior before beginning any calculation.

• ALEKS verifies if a student can choose the correct tool, use it cleanly, and move between different representations: tables, equations, graphs, diagrams, units, and word problems.

• If a question feels too easy, it must be answered perfectly. If it feels too hard, the student should first identify the topic family: linear, quadratic, rational, radical, exponential, logarithmic, geometry, or trigonometry.

High-Level ALEKS Topic Families

• Pre-Algebra: Includes whole numbers, factors, fractions, decimals, percents, ratios, rates, units, geometry, and data.

• Algebra 1: Includes expressions, equations, inequalities, lines, systems, exponents, factoring, and quadratics.

• Algebra 2: Includes functions, transformations, inverses, rational expressions, radicals, logarithms, sequences, and conics.

• Geometry and Trigonometry: Includes area, volume, coordinate geometry, circles, right triangles, the unit circle, and trigonometric graphs.

• Data and Modeling: Includes statistics, probability, counting, word-problem translation, and growth/decay models.

Numbers, Fractions, Decimals, and Percents

• Fraction Operations:

  • Addition/Subtraction: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

  • Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

  • Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

• Percent Formulas:

  • $\text{part} = \text{percent} \times \text{whole}$

  • $\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%$

• Percent Change:

  • $\% \text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

• Ratios and Proportions:

  • $\frac{a}{b} = \frac{c}{d} \implies ad = bc$

• Statistical Measures:

  • $\text{mean} = \frac{\text{sum of values}}{\text{number of values}}$

• Physics Formulas:

  • $\text{distance} = \text{rate} \times \text{time}$ ($d = rt$)

• Example: If $18$ is $30\%$ of a number, then $18 = 0.30 \times x \implies x = 60$. The whole ($60$) is logically larger than the part ($18$).

• Tutor Note: Percent questions are often nested within discounts, markups, tax, interest, mixtures, or unit rates. Always identify the "whole" before calculating.

Whole Numbers, Factors, and Pre-Algebra Essentials

• Order of Operations (PEMDAS): Parentheses, Exponents, Multiplication/Division (left to right), and Addition/Subtraction (left to right).

• Signed Numbers Warning: $-3^2 = -(3^2) = -9$, whereas $(-3)^2 = 9$.

• Absolute Value: Represented as distance from zero: $|a| \geq 0$.

• Mathematical Properties:

  • Commutative: $a + b = b + a$ and $ab = ba$

  • Associative: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$

  • Distributive: $a(b + c) = ab + ac$

  • Identity: $a + 0 = a$ and $a \times 1 = a$

  • Inverse: $a + (-a) = 0$ and $a \times \frac{1}{a} = 1$ (for $a \neq 0$)

• Number Theory Concepts:

  • Greatest Common Factor (GCF): The largest factor shared by numbers; used for simplifying or factoring.

  • Least Common Multiple (LCM): The smallest positive shared multiple; used for finding common denominators.

  • GCF-LCM Connection: For positive integers, $\text{gcd}(a, b) \times \text{lcm}(a, b) = ab$.

  • Prime Number: A whole number greater than $1$ with exactly two factors.

  • Divisibility Rules: Divisible by $2$ if even; divisible by $3$ or $9$ if the digit sum is divisible by $3$ or $9$; divisible by $5$ if the number ends in $0$ or $5$.

• Example: For $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$, the $\text{gcd} = 2^2 \times 3 = 12$ and the $\text{lcm} = 2^3 \times 3^2 = 72$.

Measurement and Unit Conversions

• Customary Length:

  • $12\,in = 1\,ft$

  • $3\,ft = 1\,yd$

  • $5280\,ft = 1\,mi$

• Customary Weight and Capacity:

  • $16\,oz = 1\,lb$

  • $2000\,lb = 1\,\text{ton}$

  • $8\,fl\,oz = 1\,\text{cup}$

  • $2\,\text{cups} = 1\,pt$

  • $2\,pt = 1\,qt$

  • $4\,qt = 1\,gal$

• Metric System:

  • Length: $10\,mm = 1\,cm$, $100\,cm = 1\,m$, $1000\,m = 1\,km$

  • Mass/Capacity: $1000\,mg = 1\,g$, $1000\,g = 1\,kg$, $1000\,mL = 1\,L$

  • Prefixes: kilo ($1000$), centi ($\frac{1}{100}$), milli ($\frac{1}{1000}$)

• Time and Temperature:

  • $60\,s = 1\,min$, $60\,min = 1\,hr$, $24\,hr = 1\,\text{day}$

  • $F = \frac{9}{5}C + 32$

  • $C = \frac{5}{9} (F - 32)$

• Unit Conversion Habit: Multiply by conversion fractions equal to $1$. Place the unit to be cancelled on the bottom and the desired unit on the top.

