FE Mathematics: Algebra and Trigonometry Review

Success on the FE exam's mathematics section isn't just about knowing formulas—it's about executing core algebraic and trigonometric manipulations with speed and accuracy under pressure. This review distills the essential concepts you must command, focusing on the rapid problem-solving techniques and common question patterns you will encounter. Mastering these foundations frees up mental bandwidth for the more complex engineering problems ahead.

Core Algebraic Manipulation and Solving

Algebraic manipulation is the process of rearranging equations and expressions to isolate a desired variable or simplify the form. On the FE exam, this often appears in problems where you are given a formula from another discipline (like thermodynamics or statics) and must solve for an unknown. The key is systematic, error-free execution. Always perform the same operation on both sides of an equation, and combine like terms carefully. A frequent task involves clearing fractions by multiplying both sides by the least common denominator, or expanding factored terms.

Factoring is the reverse of expansion and is crucial for simplifying expressions and solving equations set to zero. Look for common factors first. Remember the special forms: $$a^2-b^2=(a-b)(a+b)$$ $$a^2\pm 2ab+b^2=(a\pm b{)}^2$$ For a quadratic trinomial $a{x}^2+bx+c$, you need to find two numbers that multiply to $ac$ and sum to $b$. If factoring seems complex, the quadratic formula is your reliable alternative. The quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ solves any equation of the form $a{x}^2+bx+c=0$. The expression under the radical, $b^2-4ac$, is the discriminant. It tells you the nature of the roots: positive (two real roots), zero (one real double root), or negative (two complex conjugate roots). On the exam, be prepared to use the formula quickly and interpret the discriminant's meaning in an applied context.

Exponential and Logarithmic Functions

Exponential functions have the form $y=a\cdot b^{kx}$, where $b>0$ and $b\ne 1$. Their inverse functions are logarithmic functions, written as $y={\log }_b(x)$. The most common bases are 10 (common log, $\log$) and $e$ (natural log, $\ln$). You must be fluent in the fundamental properties that allow you to manipulate equations:

• Product Rule: ${\log }_b(MN)={\log }_b(M)+{\log }_b(N)$
• Quotient Rule: ${\log }_b(M/N)={\log }_b(M)-{\log }_b(N)$
• Power Rule: ${\log }_b(M^p)=p\cdot {\log }_b(M)$
• Change of Base: ${\log }_b(x)=\frac{{\log }_a(x)}{{\log }_a(b)}$

A classic FE question pattern gives you an equation like $3^{2x-1}=7$ and asks you to solve for $x$. The strategy is to take the logarithm (natural or common) of both sides, apply the power rule to bring the exponent down, and then solve the resulting linear equation: $(2x-1)\ln (3)=\ln (7)$.

Foundational Trigonometric Identities

Trigonometry on the FE exam heavily tests your ability to use identities to simplify expressions or prove equivalences. You must have these core identities memorized:

Pythagorean Identities: $${\sin }^2\theta +{\cos }^2\theta =1$$ $$1+{\tan }^2\theta ={\sec }^2\theta$$ $$1+{\cot }^2\theta ={\csc }^2\theta$$

Angle Sum and Difference Identities: $$\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$$ $$\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$$

Double-Angle Identities: $$\sin (2\theta )=2\sin \theta \cos \theta$$ $$\cos (2\theta )={\cos }^2\theta -{\sin }^2\theta =2{\cos }^2\theta -1=1-2{\sin }^2\theta$$

The exam often presents a complex expression that can be simplified to a single trig function or a constant. Your first step should always be to rewrite all functions in terms of sine and cosine, then apply algebraic manipulation and the identities above. Recognizing the forms of the Pythagorean identities is particularly powerful.

Laws of Sines and Cosines

These laws are your tools for solving triangles that are not right triangles. The Law of Sines relates sides and their opposite angles: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ It is most useful when you know:

• Two angles and any side (AAS or ASA), or
• Two sides and an angle opposite one of them (SSA – the ambiguous case).

