https://teddyrookbook.com/math/precalculus/Recent content in Precalculus and Trigonometry on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/precalculus/standard-position/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/standard-position/Draw angles from the positive x-axis, including negative and coterminal angles. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/radian-measure/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/radian-measure/One radian is the angle whose arc equals the radius. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/degrees-radians/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/degrees-radians/Multiply by π/180 or 180/π. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/arc-sector-radians/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/arc-sector-radians/s = rθ and A = ½r²θ when θ is in radians. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/unit-circle-sine-cosine/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/unit-circle-sine-cosine/(cos θ, sin θ) is the point at angle θ on the unit circle. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/special-angles/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/special-angles/Use 30-60-90 and 45-45-90 triangles to find exact values. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/reference-angles/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/reference-angles/Find values in any quadrant from a reference angle and a sign. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/symmetry-periodicity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/symmetry-periodicity/sin(−θ) = −sin θ, cos(−θ) = cos θ, and both repeat every 2π. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/six-trig-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/six-trig-functions/Tangent, cotangent, secant and cosecant from sine and cosine. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/pythagorean-identity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/pythagorean-identity/sin²θ + cos²θ = 1, and finding one value from another. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/sine-cosine-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/sine-cosine-graphs/Unroll the unit circle into a wave. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/amplitude-period/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/amplitude-period/In y = A sin(Bx), the amplitude is |A| and the period is 2π/B. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/phase-shift-midline/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/phase-shift-midline/y = A sin(B(x − h)) + k shifts by h and centers on y = k. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/sinusoidal-models/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/sinusoidal-models/Fit a sine curve to tides, daylight and Ferris wheels. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/tangent-reciprocal-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/tangent-reciprocal-graphs/Find asymptotes where cosine or sine is zero. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/damped-oscillation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/damped-oscillation/Multiply a sinusoid by a decaying exponential. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/inverse-trig/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/inverse-trig/Restrict sine, cosine and tangent so they have inverses. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/inverse-trig-compositions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/inverse-trig-compositions/Evaluate expressions like sin⁻¹(sin 5π/6) and cos(tan⁻¹ x). Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/fundamental-identities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/fundamental-identities/Rewrite trig expressions in terms of sine and cosine. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/verify-identities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/verify-identities/Transform one side until it matches the other. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/sum-difference-formulas/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/sum-difference-formulas/sin(a + b) = sin a cos b + cos a sin b. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/double-angle/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/double-angle/sin 2θ = 2 sin θ cos θ and three forms of cos 2θ. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/half-angle/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/half-angle/Rewrite sin²θ and cos²θ in terms of cos 2θ. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/solve-trig-basic/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/solve-trig-basic/Find every solution in one period, then all solutions. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/solve-trig-advanced/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/solve-trig-advanced/Factor, use identities, and handle 2θ and 3θ. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/ambiguous-case/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/ambiguous-case/With two sides and a non-included angle there may be 0, 1 or 2 triangles. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/triangle-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/triangle-applications/Solve navigation and surveying problems with the laws of sines and cosines. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/herons-formula/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/herons-formula/Find a triangle's area from its three sides. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/vectors-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/vectors-intro/A vector has a size and a direction. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/vector-components/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/vector-components/v = ⟨|v| cos θ, |v| sin θ⟩. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/vector-operations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/vector-operations/Add tip to tail; scaling stretches or flips. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/vector-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/vector-applications/Find a resultant for wind, current and forces. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/dot-product/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/dot-product/u · v = |u||v| cos θ. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/vector-projection/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/vector-projection/Split a vector into parts along and across another vector. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/polar-coordinates/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/polar-coordinates/Locate points by distance r and angle θ. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/polar-rectangular/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/polar-rectangular/x = r cos θ, y = r sin θ, and back again. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/polar-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/polar-graphs/Sketch circles, cardioids, roses and limaçons. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/complex-plane/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/complex-plane/Plot a + bi and find its modulus. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/complex-polar-form/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/complex-polar-form/z = r(cos θ + i sin θ). Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/de-moivre/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/de-moivre/Multiply moduli and add angles; zⁿ = rⁿ(cos nθ + i sin nθ). Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/complex-roots/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/complex-roots/A nonzero number has n nth roots spread evenly around a circle. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/parametric-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/parametric-equations/x and y both depend on a third variable t. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/eliminate-parameter/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/eliminate-parameter/Solve for t or use an identity to get one equation in x and y. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/projectile-motion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/projectile-motion/Model a launched object with x = v₀t cos θ and y = h + v₀t sin θ − 16t². Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/parabolas-conics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/parabolas-conics/Use focus, directrix and vertex for parabolas opening any way. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/ellipses/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/ellipses/The sum of distances to two foci is constant; c² = a² − b². Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/hyperbolas/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/hyperbolas/The difference of distances to two foci is constant; c² = a² + b². Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/classify-conics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/classify-conics/Complete the square to name and graph any conic. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/matrix-basics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/matrix-basics/Add matrices and multiply them by numbers. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/matrix-multiplication/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/matrix-multiplication/Each entry is a row times a column. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/gaussian-elimination/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/gaussian-elimination/Write an augmented matrix and row-reduce it. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/determinants/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/determinants/ad − bc, and expansion by minors. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/matrix-inverse/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/matrix-inverse/A⁻¹ exists exactly when det A ≠ 0. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/matrix-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/matrix-equations/Write AX = B and multiply by A⁻¹ on the left. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/matrix-transformations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/matrix-transformations/A 2 × 2 matrix moves the plane, and |det| scales area. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/partial-fractions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/partial-fractions/Split a rational expression into simpler fractions. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/induction/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/induction/Prove a base case, then show each case implies the next. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/logistic-models/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/logistic-models/Growth that slows near a carrying capacity. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/difference-quotient/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/difference-quotient/[f(x + h) − f(x)]/h is the slope of a secant line. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/limits-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/limits-intro/A limit is the value a function approaches, not necessarily its value. Precalculus and Trigonometry · Math Ladderhttps://teddyrookbook.com/math/precalculus/limits-graphs-tables/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/precalculus/limits-graphs-tables/Read limits from the left and the right. Precalculus and Trigonometry · Math Ladder