  • Example: $3\,yd \times \frac{3\,ft}{1\,yd} \times \frac{12\,in}{1\,ft} = 108\,in$.

Algebra Foundations and Literal Equations

• Equation Toolkit:

  • Like terms: Combine terms with identical variables and exponents.

  • One-step equations: $x + a = b \implies x = b - a$; $ax = b \implies x = \frac{b}{a}$.

  • Multi-step equations: Clear parentheses, combine like terms, then isolate the variable.

• Inequality Rules:

  • The inequality symbol must flip ($< \rightarrow >$) when multiplying or dividing both sides by a negative number.

  • Example: $-2x + 5 < 13 \implies -2x < 8 \implies x > -4$.

• Absolute Value Equations and Inequalities:

  • $|x - a| = b \implies x = a \pm b$ (where $b \geq 0$).

  • $|x - a| < b \implies a - b < x < a + b$ (for "less than" case).

  • $|x - a| > b \implies x < a - b \text{ or } x > a + b$ (for "greater than" case).

  • If $b < 0$, the equation $|x - a| = b$ has no solution.

• Interval and Solution Regions:

  • Union ("or"): Combine all solution regions.

  • Intersection ("and"): The overlap of solution regions.

  • Literal Equations: Solve for one variable while treating others as constants. Example: $A = \frac{1}{2}bh \implies 2A = bh \implies h = \frac{2A}{b}$.

Lines, Slope, and Systems

• Linear Formulas:

  • Slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

  • Slope-intercept form: $y = mx + b$

  • Point-slope form: $y - y_1 = m(x - x_1)$

  • Standard form: $Ax + By = C$

• Geometric Relationships:

  • Parallel lines: Have identical slopes.

  • Perpendicular lines: Slopes are negative reciprocals ($m_1 \times m_2 = -1$).

  • Midpoint formula: $M = \begin{pmatrix} \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \end{pmatrix}$

  • Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Coordinate Interpretations:

  • Slope is equivalent to the rate of change.

  • The $y$-intercept is the starting value when $x = 0$.

  • The solution to a system of equations is the intersection point of their graphs.

Parent Functions and Graph Behavior

• Core Parent Functions:

  • Constant: $f(x) = c$ (Horizontal line, slope $0$).

  • Linear: $f(x) = x$ (Domain/Range: all real numbers).

  • Absolute Value: $f(x) = |x|$ (V-shape, vertex $(0, 0)$, Range: $y \geq 0$).

  • Quadratic: $f(x) = x^2$ (Parabola, vertex $(0, 0)$, Range: $y \geq 0$).

  • Cubic: $f(x) = x^3$ (Increasing S-shape).

  • Square Root: $f(x) = \sqrt{x}$ (Domain: $x \geq 0$, Range: $y \geq 0$).

  • Cube Root: $f(x) = \sqrt[3]{x}$ (Domain/Range: all real numbers).

  • Reciprocal: $f(x) = \frac{1}{x}$ (Vertical asymptote at $x = 0$, Horizontal at $y = 0$).

  • Exponential: $f(x) = a^x$, $a > 0$, $a \neq 1$ (Domain: all real, Range: $y > 0$).

  • Logarithmic: $f(x) = \log_a x$, $a > 0$, $a \neq 1$ (Domain: $x > 0$, Range: all real).

• Function Transformations:

  • Vertical shift: $f(x) + k$ (up $k$); $f(x) - k$ (down $k$).

  • Horizontal shift: $f(x - h)$ (right $h$); $f(x + h)$ (left $h$).

  • Reflection: $-f(x)$ (reflect over $x$-axis); $f(-x)$ (reflect over $y$-axis).

  • Scaling: $af(x)$ stretches if $|a| > 1$ and shrinks if $0 < |a| < 1$.

• Behavior and Analysis:

  • Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$.

  • Intercepts: For $y$-intercept, set $x=0$. For $x$-intercept (zeros), set $y=0$.

  • Piecewise Functions: Evaluate based on the specific condition containing the input value.

