Precalculus and Trigonometry

Apply De Morbis theorem to simplify the ratio to cos 11 theta plus eta sine 11 theta.

Demonstrate simplifying (cos θ + η sin θ)^4 divided by (cos θ + η sin θ)^2 to cos 2θ + η sin 2θ, using exponent rules and de Morbis theorem.

Apply de Moivre's theorem to simplify a complex expression by transforming into cos theta plus iota sine theta, combining powers, and canceling terms to obtain a final value of one.

Using half-angle substitutions, this example shows the ratio (1+cosθ + i sinθ)/(1+cosθ − i sinθ) raised to n equals cos nθ + i sin nθ via de Moivre.

Multiply the infinite sequence x_n = (cos(pi/2^n) + i sin(pi/2^n))^n to get the product -1. Use angle addition and a geometric progression to derive cos(pi) + i sin(pi) = -1.

Derive x = cos theta plus minus i sine theta from x + 1/x = 2 cos theta, and prove x/y + y/x = 2 cos(theta - phi) using De Morbis theorem, with a power generalization.

Apply de Moivre's theorem to convert one plus eta and one minus eta, yielding left side equals two to the n over two times cos n pi over four.

Express root three plus i and root three minus i in polar form as 2(cos pi/6 ± i sin pi/6), then apply De Moivre to obtain 2^{n+1} cos(n pi/6).

Explore n roots of a complex number by expressing z in polar form and applying de Moivre's theorem to get r^(1/n)[cos((θ+2mπ)/n) + i sin((θ+2mπ)/n)], m = 0 to n−1.

Using de Moivre's theorem, find cube roots of a complex number with modulus 1 and argument pi/2, yielding (√3/2 + i/2), (-√3/2 + i/2), and -i.

Explore nth roots of unity using De Moivre's theorem, expressing roots as cos(2πr/n) + i sin(2πr/n) for r = 0 to n−1, including square and cube roots.

Find the fourth roots of unity by using n = 4, compute alpha = cos(pi/2) + i sin(pi/2) = i, and obtain the roots 1, i, -1, and -i.

Determine the added rational expression f(x)/phi(x) that turns (x^3-1)/(x^2+2) into (2x^3-x^2+3)/(x^2+2) by combining with the common denominator x^2+2, yielding (x^3-x^2+4)/(x^2+2).

Solve for y so that 1/x plus y equals x, yielding y = x - 1/x, and identify y as the quantity to be added to 1/x.

Learn how to simplify x + 1/x to (x^2+1)/x and identify its reciprocal as x/(x^2+1), showing that their product equals one.

Set x = a−b, y = b−c, z = c−a; since x+y+z = 0, x^3+y^3+z^3 = 3xyz, giving 3(a−b)(b−c)(c−a).

Combine the rational expressions, find the common denominator, and expand and simplify to cancel terms, yielding the simplified result (2x^2 + 3x + 5) / 3.

Factorize the denominators to reveal common factors, form a common denominator, and simplify the rational expression to its lowest terms by canceling shared factors and using reciprocals for division.

Learn to simplify a complex rational expression by converting division to multiplication, using l.c.m. and difference of squares, and canceling factors to reach x^2 y^2 z^2.

Prove the identity a^2/(a^2-bc) + b^2/(b^2-ca) + c^2/(c^2-ab) = 1 given ab+bc+ca=0, by substituting and simplifying to reduce to one.

Decompose 1/[(x+1)(x+2)] into partial fractions a/(x+1) + b/(x+2), compare coefficients, and determine a=1 and b=-1.

Decompose the rational expression (3x-2)/((x+1)^2(x+3)) for precalculus and trigonometry by solving for a, b, c in a/(x+1) + b/(x+1)^2 + c/(x+3).

Factorize the denominator using the remainder theorem to obtain (x-1)(x-2)(x-3). Decompose into partial fractions A/(x-1)+B/(x-2)+C/(x-3) and determine A=5/2, B=-8, C=11/2.

Decompose a rational expression into partial fractions by using a linear and a quadratic factor, solving for a, b, and c to express (2x-1)/((x+1)(x^2+2)) as -1/(x+1) + (x+1)/(x^2+2).

Decompose the given rational expression into partial fractions of the form a/(x-1)+(bx+c)/(x^2+1)+(dx+e)/(x^2+1)^2, solve for a,b,c,d,e by substitution and coefficient comparison, obtaining a=-1/4, b=c=1/4, d=1/2, e=5/2, yielding the final decomposition.

