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United StatesPrecalculus

Unit 3: Trigonometric and Polar Functions

15 dot points across 15 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do trigonometric identities let you rewrite an expression in an equivalent form?

Topic 3.12 Equivalent Representations of Trigonometric Functions: use the Pythagorean, sum and difference, and double-angle identities to rewrite trigonometric expressions in equivalent forms.
A focused answer to AP Precalculus Topic 3.12, covering the Pythagorean identities, the sum and difference formulas, and the double-angle formulas, and how to use them to rewrite trigonometric expressions and verify identities.
9 min answer →

How are the inverse trigonometric functions defined on restricted domains, and what are their ranges?

Topic 3.9 Inverse Trigonometric Functions: define arcsine, arccosine and arctangent on restricted domains, and evaluate and interpret their outputs.
A focused answer to AP Precalculus Topic 3.9, covering why trig functions must be domain-restricted to have inverses, the ranges of arcsine, arccosine and arctangent, and how to evaluate and interpret inverse trig values.
9 min answer →

What makes a relationship periodic, and how do you read its period, amplitude and midline from a graph or context?

Topic 3.1 Periodic Phenomena: identify a periodic relationship, and describe its period, amplitude and key features from a graph, table or context.
A focused answer to AP Precalculus Topic 3.1, covering what makes a relationship periodic, how to read period, amplitude and midline from a graph, table or context, and how concavity changes within a cycle.
9 min answer →

What are polar coordinates, and how do you convert between polar and rectangular form?

Topic 3.13 Trigonometry and Polar Coordinates: locate points using polar coordinates and convert between polar and rectangular coordinates.
A focused answer to AP Precalculus Topic 3.13, covering the polar coordinate system, how a point is named by radius and angle, the conversion formulas between polar and rectangular coordinates, and why polar names are not unique.
9 min answer →

How do you graph a polar function r = f(theta), and what shapes do the standard polar functions produce?

Topic 3.14 Polar Function Graphs: construct and interpret the graph of a polar function r = f(theta), including circles, roses, limacons and spirals.
A focused answer to AP Precalculus Topic 3.14, covering how to graph a polar function r = f(theta) by plotting radius against angle, the standard polar shapes (circles, roses, limacons, spirals), and how the sign of r affects the graph.
9 min answer →

How does the radius of a polar function change as the angle increases, and what does the average rate of change tell you?

Topic 3.15 Rates of Change in Polar Functions: analyze how r changes as theta increases, using the average rate of change to describe whether the curve moves toward or away from the pole.
A focused answer to AP Precalculus Topic 3.15, covering how the radius of a polar function changes with the angle, the average rate of change of r with respect to theta, and how its sign tells you whether the curve approaches or leaves the pole.
9 min answer →

What do the graphs of sine and cosine look like, and how do their period, amplitude, midline and concavity arise from the unit circle?

Topic 3.4 Sine and Cosine Function Graphs: construct and interpret the graphs of sine and cosine, identifying period, amplitude, midline, zeros and concavity.
A focused answer to AP Precalculus Topic 3.4, covering how the sine and cosine graphs are generated from the unit circle, their period, amplitude, midline, zeros, maxima and minima, and concavity within a cycle.
9 min answer →

How do sine and cosine values move around the unit circle, and how do symmetry and the Pythagorean identity connect them?

Topic 3.3 Sine and Cosine Function Values: determine sine and cosine values using the unit circle, reference angles, symmetry and the Pythagorean identity.
A focused answer to AP Precalculus Topic 3.3, covering how sine and cosine values are generated around the unit circle, reference angles, even-odd symmetry, coterminal angles, and the Pythagorean identity.
9 min answer →

How are sine, cosine and tangent defined on the unit circle, and how do they relate to right-triangle ratios?

Topic 3.2 Sine, Cosine, and Tangent: define sine, cosine and tangent using the unit circle and right-triangle ratios, and evaluate them at key angles.
A focused answer to AP Precalculus Topic 3.2, covering the unit-circle definitions of sine, cosine and tangent, their link to right-triangle ratios, radian measure, and evaluating them at the special angles.
9 min answer →

How do you build a sinusoidal model from a periodic context or data set, and how do you interpret it?

Topic 3.7 Sinusoidal Function Context and Data Modeling: construct a sinusoidal model from a periodic context or data, and use it to make and interpret predictions.
A focused answer to AP Precalculus Topic 3.7, covering how to build a sinusoidal model from a periodic context, how sinusoidal regression fits data, and how to interpret the amplitude, period, midline and phase in context.
9 min answer →

How does each transformation of a sinusoid change its graph, and how do you combine them?

Topic 3.6 Sinusoidal Function Transformations: describe how changing each parameter transforms a sinusoid, and combine vertical and horizontal stretches, reflections and shifts.
A focused answer to AP Precalculus Topic 3.6, covering how each of the four sinusoidal parameters transforms the graph, how vertical and horizontal changes combine, and how to read a transformed sinusoid back into its equation.
9 min answer →

What is the general form of a sinusoidal function, and how do its four parameters control the graph?

Topic 3.5 Sinusoidal Functions: write a sinusoidal function in the form a*sin(b(x - c)) + d (or with cosine) and relate amplitude, period, phase shift and vertical shift to the parameters.
A focused answer to AP Precalculus Topic 3.5, covering the general sinusoidal form, how amplitude, period, phase shift and vertical shift map to the parameters a, b, c and d, and how to build a sinusoid from its features.
9 min answer →

How are secant, cosecant and cotangent defined as reciprocals, and where are their asymptotes and ranges?

Topic 3.11 The Secant, Cosecant, and Cotangent Functions: define the reciprocal trigonometric functions and identify their periods, asymptotes, ranges and graphs.
A focused answer to AP Precalculus Topic 3.11, covering the reciprocal definitions of secant, cosecant and cotangent, where each has vertical asymptotes, their periods and ranges, and how their graphs relate to sine, cosine and tangent.
9 min answer →

What does the tangent function look like, and where are its period, asymptotes and zeros?

Topic 3.8 The Tangent Function: define the tangent function, graph it, and identify its period, vertical asymptotes, zeros and behavior between asymptotes.
A focused answer to AP Precalculus Topic 3.8, covering the definition of tangent as sine over cosine, its graph, period of pi, vertical asymptotes where cosine is zero, zeros where sine is zero, and its increasing behavior between asymptotes.
9 min answer →

How do you solve a trigonometric equation or inequality and find all of its solutions?

Topic 3.10 Trigonometric Equations and Inequalities: solve trigonometric equations and inequalities, using inverse functions, symmetry and periodicity to find all solutions.
A focused answer to AP Precalculus Topic 3.10, covering how to solve trigonometric equations using inverse functions, how unit-circle symmetry gives a second solution per cycle, how periodicity generates all solutions, and how to solve trig inequalities.
9 min answer →