Exponents, Radicals, and Scientific Notation

• Power Rules:

  • Product: $a^m \times a^n = a^{m+n}$

  • Quotient: $\frac{a^m}{a^n} = a^{m-n}$

  • Power of a Power: $(a^m)^n = a^{mn}$

  • Negative Exponent: $a^{-n} = \frac{1}{a^n}$

  • Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

• Radical and Rational Properties:

  • Rational Exponent Bridge: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

  • Radical Product: $\sqrt{ab} = \sqrt{a}\sqrt{b}$.

  • Scientific Notation: $a \times 10^n$ where $1 \leq a < 10$.

  • Rationalizing Denominators: Multiply by the conjugate (e.g., $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$) or multiply by the radical form to clear roots.

  • Restrictions: Even roots ($\sqrt[n]{x}$, $n$ is even) require $x \geq 0$. Odd roots allow negative numbers (e.g., $\sqrt[3]{-8} = -2$).

Polynomials, Factoring, and Quadratics

• Factoring Patterns:

  • FOIL: $(a + b)(c + d) = ac + ad + bc + bd$

  • Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

  • Perfect Square Trinomials: $a^2 \pm 2ab + b^2 = (a \pm b)^2$

  • Sum/Difference of Cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

• Quadratic Formulas:

  • Standard Form: $ax^2 + bx + c = 0$

  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  • Discriminant ($D$): $D = b^2 - 4ac$. If $D > 0$, two real solutions; if $D = 0$, one real solution; if $D < 0$, zero real (two complex) solutions.

  • Vertex Form: $y = a(x - h)^2 + k$

  • Axis of Symmetry: $x = -\frac{b}{2a}$

• Graph Analysis:

  • The sign of "$a$" determines if the parabola opens up ($a>0$) or down ($a<0$).

  • Zeros are equivalent to $x$-intercepts.

Complex Numbers and Advanced Polynomials

• Imaginary Unit: $i^2 = -1$ and $\sqrt{-a} = i\sqrt{a}$.

  • Powers of $i$ cycle every four: $i, -1, -i, 1$.

  • Complex Number: $a + bi$ (Real part $a$, Imaginary part $b$).

  • Conjugate: $a - bi$. Product: $(a + bi)(a - bi) = a^2 + b^2$.

• Higher-Degree Polynomials:

  • Division Algorithm: $P(x) = D(x)Q(x) + R(x)$, where $deg(R) < deg(D)$.

  • Remainder Theorem: Dividing $P(x)$ by $x - c$ leaves remainder $P(c)$.

  • Factor Theorem: $x - c$ is a factor if and only if $P(c) = 0$.

  • Rational Root Candidates: $\pm \frac{p}{q}$ where $p$ divides the constant and $q$ divides the leading coefficient.

  • Fundamental Theorem of Algebra: A degree-$n$ polynomial has $n$ complex roots (counting multiplicity).

  • End Behavior: Even degree implies ends point the same way; odd degree implies ends point opposite ways. Positive leading coefficient makes the right end point up.

Functions, Domain, and Inverses

• Function Notation: $f(a)$ means substitute $a$ for every occurrence of $x$.

• Composition: $(f \circ g)(x) = f(g(x))$.

• Inverse: To find $f^{-1}(x)$, swap $x$ and $y$ then solve for the new $y$.

• Function Properties:

  • Even Function: $f(-x) = f(x)$.

  • Odd Function: $f(-x) = -f(x)$.

• Domain Habits: Restrictions exist whenever a denominator might be zero or an even root might have a negative input. These restrictions must be noted before simplifying expressions.

Rational Expressions and Variation

• Operations with Rationals:

  • Cancellation: Cancel factors, not terms. $\frac{x(x+2)}{x} = x + 2$, but $\frac{x+2}{x}$ cannot be simplified to $2$.

  • Excluded Values: Set every original denominator to zero to identify undefined points. These apply even after simplifying.

• Variation Models:

  • Direct: $y = kx$

  • Inverse: $y = \frac{k}{x}$

  • Joint: $z = kxy$

Logarithms and Exponentials

• Definition: $\log_b x = y \iff b^y = x$.

• Logarithmic Rules:

  • Product Rule: $\log_b(MN) = \log_b M + \log_b N$

  • Quotient Rule: $\log_b \begin{pmatrix} \frac{M}{N} \end{pmatrix} = \log_b M - \log_b N$

  • Power Rule: $\log_b(M^p) = p \log_b M$

  • Change of Base: $\log_b a = \frac{\log a}{\log b}$

• Natural Log ($ln$): $\ln x = \log_e x$.

• Restrictions: For real logs, the argument and base must be positive; the base cannot equal $1$.

