Trigonometry Review for the FE Exam
Trigonometry is woven throughout the FE exam. You will use it in statics (resolving forces into components), dynamics (projectile motion), surveying (angle calculations), circuit analysis (AC circuits), and many other topics. A quick, confident command of trigonometric concepts saves time on nearly every section of the exam.
Right Triangle Trigonometry
The foundation of trigonometry is the right triangle. For a right triangle with angle theta, opposite side (opp), adjacent side (adj), and hypotenuse (hyp):
- sin(theta) = opp / hyp
- cos(theta) = adj / hyp
- tan(theta) = opp / adj = sin(theta) / cos(theta)
Remember the mnemonic SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
The Unit Circle
The unit circle extends trigonometry beyond right triangles. Key values to know from memory:
- sin(0) = 0, cos(0) = 1
- sin(30) = 1/2, cos(30) = sqrt(3)/2
- sin(45) = sqrt(2)/2, cos(45) = sqrt(2)/2
- sin(60) = sqrt(3)/2, cos(60) = 1/2
- sin(90) = 1, cos(90) = 0
These values are in the reference handbook, but knowing them saves lookup time on problems that use standard angles.
Essential Trigonometric Identities
Pythagorean Identities
- sin^2(theta) + cos^2(theta) = 1
- 1 + tan^2(theta) = sec^2(theta)
- 1 + cot^2(theta) = csc^2(theta)
Double Angle Formulas
- 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)
Sum and Difference Formulas
- sin(A +/- B) = sin(A)*cos(B) +/- cos(A)*sin(B)
- cos(A +/- B) = cos(A)*cos(B) -/+ sin(A)*sin(B)
Law of Sines
For any triangle (not just right triangles) with sides a, b, c opposite to angles A, B, C:
a/sin(A) = b/sin(B) = c/sin(C)
Use the law of sines when you know two angles and one side, or two sides and an angle opposite one of them.
Law of Cosines
For any triangle:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
Use the law of cosines when you know two sides and the included angle, or all three sides and need to find an angle. This is a generalization of the Pythagorean theorem (when C = 90 degrees, the last term drops out).
Inverse Trigonometric Functions
Inverse trig functions find angles from known ratios:
- arcsin (or sin^(-1)): Given sin(theta), find theta
- arccos (or cos^(-1)): Given cos(theta), find theta
- arctan (or tan^(-1)): Given tan(theta), find theta
Be aware of the principal value ranges: arcsin returns angles from -90 to 90 degrees, arccos returns 0 to 180 degrees, and arctan returns -90 to 90 degrees. You may need to adjust the result based on the quadrant of the actual angle.
Applications in Engineering
Force Resolution
Resolving forces into x and y components is one of the most common uses of trigonometry on the FE exam:
- F_x = F * cos(theta)
- F_y = F * sin(theta)
Where theta is the angle the force makes with the x-axis. This is the basis of every statics problem.
Surveying and Distances
Trigonometry is used to calculate distances, heights, and angles in surveying problems. Typical applications include finding the height of a building from angle of elevation measurements and calculating distances across rivers or valleys.
AC Circuit Analysis
Alternating current circuits use sinusoidal functions to represent voltage and current: v(t) = V_max * sin(omega*t + phi), where omega is the angular frequency and phi is the phase angle. Understanding phase relationships between voltage and current requires comfort with trigonometric functions.
Phasor Representation
In electrical engineering, phasors represent sinusoidal quantities as complex numbers or vectors. Converting between rectangular (a + jb) and polar (r at angle theta) forms requires trigonometry:
- r = sqrt(a^2 + b^2)
- theta = arctan(b/a)
- a = r * cos(theta)
- b = r * sin(theta)
Radians vs Degrees
Both systems appear on the FE exam. Key conversions:
- 180 degrees = pi radians
- To convert degrees to radians: multiply by pi/180
- To convert radians to degrees: multiply by 180/pi
Most scientific and engineering formulas use radians. Always check your calculator's angle mode before computing trigonometric functions.
Practice Tips
- Practice resolving forces into components until it is automatic
- Memorize the key unit circle values for speed
- Know when to use law of sines vs law of cosines
- Always verify your calculator is in the correct angle mode (degrees or radians)
- Use trigonometric identities to simplify expressions before calculating
Trigonometry is a tool you will use on dozens of FE exam questions. The faster and more accurately you can apply these concepts, the more time you save for the tougher engineering problems.