2.1 Algebra and Trigonometry

Algebra and trigonometry are the core math tools you'll use throughout civil engineering. Whether you're sizing a beam, surveying a site, or estimating project costs, these skills show up constantly. This guide covers equation solving, expression manipulation, trigonometry, and logarithmic/exponential functions.

Solving equations and systems

Linear and quadratic equations

A linear equation takes the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable. You solve it by isolating $x$.

• Example: Solve $2x + 5 = 13$
  • Subtract 5 from both sides: $2x = 8$
  • Divide both sides by 2: $x = 4$

A quadratic equation follows the pattern $ax^2 + bx + c = 0$. You can solve these by factoring, completing the square, or using the quadratic formula:

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

• Example: Solve $x^2 - 5x + 6 = 0$
  • Factor: $(x - 2)(x - 3) = 0$
  • Solutions: $x = 2$ or $x = 3$

The discriminant ($b^2 - 4ac$) tells you what kind of roots to expect before you even solve:

• Positive discriminant: two distinct real roots
• Zero discriminant: one repeated real root
• Negative discriminant: two complex roots (no real solutions)

Systems of equations

A system of equations has two or more equations with shared variables. The solution is the point (or points) where all equations are satisfied simultaneously. You can solve using substitution, elimination, or matrix methods.

• Example: Solve $\begin{cases} 2x + y = 7 \\ x - y = 1 \end{cases}$
  • Add the two equations directly: $3x = 8$
  • Solve for $x$: $x = \frac{8}{3}$
  • Substitute back into the second equation: $\frac{8}{3} - y = 1$, so $y = \frac{5}{3}$

Graphically, the solution to a system of linear equations is the intersection point of the lines. Plotting equations on a coordinate plane can help you visualize whether a system has one solution, no solution (parallel lines), or infinitely many solutions (same line).

Manipulating expressions and formulas

Simplifying algebraic expressions

Algebraic expressions combine variables, constants, and operations. To simplify, combine like terms and follow the order of operations (PEMDAS).

• Example: Simplify $3x^2 + 2x - 5 + 4x^2 - 3x + 7$
  • Group like terms: $(3x^2 + 4x^2) + (2x - 3x) + (-5 + 7)$
  • Result: $7x^2 - x + 2$

The distributive property lets you expand expressions: $a(b + c) = ab + ac$

• Example: $3(2x - 5) = 6x - 15$

Factoring is the reverse of distributing. You pull out common factors from each term.

• Example: Factor $12x^2 - 18x$
  • The greatest common factor is $6x$
  • Factored form: $6x(2x - 3)$

Manipulating algebraic fractions and formulas

To simplify algebraic fractions, factor the numerator and denominator, then cancel common factors.

• Example: Simplify $\frac{x^2 + 3x}{x + 3}$
  • Factor the numerator: $\frac{x(x + 3)}{x + 3}$
  • Cancel the common factor $(x + 3)$: result is $x$ (with the restriction that $x \neq -3$)

Rearranging formulas is something you'll do constantly in engineering. Use inverse operations to isolate the variable you need.

• Example: Rearrange $A = \pi r^2$ to solve for $r$
  1. Divide both sides by $\pi$: $\frac{A}{\pi} = r^2$
  2. Take the square root: $r = \sqrt{\frac{A}{\pi}}$

When working with exponents, remember the key rules apply to negative and fractional exponents too:

• Example: Simplify $(x^3 y^2)^2 \cdot (x y^{-1})^3$
  • $(x^3 y^2)^2 = x^6 y^4$
  • $(x y^{-1})^3 = x^3 y^{-3}$
  • Multiply: $x^{6+3} y^{4+(-3)} = x^9 y$

Trigonometry in problem-solving

Trigonometric functions and ratios

Trigonometry connects angles to side lengths in triangles. In a right triangle, the three primary ratios are defined relative to a chosen angle $\theta$:

• Sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

The reciprocal functions (cosecant, secant, cotangent) are just the flipped versions of these ratios. A common mnemonic is SOH-CAH-TOA.