Sequences, Series, and Counting

• Arithmetic Sequences (Repeated Addition):

  • Term: $a_n = a_1 + (n - 1)d$

  • Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Geometric Sequences (Repeated Multiplication):

  • Term: $a_n = a_1 r^{n-1}$

  • Finite Series: $S_n = \frac{a_1(1 - r^n)}{1 - r}$

  • Infinite Series: $S_{\infty} = \frac{a_1}{1 - r}$ (only if $|r| < 1$).

• Counting and Binomials:

  • Factorial: $n! = n(n - 1)(n - 2) \dots 1$; $0! = 1$.

  • Permutations (Order matters): $_nP_r = \frac{n!}{(n - r)!}$.

  • Combinations (Order does not matter): $_nC_r = \begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{r!(n - r)!}$.

  • Binomial Theorem: $(x + y)^n = \sum_{k=0}^n \begin{pmatrix} n \\ k \end{pmatrix} x^{n-k} y^k$.

Geometry and Conics

• Basic Formulas:

  • Rectangle: $A = lw$, $P = 2l + 2w$

  • Triangle: $A = \frac{1}{2}bh$

  • Circle: $A = \pi r^2$, $C = 2\pi r$

  • Standard Circle Equation: $(x - h)^2 + (y - k)^2 = r^2$

• 3D Geometry:

  • Rectangular Prism: $V = lwh$

  • Cylinder: $V = \pi r^2 h$, $SA = 2\pi r^2 + 2\pi rh$

  • Cone: $V = \frac{1}{3} \pi r^2 h$

  • Sphere: $V = \frac{4}{3} \pi r^3$, $SA = 4\pi r^2$

• Conic Sections Decoding:

  • One squared variable: Parabola.

  • Two squared variables (added): Circle or Ellipse.

  • Two squared variables (subtracted): Hyperbola.

  • Ellipse Foci: $c^2 = a^2 - b^2$.

  • Hyperbola Foci: $c^2 = a^2 + b^2$.

Trigonometry Essentials

• Right Triangle Ratios (SOH-CAH-TOA):

  • $\sin(\theta) = \frac{\text{opp}}{\text{hyp}}$

  • $\cos(\theta) = \frac{\text{adj}}{\text{hyp}}$

  • $\tan(\theta) = \frac{\text{opp}}{\text{adj}}$

• Identities:

  • Pythagorean: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, $1 + \cot^2(\theta) = \csc^2(\theta)$.

  • Tangent: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

  • Reciprocals: $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, $\cot(\theta) = \frac{1}{\tan(\theta)}$.

• Angles and Graphs:

  • Radians/Degrees: $180^{\circ} = \pi\, \text{radians}$.

  • Arc Length: $s = r\theta$

  • Sector Area: $A = \frac{1}{2}r^2\theta$

  • Sine/Cosine Graph Model: $y = A \sin(B(x - C)) + D$. Amplitude = $|A|$, Period = $\frac{2\pi}{|B|}$, Phase Shift = $C$, Midline $y = D$.

• Laws for Non-Right Triangles:

  • Law of Sines: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

  • Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos(C)$

Statistics, Matrices, and Data

• Statistics:

  • Weighted Average: $\frac{\sum w_i x_i}{\sum w_i}$.

  • Probability: $P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$. Range must be $0$ to $1$.

  • Complement: $P(\text{not } E) = 1 - P(E)$.

  • Standard Deviation: Square root of variance ($\sigma^2 = \frac{\sum(x - \mu)^2}{N}$).

  • Z-score: $z = \frac{x - \mu}{\sigma}$.

• Matrices:

  • Multiplication ($AB$) is defined only when the number of columns in $A$ matches rows in $B$. Order matters: $AB \neq BA$.

  • Determinant for 2x2: $ad - bc$. If determinant is zero, no inverse exists.

  • Matrix Equation: $AX = B \implies X = A^{-1}B$.

Final ALEKS Preparation Checklist

• Questions and Discussion:

  • How to handle complex word problems? Translate triggers first: "per" (rate), "of" (multiply), "is" (equals), "at least" ($\geq$), "no more than" ($\leq$).

  • When is a function undefined? Watch for denominators of $0$, invalid log arguments ($\leq 0$), or negative inputs in even roots.

  • What are "zeros" or "roots"? They are the solution where the expression or function equals $0$.

  • What is "Rate of Change"? In linear contexts, it is the slope; ingeneral functions, it is the average rate of change between two points.

  • Strategic Practice: When a problem is missed, do not just fix the answer. Identify the formula family and review the two prerequisite skills below it to close the knowledge gap definitively.

