Trigonometry
Trigonometry develops a deep understanding of the six trigonometric functions through the lens of the unit circle, explores their graphs and transformations, and builds a powerful toolkit of identities for simplification and equation solving. The course also covers oblique triangle applications using the Law of Sines and Law of Cosines, and concludes with the polar form of complex numbers and De Moivre's Theorem.
Who Should Take This
This course is designed for students who have completed Geometry and Algebra II and need a focused treatment of trigonometric concepts before entering Pre-Calculus or AP Calculus. It is also ideal for college students who need a refresher on circular functions, identity proofs, and triangle solving for physics, engineering, or any quantitative field.
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
13 Activity Formats
Course Outline
1Unit Circle and Radian Measure 7 topics
Describe radian measure as the ratio of arc length to radius and state the conversion factor between degrees and radians, converting angle measures fluently in both directions
Identify the coordinates of all 16 standard angles on the unit circle (multiples of 30 and 45 degrees in all four quadrants) from memory, including the signs of x and y in each quadrant
Calculate arc length s equals r theta and sector area A equals one-half r squared theta for a circle with given radius and central angle in radians, applying dimensional analysis when angles are in degrees
Apply the reference angle concept to find the acute angle formed by a terminal side and the x-axis for any angle in standard position, using it to evaluate trig functions in any quadrant
Calculate linear speed and angular speed in problems involving rotating objects using the relationships v equals r omega and s equals r theta, converting between units as needed
Analyze the periodicity of trigonometric values by explaining why coterminal angles produce identical function values and how this periodicity extends the unit circle to all real number inputs
Solve problems involving the relationship between angle of rotation in degrees and in radians, central angle, radius, arc length, and sector area when any two of these values are given
2Six Trigonometric Functions 7 topics
State the definitions of the six trigonometric functions as right-triangle ratios (SOH-CAH-TOA, cosecant, secant, cotangent) and as unit-circle coordinates (sin is y, cos is x, tan is y over x)
Evaluate the six trigonometric functions at all standard unit-circle angles without a calculator and determine the sign of each function based on the quadrant of the terminal side
Apply the reciprocal, Pythagorean, and quotient identities to find all six trig function values given one value and the quadrant of the angle, without knowing the angle itself
Evaluate trigonometric function values for angles outside the first quadrant using the ASTC (All Students Take Calculus) quadrant-sign rule and reference angle approach
Analyze the domain, range, and undefined values for all six trigonometric functions by identifying where each function's denominator equals zero and where the output is unbounded
Apply even-odd function properties of the trigonometric functions to simplify expressions involving negative angles, stating which functions are even (cosine, secant) and which are odd (sine, tangent, cosecant, cotangent)
Identify cofunction relationships (sin and cos, tan and cot, sec and csc) using the complementary angle identity f(theta) equals co-f(90 minus theta) and apply them to simplify expressions
3Graphs of Trigonometric Functions 8 topics
Identify the key graph features of y equals sine x and y equals cosine x including amplitude, period, x-intercepts, maximum and minimum points, and symmetry type (odd vs. even)
Graph transformations of sine and cosine of the form y equals A sin(Bx minus C) plus D by computing amplitude A, period 2pi over B, phase shift C over B, and vertical shift D
Graph y equals tangent x and y equals cotangent x by identifying period pi, vertical asymptotes, zero crossings, and the direction of increase or decrease on each branch
Graph y equals secant x and y equals cosecant x as reciprocals of cosine and sine respectively, marking vertical asymptotes where the base function equals zero and U-shaped arcs between them
Write the equation of a sinusoidal function from its graph by reading amplitude, period, phase shift, and vertical shift directly from the graph's key features
Analyze real-world periodic phenomena such as tidal heights, temperature cycles, and sound waves by fitting a sinusoidal model and interpreting the parameters in physical terms
Graph transformations of tangent, cotangent, secant, and cosecant by applying horizontal shifts, vertical shifts, and period changes and identify asymptote locations in the transformed graph
Identify the equation of a sinusoidal function in the form y equals A sin(Bx plus C) plus D or cosine equivalent from a verbal description of maximum, minimum, midline, and period
4Trigonometric Identities 8 topics
State the eight fundamental identities: three Pythagorean (sin squared plus cos squared equals one and its two rearrangements), two reciprocal pairs (csc, sec), two quotient (tan, cot), and one negative angle pair
Verify trigonometric identities by transforming one side of the equation into the other using fundamental identities, factoring, combining fractions, or multiplying by a conjugate
Apply the sum and difference formulas for sine, cosine, and tangent to find exact values of trig functions at non-standard angles (such as 15 or 75 degrees) by combining standard angles
Apply the double-angle formulas (three forms for cosine, one for sine, one for tangent) and half-angle formulas to simplify expressions and find exact function values
Apply product-to-sum and sum-to-product identities to rewrite products of sines and cosines as sums and vice versa for use in equation solving and simplification
Analyze multi-step identity verifications that require selecting the non-obvious manipulation path, such as multiplying by a conjugate or converting all functions to sines and cosines first
Evaluate a problem's efficiency by comparing identity proof strategies including working from the more complex side versus both sides, explaining why common algebraic operations are not valid for identity verification
Apply the power-reducing formulas derived from double-angle identities to rewrite sin squared theta and cos squared theta in terms of cosine of twice the angle for use in simplification
5Inverse Trigonometric Functions 7 topics
