Precalculus

Use this for “Find cos⁻¹(−1/2) in degrees and radians.”

Use this for “A road rises 3 m over 40 m of ground. What angle does it make with the level?”

Use this for “Find the exact value of cos(sin⁻¹(3/5)).”

Use this for “Find the exact value of cos 15° as cos(45° − 30°).”

Use this for “sin A = 3/5 with A in Quadrant II. Find sin 2A and cos 2A.”

Use this for “cos A = 7/25 and A/2 is in Quadrant I. Find sin(A/2) and cos(A/2).”

Use this for “Solve 2 sin x + 3 = 4 for 0° ≤ x < 360°.”

Use this for “Solve 3 tan x + 1 = 4 for 0° ≤ x < 360°.”

Use this for “Solve 2 sin²x − sin x − 1 = 0 for 0° ≤ x < 360°.”

Use this for “Find u + v for u = ⟨3, 1⟩ and v = ⟨1, 2⟩, and its magnitude.”

Use this for “Find −2u for u = ⟨3, 4⟩ and its magnitude.”

Use this for “Find the angle between ⟨2, 1⟩ and ⟨1, 3⟩” or “Are these vectors perpendicular?”

Use this for “Forces of 30 N at 0° and 40 N at 60° pull on a ring. Find the resultant.”

Use this for “Find u × v for u = ⟨1, 2, 3⟩ and v = ⟨2, 0, 1⟩, and the areas of the parallelogram and the triangle with sides u and v from one vertex.”

Use this for “Find the volume of the box that u = ⟨1, 2, 0⟩, v = ⟨0, 1, 3⟩ and w = ⟨2, 0, 1⟩ span.”

Use this for “Find the distance between P(1, 2, 3) and Q(3, 5, 9) and the midpoint of PQ.”

Use this for “Write 2(cos 60° + i sin 60°) as a + bi,” or back.

Use this for “Multiply 2(cos 30° + i sin 30°) by 3(cos 60° + i sin 60°).”

Use this for “Find (1 + i)⁸ with De Moivre’s theorem.”

Use this for “Find the cube roots of 8i” or “the fourth roots of −16.”

Use this for “How many petals does r = 4 cos 2θ have? Find r at θ = 30°.”

Use this for “Find r on r = 2 + 2 cos θ at θ = 60°” and the curve’s shape.

Use this for “Graph r = 6 cos θ: find r at θ = 60°, the center and the radius.”

Use this for “Graph r = 0.5θ: find r at θ = 270°” or “At what angle is r = 6?”

Use this for “Eliminate the parameter from x = 1 + 3 cos t, y = −2 + 2 sin t.”

Use this for “A ball is thrown at 20 m/s, 30° up. Where is it after 1 s, and when does it land?”

Use this for “Multiply [[2, 1], [3, 4]] by the column (5, −1).”

Use this for “Find the determinant of [[2, 0, 1], [1, 3, 2], [1, 1, 4]].”

Use this for “Find the inverse of [[4, 7], [2, 6]].”

Use this for “Solve 3x + 2y = 7 and 5x + 4y = 13 with the inverse matrix.”

Use this for “Solve 2x + 3y = 13 and x − y = −1 by Cramer’s rule.”

Use this for “Where does [[2, 1], [1, 1]] send (3, −1)? Which point does it send to (5, 2)?”

Use this for “A shape of area 4 is transformed by [[3, 1], [1, 2]]. What is the new area?”

Use this for “Turn (4, 2) by 30° and then by 60°. Which single matrix does both?”

Use this for “Which transformation does [[0, 1], [1, 0]] make?”

Use this for “Find the focus and directrix of (x − 2)² = 8(y + 1).”

Use this for “Find the foci and asymptotes of x²/9 − y²/16 = 1.”

Use this for “Which conic is 4x² − 9y² = 36?”

Use this for “Why are circles, ellipses, parabolas and hyperbolas called conic sections?”

Use this for “Prove 1 + 3 + 5 + … + (2n − 1) = n² by induction.”

Use this for “Prove 1 + 2 + 4 + … + 2ⁿ⁻¹ = 2ⁿ − 1 by induction” (r = 2), or any ratio r.

Use this for “Prove 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 by induction.”

Use this for “Prove that n³ − n is divisible by 3 for every whole number n ≥ 1.”

Use this for “Prove by induction that 1 + 3 + 5 + … + (2n − 1) = n² for every n ≥ 1.”

Use this for “Write (3x − 1) ÷ (x − 2)² as partial fractions.”

Use this for “Write (3x² − 2x + 3) ÷ ((x − 1)(x² + 1)) as partial fractions.”

Use this for “Name the conic r = 4 ÷ (1 + sin θ), its eccentricity and its directrix.”

Use this for “For r = 3 ÷ (1 − 0.5 cos θ), find the vertices, the center and the lengths of the axes.”

Use this for “Find the vertex and the directrix of r = 4 ÷ (1 − cos θ), and the point at θ = 90°.”

Use this for “Through what angle should the axes turn to remove the xy term of 4x² + 2xy + 2y² = 1? Which conic is it?”

Use this for “Turn the axes to remove the xy term of 4x² + 2xy + 2y² = 1 and write the equation in x′ and y′.”

Use this for “f(x) = x + 1 for x < 2 and 2x − 3 for x ≥ 2. Is f continuous at 2?”

Use this for “Find the slope of the secant of x² from 3 to 3.1, then the derivative at 3.”

Use this for “Find the limit of (2x + 1) ÷ (x − 3) as x → ∞.” Other powers of x are on “Limits at infinity by degree.”

Use this for “Estimate the area under y = 2x + 1 from 0 to 4 with 8 rectangles, then find it exactly.”

Use this for “Find the limit of (3n + 1) ÷ (2n − 1) as n → ∞” or “of (2n² + 1) ÷ (n² − 3), or say there is none.”