Become a Trigonometry & Precalculus Master

Explore angles using a pizza analogy to grasp angle measures, from 45 degrees to a full circle of 360 degrees, and learn names based on measure.

Define an angle as a wedge of a circle bounded by two rays at a vertex, measured in degrees or radians, with theta used to name and classify angles.

Determine an angle’s quadrant from its terminal side in standard position, using positive or negative rotations and degree-radian conversions with examples like 400 degrees and 10 radians.

Learn how degrees, minutes, and seconds (DMX) relate to radians and degrees, and master conversions via a degrees intermediary, including 1 degree = 60 minutes and 3600 seconds.

The unit circle defines sine as y and cosine as x; tangent equals sine over cosine. Use quotient identities to find tangent and cotangent from sine and cosine.

Derive the three Pythagorean identities from the unit circle and the Pythagorean theorem, linking sine, cosine, tangent, secant, and cosecant; show how dividing by sine^2 or cosine^2 yields the others.

Explore how the unit circle and right triangle determine signs of the six trig functions across quadrants, and derive all six via one value using pythagorean and reciprocal identities.

Angles on the major axes render sine and cosine values that cause tangent, cotangent, secant, and cosecant to be undefined where the denominators vanish.

Explore the unit circle, the circle of radius one at the origin, and learn the common angles in degrees and radians with their coordinate points for trigonometry, precalculus, and calculus.

Learn how to find all six trig functions for angles outside the principal interval by using coterminal angles on the unit circle, including negative angles and angles beyond one rotation.

Identify coterminous angles within a given interval by adding multiples of 360 degrees (or 2 pi) to a base angle. It shows when a solution exists and when not.

Learn to find reference angles across quadrants using radians and degrees, and convert negative angles to positive coterminal angles.

Explore symmetry across the axes and reference angles on the unit circle to build all quadrant points, and apply this to evaluate sine and cosine at theta minus pi.

Explore even-odd identities by showing cosine is even and sine is odd on the unit circle, then derive all trig functions, including tangent, cosecant, and cotangent, using identities and reciprocals.

Identify all angles theta where sine theta equals zero, using coterminal angles and the unit circle, and express the solution as theta = k pi with k in integers.

Learn how to compute sine, cosine, tangent, and their reciprocals for a point off the unit circle using similar triangles, the distance formula, and x,y coordinates.

Explore how angles in circles connect to the unit circle and radians. Use radians to find sector areas and analyze linear and angular velocity as rate of change.

Learn radians and arc length on the circle, using S = R theta with radius R and theta in radians, and convert degrees to radians for practical problems.

Explore circular sectors, central angles, and minor versus major arcs, and learn to find sector area using radians or degrees with the formula 1/2 r^2 theta.

Evaluate all six trig functions at real numbers expressed in radians. Use a calculator in radian mode to evaluate values like 6.2832 and 1.5708 for unit circle results.

Relate linear and angular velocity by substituting arc length into linear velocity and using theta over t as angular velocity, yielding V = R omega and omega = V/R.

Explore the graphs of the six trig functions, starting with sine and cosine. Learn to sketch one period, apply transforms like stretches, compressions, reflections, and shifts, and graph combinations.

Explain sine theta and cosine theta graphs from the unit circle. Show the forms y = a sin(bx + c) + d and y = a cos(bx + c) + d; explain vertical and horizontal scaling, reflections, and periodicity.

Explore amplitude and period in trig graphs: amplitude equals the absolute value of a for sine and cosine, and period equals 2π/|b|; tangent and cotangent use π/|b|.

Learn horizontal and vertical shifts of trig graphs. Use y = a sin(bx + c) + d to see how c and d move graphs left, right, up, and down.

Learn to graph trig transformations in standard form by applying sequential horizontal and vertical shifts, stretches, and reflections to a base sine or cosine function, resulting in the target function.

Graph combinations of trig functions by graphing each function separately, then applying sum, difference, product, or quotient to form the final curve, using point evaluations to guide plotting.

Explore inverse trig relations, including why some inverses are not functions, using the vertical line test, reflections over y=x, and arc sine notation.

Turn inverse trig relations into functions by restricting their ranges, using capital notation for inverse functions and lowercase for inverse relations, with sine, cosine, and tangent ranges noted.

Analyze how trig functions act on inverse trig functions, using right-triangle reasoning and Sohcahtoa to derive sine of inverse tangent and related secant, cosecant, and cotangent forms.

Learn how trig identities enable substitutions to simplify expressions, starting with sine and cosine relations such as sin^2 theta plus cos^2 theta equals one.

Explore co function identities that link sine to cosine and tangent to cotangent, and learn to rewrite functions in terms of their co functions using pi/2.

Compute tangent values using sum and difference identities by decomposing angles, applying the tangent addition formula, and deriving the tangent identity from sine and cosine on the unit circle.

Explore product-to-sum identities derived from sine and cosine sum and difference formulas, converting products to sums or differences and vice versa with examples.

Apply the law of sines and the law of cosines to determine interior angles, side lengths, and the area of any oblique triangle, extending beyond right triangles.

Learn to solve oblique triangles using the law of sines, finding all three sides and angles from two angles and one side, including ASA and SA cases.

Explore the ambiguous SSA case of the law of sines, solving for angles and sides when two sides and a nonincluded angle are known, revealing one or two triangles.

Compute the area of any triangle using the included angle: area equals one-half times the product of two sides times the sine of the included angle, in three equivalent forms.

Apply the law of cosines to oblique triangles in SAS and SSS cases to find sides and angles using three equivalent formulas, and identify impossible triangles.

Apply Heron's formula to find a triangle’s area from side lengths a, b, and c by using S as half the perimeter, then plug into sqrt(S(S-a)(S-b)(S-c)).