2.1 Algebra and Trigonometry

Algebra and trigonometry are crucial tools for civil engineers. These mathematical foundations help solve complex problems in structural design, surveying, and project planning. From linear equations to advanced trigonometric concepts, these skills are essential for accurate calculations and efficient problem-solving in the field.

Logarithmic and exponential functions play a vital role in civil engineering applications. These mathematical tools are used to model various phenomena, such as population growth, structural decay, and compound interest calculations. Understanding these functions enables engineers to make precise predictions and informed decisions in their projects.

Solving equations and systems

Linear and quadratic equations

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• Linear equations take the form $ax + b = 0$ (a and b are constants, x is the variable)
  • Solve by isolating x on one side of the equation
  • Example: Solve $2x + 5 = 13$
    • Subtract 5 from both sides: $2x = 8$
    • Divide both sides by 2: $x = 4$
• Quadratic equations follow the pattern $ax² + bx + c = 0$ (a, b, and c are constants, x is the variable)
  • Solve using factoring, completing the square, or the quadratic formula
  • Quadratic formula: $x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}$
  • Example: Solve $x² - 5x + 6 = 0$
    • Factoring: $(x - 2)(x - 3) = 0$
    • Solutions: $x = 2$ or $x = 3$
• Discriminant ($b² - 4ac$) determines the nature of quadratic roots
  • Positive discriminant yields two real roots
  • Zero discriminant produces one real root
  • Negative discriminant results in two complex roots

Systems of equations

• Systems of linear equations solved through substitution, elimination, or matrix methods
  • Solution represents the intersection point of the lines
  • Example: Solve the system $\begin{cases} 2x + y = 7 \\ x - y = 1 \end{cases}$
    • Using elimination:
      • Multiply second equation by 2: $2x - 2y = 2$
      • Add to first equation: $4x = 9$
      • Solve for x: $x = \frac{9}{4}$
      • Substitute into either original equation to find y
• Graphical methods visualize solutions for linear, quadratic, and systems of equations
  • Plot equations on a coordinate plane
  • Identify points of intersection

Manipulating expressions and formulas

Simplifying algebraic expressions

• Algebraic expressions combine variables, constants, and mathematical operations
  • Simplify by combining like terms and applying order of operations (PEMDAS)
  • Example: Simplify $3x² + 2x - 5 + 4x² - 3x + 7$
    • Combine like terms: $7x² - x + 2$
• Distributive property expands expressions
  • $a(b + c) = ab + ac$
  • Example: Expand $3(2x - 5)$
    • $3(2x) - 3(5) = 6x - 15$
• Factoring identifies common factors in algebraic expressions
  • Example: Factor $12x² - 18x$
    • Common factor: $6x$
    • Factored form: $6x(2x - 3)$

Manipulating algebraic fractions and formulas

• Simplify algebraic fractions by canceling common factors in numerator and denominator
  • Example: Simplify $\frac{x² + 3x}{x + 3}$
    • Factor numerator: $\frac{x(x + 3)}{x + 3}$
    • Cancel common factor: $x$
• Rearrange formulas by isolating specific variables using inverse operations
  • Example: Rearrange $A = πr²$ to solve for r
    • Divide both sides by π: $\frac{A}{π} = r²$
    • Take square root of both sides: $r = \sqrt{\frac{A}{π}}$
• Apply rules for exponents when manipulating expressions with powers
  • Include negative and fractional exponents
  • Example: Simplify $(x³y²)² \cdot (xy^{-1})³$
    • $(x³y²)² = x^6y^4$
    • $(xy^{-1})³ = x³y^{-3}$
    • Result: $x^9y$

Trigonometry in problem-solving

Trigonometric functions and ratios

• Primary trigonometric functions sine, cosine, and tangent
  • Reciprocals cosecant, secant, and cotangent
  • In right triangles:
    • Sine ratio of opposite side to hypotenuse
    • Cosine ratio of adjacent side to hypotenuse
    • Tangent ratio of opposite side to adjacent side
• Pythagorean theorem ($a² + b² = c²$) relates sides of right triangles
  • Fundamental to many trigonometric applications
  • Example: Find the hypotenuse of a right triangle with sides 3 and 4
    • $3² + 4² = c²$
    • $9 + 16 = c²$
    • $c = \sqrt{25} = 5$
• Law of sines and law of cosines extend problem-solving to non-right triangles
  • Law of sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
  • Law of cosines: $c² = a² + b² - 2ab \cos C$

Advanced trigonometric concepts

• Trigonometric identities simplify and solve trigonometric equations
  • Example: $\sin²θ + \cos²θ = 1$
  • Use to verify or simplify complex trigonometric expressions
• Unit circle provides understanding of trigonometric functions beyond first quadrant
  • Includes negative angles
  • Example: Find $\sin 150°$ using the unit circle
    • $\sin 150° = \sin (180° - 30°) = \sin 30° = \frac{1}{2}$
• Inverse trigonometric functions (arcsin, arccos, arctan) find angles given trigonometric ratios
  • Example: Find θ if $\sin θ = 0.5$
    • $θ = \arcsin 0.5 = 30°$ (in the first quadrant)

Logarithmic and exponential functions

Exponential functions and properties

• Exponential functions follow form $f(x) = a^x$ (a is base, x is exponent)
  • Common bases e (natural exponential) and 10
  • Example: Graph $f(x) = 2^x$ for $-2 \leq x \leq 2$
    • Plot points ((-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4))
• Applications include compound interest, population growth, radioactive decay
  • Example: Calculate compound interest using $A = P(1 + r)^t$
    • A is final amount, P is principal, r is interest rate, t is time

Logarithmic functions and properties

• Logarithmic functions inverse of exponential functions
  • Expressed as $y = \log_a(x)$ (a is base, x is argument)
  • Natural logarithm (ln) uses base e
  • Common logarithm (log) typically uses base 10
• Change of base formula converts between logarithms of different bases
  • $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$
  • Example: Express $\log_2(8)$ in terms of natural logarithm
    • $\log_2(8) = \frac{\ln(8)}{\ln(2)}$
• Logarithmic properties aid in simplification and problem-solving
  • $\log_a(xy) = \log_a(x) + \log_a(y)$
  • $\log_a(x/y) = \log_a(x) - \log_a(y)$
  • $\log_a(x^n) = n \log_a(x)$
  • Example: Simplify $\log_3(27) - \log_3(9)$
    • $\log_3(27) - \log_3(9) = \log_3(\frac{27}{9}) = \log_3(3) = 1$
• Solve exponential and logarithmic equations using definitions and algebraic techniques
  • Example: Solve $2^x = 8$
    • Take log base 2 of both sides: $\log_2(2^x) = \log_2(8)$
    • Simplify: $x = 3$