Trigonometry

Course Description

MAC1114 / MAC1114C – Trigonometry (also titled College Trigonometry, Precalculus Trigonometry, or Plane Trigonometry) is a 3-credit lecture course in the Mathematics: Calculus and Pre-Calculus taxonomy of Florida's Statewide Course Numbering System (SCNS). The course serves as a calculus-preparatory course in trigonometry with emphasis on functions. Topics include angular measure (degrees and radians); the unit circle; right-triangle and unit-circle trigonometry; trigonometric (circular) and inverse trigonometric functions and their graphs; trigonometric identities; conditional trigonometric equations; the Law of Sines and Law of Cosines for solving oblique triangles; vectors; complex numbers in trigonometric (polar) form including De Moivre's Theorem; polar coordinates and graphs; and parametric equations and graphs. The use of graphing calculators is incorporated throughout the course.

MAC1114 is part of Florida's state-mandated General Education Core in Mathematics at the institutions where it is offered as a Gen-Ed core course. The course fulfills the Gordon Rule computation requirement (Florida State Board of Education Rule 6A-10.030) and must be completed with a grade of C or higher. Offered at 43 Florida public institutions, MAC1114 transfers as equivalent across the state. The course is required preparation for MAC2311 (Calculus I) and the engineering/STEM calculus sequence; together with MAC1140 (Precalculus Algebra), MAC1114 provides the algebra and trigonometry foundation that engineering, physics, chemistry, computer science, and mathematics majors need before entering MAC2311.

Learning Outcomes

Required Outcomes

Upon successful completion of this course, students will be able to:

• Convert between degree and radian measure; compute arc length and sector area; analyze angular and linear velocity.
• Apply right-triangle trigonometry to compute side lengths and angles using the six trigonometric ratios; solve real-world applications including angles of elevation and depression.
• Use the unit circle to evaluate trigonometric functions for angles in standard position, including reference angles in all four quadrants.
• Define and evaluate the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) and recognize their reciprocal, quotient, and Pythagorean identities.
• Sketch and analyze graphs of trigonometric functions, identifying amplitude, period, phase shift, vertical shift, and asymptotes.
• Define and evaluate inverse trigonometric functions; identify their domains and ranges; sketch their graphs.
• Verify and apply trigonometric identities: fundamental identities, sum and difference identities, double-angle and half-angle identities, product-to-sum and sum-to-product identities.
• Solve conditional trigonometric equations on specified intervals or over the real numbers, including equations involving multiple angles, factoring, and identity substitution.
• Apply the Law of Sines and the Law of Cosines to solve oblique triangles, including the ambiguous case (SSA), and compute triangle areas.
• Perform basic operations on vectors in two dimensions: addition, scalar multiplication, computing magnitude and direction, finding components, and applying the dot product to compute angles between vectors and projections.
• Express complex numbers in trigonometric (polar) form; multiply, divide, and find powers and roots using De Moivre's Theorem.
• Plot points and graph equations using polar coordinates; convert between rectangular and polar coordinates and equations; recognize common polar curves (circles, cardioids, limaçons, rose curves).
• Eliminate the parameter and graph equations given in parametric form; convert between parametric and rectangular forms.
• Use a graphing calculator (TI-84 or equivalent) effectively for trigonometric function evaluation, graphing, and numerical solution of equations.
• Apply trigonometry to real-world problems in physics, engineering, navigation, surveying, and astronomy.

Optional Outcomes

Depending on institutional emphasis, students may also:

• Apply trigonometric functions to simple harmonic motion and oscillatory phenomena.
• Apply three-dimensional vector concepts (introductory) and the cross product.
• Examine conic sections in polar form (focus-directrix definition).
• Use Desmos, GeoGebra, Wolfram Alpha, or Python (with libraries like NumPy/Matplotlib) to visualize trigonometric functions and verify computations.
• Apply trigonometric concepts to navigation problems using bearings.