Describe the restricted domains and ranges of arcsine, arccosine, and arctangent that make each function a valid inverse, and explain why restriction is necessary for a periodic function
Evaluate arcsine, arccosine, and arctangent at standard values by identifying the angle in the restricted range whose trig function equals the given value, without a calculator
Simplify composite expressions such as sin(arctan(x)) and arccos(sin(theta)) by drawing a right triangle or applying cancellation rules, paying careful attention to domain restrictions
Solve right-triangle applications using inverse trig functions to find angles of elevation, depression, and bearing, expressing answers in both degrees and radians
Analyze the domain restrictions on inverse trig compositions and explain when the cancellation rule f-inverse(f(theta)) equals theta holds versus when it does not, with concrete examples
Apply inverse trigonometric functions to write algebraic expressions for all angles satisfying a given right-triangle relationship, distinguishing the principal value from the complete solution set
Calculate the exact value of expressions like tan(arcsin(3/5)) or cos(arctan(x)) by drawing a reference right triangle, labeling the known sides and angle, and computing the required ratio
6Solving Trigonometric Equations 7 topics
Solve basic trigonometric equations of the form sin(theta) equals k on a specified interval by finding all reference angles and applying quadrant signs to list all solutions
Solve trigonometric equations involving multiple angles (such as sin(2x) equals one-half) by first solving for the compound angle argument and then dividing to find x within the specified interval
Solve trigonometric equations that require factoring or using identities to rewrite the equation in terms of a single trig function before finding solutions
Write the general solution to a trigonometric equation by adding all period multiples to the particular solutions, expressing the answer using appropriate k notation for integer multiples of the period
Analyze trigonometric equations solvable by the quadratic formula by substituting u equals sin(x) or cos(x) to create a quadratic in u, then back-substituting to find all angle solutions
Solve trigonometric equations that require applying sum, difference, or double-angle formulas to rewrite the equation in a solvable form before finding all solutions in the specified domain
Evaluate whether a proposed solution to a trigonometric equation is extraneous by substituting back into the original equation and checking whether domain violations introduced by transformations invalidate the root
7Solving Oblique Triangles 8 topics
State the Law of Sines (a over sin A equals b over sin B equals c over sin C) and identify which triangle configurations (AAS, ASA, SSA) call for its application
Apply the Law of Sines to find unknown sides and angles in AAS and ASA triangles, checking that angle sums equal 180 degrees and all side lengths are positive
Analyze the ambiguous case (SSA) by determining whether zero, one, or two triangles exist for given values using the height h equals b sin A as a threshold and solving all valid configurations
State the Law of Cosines (c squared equals a squared plus b squared minus 2ab cos C) and identify which triangle configurations (SAS, SSS) require its use
Apply the Law of Cosines to solve SAS and SSS triangles for all missing parts and verify that the largest angle is opposite the longest side
Calculate the area of a triangle using the formula one-half a b sin C when two sides and the included angle are known, and using Heron's formula when all three sides are known
Solve multi-step navigation, surveying, and force decomposition problems by choosing between the Law of Sines and Law of Cosines based on the given information and interpreting the solution in context
Apply the two oblique triangle formulas for area, one-half base times height and one-half a b sin C, to solve problems where height is not given directly, demonstrating their equivalence
8Polar Form and Complex Numbers 8 topics
Describe the polar form of a complex number r(cos theta plus i sin theta), also written as r cis theta, and explain how modulus r and argument theta relate to the rectangular form a plus bi
Convert complex numbers between rectangular form a plus bi and polar form r cis theta using r equals sqrt(a squared plus b squared) and theta equals arctan(b over a), adjusted for quadrant
Multiply and divide complex numbers in polar form using the rules that moduli multiply or divide and arguments add or subtract, then convert the result back to rectangular form if needed
Apply De Moivre's Theorem to raise a complex number in polar form to an integer power n, computing r to the n times cis(n theta), and expand or simplify the result
Find the nth roots of a complex number using De Moivre's Theorem by computing the n distinct root modulus and equally-spaced root arguments and plotting them on the complex plane
Analyze the geometric interpretation of complex number multiplication in polar form as a rotation and scaling transformation in the complex plane, connecting to trigonometric identities
Convert equations between polar and rectangular form using substitution relationships and graph classic polar curves including the limaçon, cardioid, and rose curve by plotting key points
Compare the rectangular and polar representations of complex numbers, explaining which form is more efficient for multiplication and division versus for addition and subtraction
Scope
Included Topics
- Unit circle (radian and degree measures, coordinates, reference angles), six trigonometric functions (definitions, values at standard angles, reciprocal relationships), radian measure and arc length, graphs of trig functions (period, amplitude, phase shift, vertical shift for sine, cosine, tangent, secant, cosecant, cotangent), Pythagorean identities, reciprocal identities, quotient identities, co-function identities, even-odd identities, sum and difference formulas, double-angle formulas, half-angle formulas, inverse trigonometric functions (arcsine, arccosine, arctangent and their restricted domains), solving trigonometric equations, Law of Sines (including ambiguous case), Law of Cosines, area of a triangle formulas, oblique triangle applications, polar form of complex numbers, De Moivre's Theorem
Not Covered
- Calculus of trigonometric functions (derivatives and integrals)
- Three-dimensional trigonometry beyond basic applications
- Hyperbolic trigonometric functions
- Fourier series and transforms
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