Unit notes

Radians measure angle by arc length divided by radius. On the unit circle, cosine is the horizontal coordinate and sine the vertical coordinate. A trigonometric equation can have several solutions in a stated interval, and infinitely many when no interval restriction is imposed.

A radian measure is arc length divided by radius. One full turn is 2π radians, equivalent to 360°. Using degrees in a formula that assumes radians creates a scale error. The unit circle gives cosθ as the horizontal coordinate and sinθ as the vertical coordinate of the point at angle θ.

Reference angles and quadrant signs extend exact values beyond the first quadrant. Tangent is sinθcosθ where the denominator is nonzero. Periodicity means many angles can produce the same function value, so equation solutions must be matched to the requested interval.

For y = A sin(B(x − C)) + D, the amplitude is |A|, the midline is D, and the period is 2π/|B| when B ≠ 0. C is a horizontal shift in this factored form. A negative A reflects the output relative to the midline.

A sinusoidal model can represent a tide, seasonal quantity, or rotating point when a repeating smooth cycle is appropriate. Determine maximum, minimum, period, and a phase reference from context. The first visible peak is not automatically at x = 0, and a measured phase should be checked by substitution.

Use identities or algebra to isolate a basic trig expression, then find all solutions in the domain. A calculator inverse-trig output gives a principal value, not every angle. Symmetry and periodicity generate other solutions. State whether endpoints are included.

Inverse sine and cosine use restricted ranges so they are functions. Consequently, arcsin(sinθ) does not always equal θ for every real θ. Choose the equivalent angle in the inverse function’s principal range. When dividing by a trig expression, check whether zeros of that expression would be lost solutions.

The Pythagorean identity, reciprocal relationships, and angle-sum identities help rewrite expressions and solve equations. An identity is true throughout its common domain, while an equation may hold only for selected inputs. Simplifying an expression must preserve domain restrictions.

Choose a transformation based on the target. Rewriting tan as sin/cos can expose a common denominator; replacing sin² with 1 − cos² can turn a problem into a quadratic in cosine. Verify candidates in the original equation after operations that may add or remove solutions.

A polar point (r,θ) locates a point along the direction θ at signed distance r. A negative radius places the point in the opposite direction, equivalent to adding π to the angle and using positive radius. Many polar pairs represent the same Cartesian point.

Convert using x = r cosθ and y = r sinθ. Recovering an angle from x and y requires quadrant awareness; arctan(yx) alone can select the wrong quadrant. At the origin, angle is not uniquely determined. A polar function r = f(θ) describes radius as angle changes, not a standard Cartesian y-versus-x graph.

A positive change in r does not always mean the point moves farther from the origin if r is negative; distance is |r|. Zeros of r identify passages through the pole. Symmetry and repeated values can help sketch a curve, but test points are useful when the sign changes.

Average rate of change of r with respect to θ has units of distance per radian when radius has distance units. A polar graph and the rectangular graph of r versus θ show different information. The latter makes radial rate patterns easier to inspect; the former shows the geometric path.

For A sin(B(t−h))+D, the midline is D, amplitude is |A|, and period is 2π/|B| in radian measure. Find the midline by averaging maximum and minimum outputs, and find amplitude as half their difference. A complete cycle must return to the same phase, not merely the same output. A rising midline crossing and a falling midline crossing are half a cycle apart.

Choose sine or cosine according to a convenient landmark. Cosine is useful when the data start at a maximum; a negative cosine can start at a minimum. State units for time and output and restrict the model to a sensible interval. Seasonal or mechanical data are often only approximately sinusoidal, so compare the model with observations rather than assuming periodicity guarantees a perfect sine curve.

Isolate the trigonometric expression when possible, find reference angles, and use quadrant signs to locate all angles in the required interval. An inverse-trigonometric calculator value gives a principal value, not automatically every solution. If the argument is 2x or x−h, first solve for that full argument over its corresponding transformed interval.

For an identity-based equation, factor or use an appropriate identity before dividing. Dividing by sin x can lose solutions where sin x=0. Check candidate values in the original equation, especially after squaring. Exact special-angle values should remain exact when requested; round only at the end of a calculator-based calculation.

In polar coordinates, r is a signed distance and θ is an angle. The same point has multiple representations: adding 2π to the angle leaves the point unchanged, and changing the sign of r while adding π also gives the same point. Convert to Cartesian coordinates with x=r cosθ and y=r sinθ when that clarifies a location.

For a polar function r=f(θ), a table lists radius outputs for angle inputs. A positive change in r does not always mean increasing distance from the origin, because distance is |r|. If r rises from −5 to −2, the point moves closer to the origin in radial distance. Analyze zeros, signs, symmetry, and repeated patterns while distinguishing the graph in polar space from a graph of radius versus angle.

On the unit circle, cos θ is the horizontal coordinate and sin θ is the vertical coordinate. Signs follow the quadrant. Radians measure angle as arc length divided by radius, so one full turn is 2π radians. A calculator in the wrong mode can produce plausible-looking but incorrect results.

Reference angles connect quadrant signs with familiar exact values. Tangent is sin θcos θ where cosine is nonzero. Reciprocal functions inherit their own restrictions. An identity is valid wherever both sides are defined; simplifying an expression does not automatically remove original domain exclusions.

The identity sin²θ+cos²θ=1 follows the unit-circle equation. Use it to replace a squared function when solving or simplifying, while remembering that taking a square root introduces a sign choice. Choose that sign from the quadrant or stated interval, not from habit.

For y=D+A sin(B(x−C)), the midline is D, amplitude is |A|, and period is 2π/|B| in radians. Determine maximum and minimum first: their average gives the midline and half their difference gives amplitude. Then use a full cycle to determine period.

Phase is best anchored to a known event, such as a maximum or an upward midline crossing. Several equivalent sine and cosine equations may represent the same graph. Check at least two characteristic points instead of assuming one symbolic form is uniquely correct.

Context restricts the model. A temperature cycle may be approximately periodic but change with weather; a rotating point has a more controlled geometry. State what the input and output measure and use their units. Amplitude is not the same as maximum output unless the midline is zero.

Inverse trigonometric functions return principal values because unrestricted trigonometric functions are not one-to-one. Solving sin x=c over a full cycle usually requires another angle in addition to arcsin(c). Use symmetry and periodicity to find all solutions in the specified interval.

For equations involving identities, rewrite into a manageable form, solve algebraically, and then recover angles. Dividing by sin x or cos x can discard solutions where that factor is zero. Factoring and treating each factor separately is often safer.

Check endpoints and units. An interval may include one endpoint but exclude another, and equivalent angles can be double-counted. A numerical root finder can miss additional roots; use the graph and known period to organize the search.

A polar point uses a directed distance r and angle θ. Negative r places the point opposite the ray for θ, so one location has multiple coordinate descriptions. Convert with x=r cos θ and y=r sin θ when Cartesian coordinates make a question easier.

For a polar function r=f(θ), sample angles carefully and use symmetry to understand the curve. A negative output changes the plotted direction rather than creating a negative physical distance. Zeros indicate passage through the pole, but several angles can represent that same point.

Distinguish change in radial distance from movement along the curve. A point can rotate while its radius stays constant. The assessed precalculus work focuses on interpreting and analyzing these functions; area accumulation and arc-length integrals belong to later calculus work.