Major Topics

Required Topics

• Angles and Their Measure: Angle definition; standard position; degree and radian measure; conversion between degrees and radians; coterminal angles; complementary and supplementary angles; arc length; area of a circular sector; angular and linear velocity.
• Right-Triangle Trigonometry: Six trigonometric ratios (SOH-CAH-TOA); special right triangles (30-60-90, 45-45-90); applications including angles of elevation and depression; solving right triangles.
• The Unit Circle and Circular Functions: Defining trigonometric functions on the unit circle; trigonometric values for special angles (multiples of 30, 45, 60, 90); reference angles; signs of trigonometric functions in each quadrant; periodicity.
• Graphs of Trigonometric Functions: Graphs of y = sin x, y = cos x, y = tan x, y = csc x, y = sec x, y = cot x; transformations: amplitude, period, phase shift, vertical shift; sketching y = A sin(Bx - C) + D and similar.
• Inverse Trigonometric Functions: Definition and domain restrictions for inverse sine, cosine, tangent (and inverse cosecant, secant, cotangent); graphs; evaluation; composition of trigonometric and inverse trigonometric functions.
• Trigonometric Identities: Fundamental (reciprocal, quotient, Pythagorean) identities; sum and difference identities for sine, cosine, tangent; double-angle identities; half-angle identities; product-to-sum and sum-to-product identities; verifying identities algebraically.
• Trigonometric Equations: Solving basic conditional equations; solving on a specified interval; solving equations with multiple angles; using identities to simplify before solving; equations involving inverse functions.
• Solving Triangles (Oblique): Law of Sines (including the ambiguous case SSA — no triangle, one triangle, two triangles); Law of Cosines (SSS, SAS); area of a triangle using trigonometry (1/2 ab sin C and Heron's formula); applications.
• Vectors: Vector definition; geometric and component representations; vector addition and scalar multiplication; magnitude and direction; unit vectors; the dot product; angle between vectors; vector projections; applications to forces and velocities.
• Complex Numbers in Trigonometric Form: Rectangular form (a + bi); trigonometric (polar) form r(cos θ + i sin θ); multiplication and division in polar form; De Moivre's Theorem for powers; finding nth roots of complex numbers.
• Polar Coordinates and Graphs: Polar coordinate system; converting between polar and rectangular coordinates; graphing polar equations; common polar curves (circles, cardioids, limaçons, rose curves, lemniscates); symmetry tests.
• Parametric Equations: Definition of parametric equations; eliminating the parameter; converting between parametric and rectangular forms; sketching parametric curves; basic applications (projectile motion, cycloids).

Optional Topics

• Simple Harmonic Motion: Modeling oscillating systems with sine and cosine functions; amplitude, period, frequency.
• 3D Vectors and the Cross Product (Introduction): Vectors in space; the cross product; applications.
• Conic Sections in Polar Form: Focus-directrix definitions; eccentricity.
• Computational Tools: Using Desmos and GeoGebra for visualization; introductory Python (NumPy, Matplotlib) for plotting trigonometric functions and curves.
• Navigation and Surveying: Bearings; relative motion; applications to land surveying.

Resources & Tools

• Standard Textbooks: Trigonometry by Lial, Hornsby, Schneider, and Daniels (Pearson — widely adopted in Florida); Algebra and Trigonometry by Stewart, Redlin, and Watson (Cengage); Trigonometry by Sullivan (Pearson); Trigonometry by Aufmann and Nation (Cengage); OpenStax Algebra and Trigonometry 2e (free, open-access at openstax.org); Precalculus by Stitz and Zeager (free, open-access).
• Online Homework Platforms: Pearson MyLab Math (most common); WebAssign (Cengage); Hawkes Learning; ALEKS
• Required Calculator: Texas Instruments TI-84 Plus or TI-84 Plus CE graphing calculator at most institutions. TI-89 and TI-Nspire CX CAS may be allowed in some contexts but are often prohibited on exams. Casio fx-9750GIII is a less expensive alternative accepted at many institutions.
• Free Online Tools: Desmos (desmos.com — exceptional for graphing trigonometric functions, polar curves, and parametric equations); GeoGebra (geogebra.org); Wolfram Alpha for symbolic verification; Symbolab for step-by-step solutions; UnitCircleGame (interactive unit-circle drilling).
• Tutoring Resources: Free college tutoring centers; Khan Academy Trigonometry (free, comprehensive); Paul's Online Math Notes (tutorial.math.lamar.edu); Professor Leonard's Trigonometry video series; Patrick Just Math Tutorials.
• State Resources: Florida General Education Core Mathematics outcomes; Florida SCNS course descriptions; published syllabi from UF, FSU, USF, UCF, FIU, and other Florida public institutions (often available via the institution's math department web site).