Numbers, Fractions, Decimals, and Percents

Fraction Operations

1. Addition/Subtraction Example: $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$

   • First find a common denominator (12) for both fractions.

   • Convert fractions and add numerators.

2. Multiplication Example: $\frac{3}{5} \times \frac{4}{7} = \frac{12}{35}$

   • Multiply the numerators and denominators directly.

3. Division Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

   • Remember to multiply by the reciprocal.

Whole Numbers, Factors, and Pre-Algebra Essentials

Order of Operations

1. Complex Calculation: Evaluate: $3 + 4 \times (2^2 - 1)$

   • Step 1: Evaluate the exponent: $(2^2 = 4)$.

   • Step 2: Inside parentheses: $(4 - 1 = 3)$.

   • Step 3: Multiplication next: $4 \times 3 = 12$.

   • Step 4: Finally, add: $3 + 12 = 15$.

2. Nested Operations: Evaluate: $6 + 2[3 + 5(2 - 1)]$

   • Step 1: Calculate the innermost parentheses: $(2 - 1 = 1)$.

   • Step 2: Multiply: $5 \times 1 = 5$.

   • Step 3: Sum in brackets: $(3 + 5 = 8)$.

   • Step 4: Multiply: $2 \times 8 = 16$.

   • Step 5: Final addition: $6 + 16 = 22$.

3. Including Exponents and Roots: Evaluate: $5 + 2^3 - \sqrt{16}$

   • Step 1: Evaluate the exponent: $(2^3 = 8)$.

   • Step 2: Calculate the square root: $\sqrt{16} = 4$.

   • Step 3: Combine: $5 + 8 - 4 = 9$.

Algebra Foundations and Literal Equations

One-step equations

1. Solving Simple Equation: $x + 4 = 10$

   • Subtract 4 from both sides:
     $x = 10 - 4$

   • Therefore, $x = 6$.

2. Including Negative Numbers: $-3x = 9$

   • Divide by -3:
     $x = \frac{9}{-3}$

   • Therefore, $x = -3$.

3. Involving Fractions: $\frac{x}{4} = 5$

   • Multiply by 4 to isolate x:
     $x = 5 \times 4$

   • Therefore, $x = 20$.

Linear Formulas

Slope Calculation

1. Slope for Two Points (Simple Example): Find the slope between points (2, 3) and (4, 7).

   • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2$.

2. Three Points Slope (Complex Calculation): Find the average slope between points (1, 2), (2, 4), and (3, 5).

   • Slope between (1, 2) and (2, 4): $m_1 = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2$.

   • Slope between (2, 4) and (3, 5): $m_2 = \frac{5 - 4}{3 - 2} = \frac{1}{1} = 1$.

   • Average slope: $\text{average} = \frac{m<em>1 + m</em>2}{2} = \frac{2 + 1}{2} = 1.5$.

3. Negative Slope Example: Determine the slope of the line through (4, 1) and (2, 3):

   • $m = \frac{3 - 1}{2 - 4} = \frac{2}{-2} = -1$.

Geometry and Conics

Area Calculations

1. Area of a Rectangle:
   Given: Length = 5 units, Width = 3 units.
   $A = \text{Length} \times \text{Width} = 5 \times 3 = 15 \, \text{units}^2$.

2. Area of Triangle:
   Given: Base = 6 units, Height = 4 units.
   $A = \frac{1}{2}bh = \frac{1}{2} \times 6 \times 4 = 12 \, \text{units}^2$.

3. Area of Circle:
   Given: Radius = 3 units.
   $A = \pi r^2 = \pi \times 3^2 = 9\pi \, \text{units}^2 (approximately 28.27)$.

Trigonometry Essentials

Right Triangle Ratios

1. Basic Calculation: Find $\sin(30^{\circ})$:

   • $\sin(30^{\circ}) = \frac{\text{opp}}{\text{hyp}} = \frac{1}{2}$.

2. Using Pythagorean Identity: If $\sin(\theta) = \frac{3}{5}$, find $\cos(\theta)$.

   • Using $\sin^2(\theta) + \cos^2(\theta) = 1$:

   • $\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1 \rightarrow \frac{9}{25} + \cos^2(\theta) = 1$.

   • $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25} \rightarrow \cos(\theta) = \frac{4}{5}$.

3. Finding Angle with Tangent: If $\tan(\theta) = \frac{3}{4}$, find $\theta$:

   • $\theta = \tan^{-1}(\frac{3}{4}) \approx 36.87^{\circ}$.