The Law of Cosines is essentially a generalized Pythagorean theorem: $$a^2=b^2+c^2-2bc\cos A$$ $$b^2=a^2+c^2-2ac\cos B$$ $$c^2=a^2+b^2-2ab\cos C$$ Use it when you know:

• Three sides (SSS), or
• Two sides and the included angle (SAS).

For problem-solving, a quick flowchart works: Is it a right triangle? Use SOH-CAH-TOA. If not, do you know SAS or SSS? Use Law of Cosines. Do you know AAS, ASA, or are trying SSA? Start with Law of Sines. Expect questions involving vector components, force resolution, or surveying distances, where these laws are applied directly.

Inverse Trigonometric Functions and Their Domains

Inverse trigonometric functions (e.g., $\arcsin (x)$, $\arccos (x)$, $\arctan (x)$) answer the question: "What angle has this sine, cosine, or tangent value?" They are crucial for solving equations like $\sin (\theta )=0.5$ for $\theta$. However, because trig functions are periodic, their inverses must be restricted to principal values to be functions. You must know these standard restricted ranges:

• $\arcsin (x)$ returns an angle in $[-\pi /2,\pi /2]$ or $[-90^{\circ },90^{\circ }]$.
• $\arccos (x)$ returns an angle in $[0,\pi ]$ or $[0^{\circ },180^{\circ }]$.
• $\arctan (x)$ returns an angle in $(-\pi /2,\pi /2)$ or $(-90^{\circ },90^{\circ })$.

An FE exam trap is asking for the value of $\arccos (-\frac{1}{2})$. The answer is $120^{\circ }$ or $2\pi /3$, not $-120^{\circ }$, because the range of $\arccos$ is $[0,\pi ]$. Always check that your calculator is in the correct mode (degrees vs. radians) as specified by the problem.

Common Pitfalls

1. Sign Errors in Algebraic Manipulation: When moving terms across an equals sign or distributing a negative sign, errors are common. Correction: Work methodically, write each step, and consider plugging your answer back into the original equation to verify.

1. Misapplying Logarithm Rules: A frequent mistake is treating ${\log }_b(M+N)$ as if it were ${\log }_b(M)+{\log }_b(N)$. Correction: Remember, logs convert multiplication into addition ($\log (MN)=\log M+\log N$), but there is no identity for the log of a sum. You cannot simplify $\ln (x+1)$.

1. Forgetting the Ambiguous Case (SSA): When using the Law of Sines with two sides and a non-included angle, zero, one, or two valid triangles may exist. Correction: Always check for the possibility of an obtuse solution. If the given angle is acute and the side opposite it is shorter than the adjacent side, calculate the altitude ($b\sin A$) and compare to the given side to determine the number of triangles.

1. Ignoring Domain/Range of Inverses: Selecting an angle outside the principal range for an inverse trig function is a guaranteed wrong answer. Correction: Memorize the ranges. For $\arcsin$ and $\arctan$, think "Quadrants I and IV." For $\arccos$, think "Quadrants I and II."

Summary

• Algebraic fluency is non-negotiable. Be able to factor, expand, and apply the quadratic formula without hesitation. The discriminant reveals root behavior.
• Logarithms are exponents. Use the product, quotient, and power rules to solve exponential equations by "taking the log of both sides."
• Trig identities are for simplification. Start by converting to sines and cosines, then strategically apply Pythagorean and angle-sum identities to condense expressions.
• Solve non-right triangles systematically. Use Law of Cosines for SAS/SSS and Law of Sines for AAS/ASA. Be vigilant for the ambiguous SSA case.
• Inverse trig functions have restricted outputs. $\arcsin$ and $\arctan$ output angles between $-90^{\circ }$ and $90^{\circ }$; $\arccos$ outputs angles between $0^{\circ }$ and $180^{\circ }$. This is critical for getting unique answers.