Career Pathways

MAC1114 is foundational for STEM, engineering, and analytical pathways:

• Associate in Arts (A.A.) Transfer Pathway – Required Gen-Ed mathematics course satisfying the math core for transfer to all Florida public universities (where designated as a Gen-Ed core option).
• Engineering, Physics, Chemistry, Computer Science, Mathematics Majors – Required preparation for MAC2311 (Calculus I) at all Florida public universities. Together with MAC1140 (Precalculus Algebra), MAC1114 forms the standard precalculus preparation. UF, FSU, USF, UCF, FIU, FAU, FAMU, UNF, and FGCU all require MAC2311 (which requires this trigonometry preparation) for engineering, physics, chemistry, computer science, mathematics, and chemical/biological engineering majors.
• Architecture, Surveying, Construction Management – Required or recommended for these technical pathways.
• Workforce Application – Trigonometry-based reasoning supports Florida's surveying, construction, manufacturing (especially aerospace and defense at Lockheed Martin, Boeing, L3Harris), navigation (commercial maritime industry, aviation), and engineering technology sectors.
• Alternative Path: MAC1147 – Some students choose MAC1147 (Precalculus Algebra and Trigonometry) as a single-semester combined course covering MAC1140 + MAC1114 content. MAC1147 is more demanding (4-5 credits, faster pace) but saves a semester.

Special Information

Gen-Ed Core and Gordon Rule

MAC1114 satisfies Florida's General Education Core Mathematics requirement at institutions where it is so designated, and the Gordon Rule computation requirement (Florida State Board of Education Rule 6A-10.030). Students must earn a grade of C or better for the course to satisfy these requirements.

Prerequisite

Students must demonstrate readiness through one of the following: minimum grade of C in MAC1140 (Precalculus Algebra), MAC1105 (College Algebra), or MAC1105C; or qualifying placement score (SAT Math, ACT Math, PERT, or ALEKS PPL). A strong high-school algebra background is recommended; students who struggled in MAC1105 should consider taking MAC1140 before attempting MAC1114.

Course Variants and Sequence

MAC1114 is offered as MAC1114 (3 credits, lecture only) and MAC1114C (with integrated lab/recitation). The two forms are equivalent for transfer and Gen-Ed credit. The standard precalculus sequence is MAC1140 (Precalculus Algebra) followed by MAC1114 (Trigonometry), though many students take MAC1114 concurrently with or before MAC1140 depending on placement and program. The accelerated alternative is MAC1147 (Precalculus Algebra and Trigonometry) as a single 4-5 credit combined course.

MAC1114 vs. MAC2233

Students should note: MAC1114 is required for the engineering/STEM calculus sequence MAC2311/MAC2312/MAC2313. Students taking MAC2233 (Calculus for Business and Social Sciences) typically do not need MAC1114, since MAC2233 does not include trigonometric calculus. Confirm the requirement with your academic advisor based on intended major.

Workload and Time Expectations

Most institutions expect 6-9 hours of weekly out-of-class work, including online homework completion (MyLab Math or equivalent) and practice problem sets. Most courses include 3-4 mid-term examinations plus a comprehensive final examination. Memorizing the unit circle, common identities, and special-angle values is essential and should not be deferred — these are foundational for every subsequent topic and for calculus.

Foundation for Calculus

MAC1114 is the direct preparation for the trigonometric portions of MAC2311 (Calculus I), including derivatives and integrals of trigonometric and inverse trigonometric functions, trigonometric substitution (in MAC2312), and applications to physics. Strong fluency with identities, the unit circle, and inverse trigonometric functions is essential — students who cannot quickly recall sin(π/3) or use a Pythagorean identity will struggle in calculus.