Compute the partial fractions decomposition of (2x+3)/((x+1)(x-3)) by matching the numerator to a(x-3)+b(x+1). Substitute x=3 to find b=9/4 and x=-1 to find a=-1/4, so a+b=2.

Use partial fractions to decompose 3x+alpha over x^2-3x+2 into a/(x-2) plus 10/(x-1), solve by plugging x=1 and x=2, yielding alpha=13 and a=19.

Apply f(x)=x^2+3 on the real domain to find x for f(x)=28 and f(x)=39, and conclude no real preimage exists for f(x)=2.

Apply function composition with f(x)=x^3 and g(x)=3x-1 to compute f∘g(x)=(3x-1)^3. Then compute g∘f(x)=3x^3-1 and show that f∘g ≠ g∘f.

Show that f(f(f(x))) = x / square root(1 + 3x^2) by computing f(x) = x / square root(1 + x^2) and applying substitution and simplification.

Explore inverse functions with f(x)=x^2 and g(x)=√x, showing f(g(x))=x and g(f(x))=x, and learn the f^{-1} notation, including log base e and e^x.

Find the inverse of the function f defined by f(x)=3x from set A to set B by reversing the diagram’s ordered pairs: (0,0), (-3,-1), (-9,-3), (6,2).

Discover sequences and arithmetic progression, where numbers follow a rule, define the first term and common difference, and derive the nth term formula a_n = a + (n−1)d.

Demonstrate that the sequence a_n = 2n^2 + 1 is not an arithmetic progression by computing a_{n+1} and showing the difference a_{n+1}-a_n equals 4n+2, which is not constant.

Identify the arithmetic progression with first term four and common difference five. Solve 124 = 4 + (n−1)×5 to find n = 25, so 124 is the 25th term.

Solve for n in the arithmetic progression 3, 6, 9, ..., 111 using a_n = a + (n−1)d with 111 = 3 + (n−1)·3 to obtain n = 37.

Verify if 184 is a term in arithmetic progression 3 7 11; a_n = a + (n−1)d shows n must be natural, 184 yields non-integer n, so not a term.

Identify the given complex sequence as an arithmetic progression, compute the first term and common difference. Derive the nth term and determine which terms are purely real or purely imaginary.

Learn to select terms in an AP to simplify sums, choosing three, four, five, or six terms so the common difference cancels and the sum depends on a.

determine three numbers in an arithmetic progression with sum minus three and product eight, yielding a equals minus one and d equals plus or minus three, producing -4, -1, 2.

Learn to identify the arithmetic mean between two numbers in an arithmetic progression, compute am as (a+b)/2, and verify with examples like 13 and 19.

Identify n so that 2, n−1, 3, n+2, 6n−1 form an arithmetic progression, compute the common difference, solve for n, and obtain the numbers 5, 11, 17.

Determine k so that the terms 3k-2, 4, k-6, and k+2 form an arithmetic progression, equating second minus first to third minus second, yielding k=3.

Learn to find the sum of the first n terms of an arithmetic progression using s_n = n/2 (a + l) or s_n = n/2 [2a + (n-1)d].

Calculate the sum of an arithmetic progression with first term 5, common difference 8, last term 181, yielding 23 terms and a total of 2139.

Compute the sum of natural numbers between 250 and 1000 divisible by three using an arithmetic progression with first term 252 and common difference 3; the result is 156,375.

Identify odd multiples of three between 2 and 100, form an arithmetic progression with first term 3 and common difference 6, determine 17 terms, and compute their sum as 867.

Demonstrates that the sum of the n terms of an AP can be written as a n square plus b n, with a and b constants.

Solve for the first term and common difference of an arithmetic progression from t3 and t7 conditions, then compute the sum of the first 20 terms, which equals 740.

Demonstrate the equal-sum property of arithmetic progressions, where equidistant terms sum to the same value, and compute S24 using the S_n formula to 900.

Derive the sum to n terms for a_n = 5 - 6n, with a1 = -1 and d = -6, yielding S_n = n(2 - 3n).

Show that if a, b, c are in arithmetic progression, then b+c, c+a, a+b are in arithmetic progression by equating the common difference.