The Pythagorean theorem ($a^2 + b^2 = c^2$) relates the sides of any right triangle, where $c$ is the hypotenuse.

• Example: A right triangle has legs of 3 m and 4 m. Find the hypotenuse.
  • $3^2 + 4^2 = c^2$
  • $9 + 16 = 25$
  • $c = 5$ m

For non-right triangles, you'll need the law of sines or the law of cosines:

• Law of sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of cosines: $c^2 = a^2 + b^2 - 2ab\cos C$

Use the law of sines when you know an angle and its opposite side. Use the law of cosines when you know two sides and the included angle (SAS) or all three sides (SSS).

Advanced trigonometric concepts

Trigonometric identities are equations that hold true for all valid angle values. The most fundamental is the Pythagorean identity:

$\sin^2\theta + \cos^2\theta = 1$

You can use identities to simplify complex trig expressions or verify that two expressions are equivalent.

The unit circle extends trig functions beyond acute angles to any angle, including negative angles and angles greater than $360°$. Each point on the unit circle has coordinates $(\cos\theta, \sin\theta)$.

• Example: Find $\sin 150°$
  • $150°$ is in the second quadrant, with a reference angle of $30°$
  • Sine is positive in the second quadrant, so $\sin 150° = \sin 30° = \frac{1}{2}$

Inverse trigonometric functions (arcsin, arccos, arctan) work backwards: given a ratio, they return the angle.

• Example: If $\sin\theta = 0.5$, then $\theta = \arcsin(0.5) = 30°$ (in the first quadrant)

Be careful with inverse trig functions. They return only one value from a restricted range, but there may be additional valid angles depending on the context.

Logarithmic and exponential functions

Exponential functions and properties

An exponential function has the form $f(x) = a^x$, where $a$ is the base and $x$ is the exponent. The two most common bases are $e \approx 2.718$ (the natural exponential) and 10.

These functions grow (or decay) rapidly, which is why they model real-world phenomena like population growth, radioactive decay, and compound interest.

• Example: For $f(x) = 2^x$, plotting from $x = -2$ to $x = 2$ gives points $(-2, 0.25)$, $(-1, 0.5)$, $(0, 1)$, $(1, 2)$, $(2, 4)$. Notice how the output doubles each time $x$ increases by 1.

A common civil engineering application is the compound interest formula: $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate per period, and $t$ is the number of periods. This shows up in project cost estimation and long-term infrastructure planning.

Logarithmic functions and properties

A logarithm is the inverse of an exponential. The expression $y = \log_a(x)$ asks: "What power do you raise $a$ to in order to get $x$?" So $\log_2(8) = 3$ because $2^3 = 8$.

• Natural logarithm ($\ln$) uses base $e$
• Common logarithm ($\log$) uses base 10

The change of base formula lets you convert between bases:

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

• Example: $\log_2(8) = \frac{\ln(8)}{\ln(2)} = \frac{2.079}{0.693} = 3$

Three logarithmic properties you should know:

• Product rule: $\log_a(xy) = \log_a(x) + \log_a(y)$
• Quotient rule: $\log_a(x/y) = \log_a(x) - \log_a(y)$
• Power rule: $\log_a(x^n) = n\log_a(x)$

These let you break apart or combine log expressions.

• Example: Simplify $\log_3(27) - \log_3(9)$
  • Apply the quotient rule: $\log_3(27/9) = \log_3(3) = 1$

To solve exponential equations, take the log of both sides:

• Example: Solve $2^x = 8$
  • Recognize that $8 = 2^3$, so $x = 3$
  • Alternatively, take $\log_2$ of both sides: $x = \log_2(8) = 3$

For equations where the answer isn't obvious, taking the natural log of both sides and using log properties is the standard approach.