Prove that if a, b, c are in arithmetic progression, then a^2(b+c), b^2(c+a), and c^2(a+b) are in arithmetic progression.

If a, b, c are in AP, 1/(√b+√c), 1/(√c+√a), and 1/(√a+√b) form an AP. The condition 2b = a + c characterizes this.

demonstrates that if a, b, c are in AP, then the terms (b+c)^2 - a^2, (c+a)^2 - b^2, (a+b)^2 - c^2 form an AP, reducing to a - b = b - c.

Shows that if a^2, b^2, c^2 are in AP, then 1/(b+c), 1/(c+a), and 1/(a+b) are in AP, by cross-multiplication leading to 2b^2 = a^2 + c^2.

Solve for x in a logarithmic arithmetic progression using base-ten logs and the base-change formula, yielding x = log base 2 of 5.

Learn how to insert arithmetic means between two numbers to form an arithmetic progression. Derive the common difference d = (b − a)/(n+1) and the means a1 through an.

Learn to identify geometric progressions, derive the nth term ar^{n-1}, compute sums (finite and infinite), and apply the from end term ar^{m-n}.

Determine the GP’s common ratio with first term 1 given a3 + a5 = 90. Let x = r^2; x^2 + x - 90 = 0, giving r = ±3.

Solve a GP from the fourth term 54 and ninth term 13,122 using ar^3 = 54 and ar^8 = 13,122, yielding a = 2 and r = 3.

Find a geometric progression with the sum of the first two terms -4 and fifth term four times the third; solve for r = ±2 and a, giving two sequences.

Derive the common ratio r from the first term a and nth term b in a geometric progression, and prove p^2 = (ab)^n for product of the first n terms.

Identify the third term ar^2 = 4 and compute the product of the first five terms in this geometric progression as (ar^2)^5 = 4^5 = 1024.

Demonstrate that, for a geometric progression with first term A and common ratio R, the expression equals zero by applying log properties to the p, q, r terms.

Show that for a gp with first term a and ratio r, a_p = a r^{p-1}, a_q = b, a_r = c reduce to one, proving LHS = RHS.

Proves that in a finite geometric progression, the product of equidistant terms from the beginning and end equals the product of the first and last terms.

Solve for n by equating nth terms of two geometric progressions, using a_n = a r^{n-1} with a=5, r=2, and a second progression, demonstrating how powers of two determine n.

Use the gp sum formulas sn=a(r^n-1)/(r-1) or sn=(1-r^n)/(1-r) with r≠1; if r=1, sn=na, for the sum of the first n terms.

Compute the sum of seven terms of a geometric progression with first term 3 and ratio 2. Use S_n = a(r^n - 1)/(r - 1) to obtain 381.

Compute the sum of the first ten terms of a geometric progression with a=1 and r=1/2 using the geometric series formula. The sum equals 1023/512.

Sum the first n terms of the sequence, expressed as two geometric progressions plus 2n, yielding a closed form S = (x^{2n-1}/(x^2-1)) (x^2 + 1/x^{2n} + 2n).

Sum the series by splitting into two geometric progressions: x^{2k} terms and (xy)^k terms from k=1 to n. The total is S = x^2 (x^{2n}-1)/(x^2-1) + xy ((xy)^n-1)/(xy-1).

Learn how to sum the series 5, 55, 555, ... up to n terms by factoring five, dividing by nine to form a GP, and applying the GP sum.

Determine the number of terms in a geometric progression with a1=3, an=96, and sn=189 by solving for r using an=ar^(n-1) and sn=a(r^n-1)/(r-1); find r=2 and n=6.

Determine first term and common ratio of a geometric progression from sums of the first three and next three terms, obtaining r=2 and a=16/7, then compute sum of n terms.

Determine the least n such that the geometric series 1 + 3 + 3^2 + ... + 3^{n-1} exceeds 7000; use s_n = (3^n - 1)/2 and conclude n = 9.

Compute the sum to infinity of a geometric progression with first term -5/4 and common ratio -1/4, using S infinity = a/(1−r), yielding -1.

Show how to express two infinite geometric series with ratios a and b, derive a=(x-1)/x and b=(y-1)/y, and prove that the infinite sum with ratio ab equals xy/(x+y-1).

Derive r from the sum of an infinite geometric progression with first term one, showing r = ((a-1)/a)^(1/a) = ((b-1)/b)^(1/b) and proving their equality.

Using geometric sums of cos^2n theta and sin^2 phi, this example derives equations for x, y, z and proves x z + y z − z = x y.

Solve for the common ratio in a geometric progression using the infinite sum formula; with first term two and sum six, r equals two thirds.

Solve for the common ratio in an infinite geometric progression where every term is twice the sum of all subsequent terms; the ratio is 1/3.

determine the rational number for the decimal 0.356 with 56 repeating by decomposing it into a finite part and an infinite geometric progression, yielding 353/990.

We prove that if p, q, r are in AP, then the pth, qth, and rth terms of any GP are in GP, using first term A and ratio R.

Show that for a, b, c in AP and x, y, z in GP, x^(b−c) y^(c−a) z^(a−b) equals 1, given y^2 = xz and 2b = a + c.

Proves that if a, b, c are in gp, then log a^n, log b^n, log c^n form an ap using b^2 = ac and log m^n = n log m.

Show that if mth, nth, and pth terms of a gp are consecutive, then m, n, p form AP, since a_m a_p = a_n^2 implies 2n = m + p.

Explore how a, b, c in gp with x=(a+b)/2 and y=(b+c)/2 lead to the equalities a/x+c/y=2 and 1/x+1/y=2/b.

Explore two geometric progression identities: substitute b=ar, c=ar^2, d=ar^3 to prove LHS equals RHS, and show (ab+bc+cd)^2 matches the corresponding GP-based expression by factoring common terms.

Show that if a^2 + b^2, ab + bc, and b^2 + c^2 form a geometric progression, then b^2 equals ac, so a, b, c are in gp.

Find n so that (a^(n+1)+b^(n+1))/(a^n+b^n) equals the geometric mean sqrt(ab). The solution uses cross multiplication and factoring to show n = -1/2.

Solve a problem where two positive numbers differ by 12 and their arithmetic mean exceeds their geometric mean by 2, yielding numbers 16 and 4.

Solve for two numbers with arithmetic mean 34 and geometric mean 16 by using a+b=68 and ab=256, yielding 64 and 4.

In precalculus and trigonometry, Example-5 shows that if a, b, c are in GP and the quadratics ax^2+2bx+c=0 and dx^2+2ex+f=0 share a root, then d/a, e/b, f/c are in AP.

This proof shows that when the arithmetic and geometric means of two numbers are in ratio m to n, the numbers have ratio m + root(m^2-n^2) to m - root(m^2-n^2).

Derive that the two geometric means between any two positive numbers satisfy (y^3+z^3)/(x y z)=2, using x=(a+b)/2 and y=a r, z=a r^2 in a geometric progression.

Show that if a is the arithmetic mean of b and c and b, g1, g2, c form a geometric progression, then g1^3 + g2^3 equals 2abc.

From a and b, a1 = (2a + b)/3 and a2 = (a + 2b)/3 form arithmetic progression, and (2a1 - a2)(2a2 - a1) = g^2 with g = sqrt(ab).

Construct a quadratic equation with the arithmetic mean a and geometric mean g of its roots, using alpha+beta=2a and alpha beta=g^2 to get x^2-2ax+g^2=0.

learn to work with harmonic progression by converting to the reciprocal arithmetic progression, find its nth term, then take reciprocals; compute the harmonic mean h = 2ab/(a+b).

Show the reciprocal relation between arithmetic progression a_p and h_p; d = 1/(mn) and a = 1/(mn); r-th term of a_p is r/(mn) and r-th term of h_p is mn/r.

Identify the series as a harmonic progression and derive the corresponding arithmetic progression with a = 3 and d = -1/8. The fifth term of the harmonic progression is 2/5.

Prove that the mth term of a harmonic progression equals one when the mth term is n and the nth term is m, via reciprocals forming an arithmetic progression.

Learn to find four harmonic means between 1 and 30 by constructing an arithmetic progression from 1 to 1/30 and taking reciprocals.

this lecture shows that the reciprocals of n harmonic means between a and b form an arithmetic progression, deriving h1 and hn.

Compute the harmonic mean of the roots alpha and beta using the sum and product of roots, yielding HM = 2 alpha beta/(alpha+beta) = 4.

Explore the relationship among arithmetic, geometric, and harmonic means. Derive G = sqrt(AB) and H = 2AB/(A+B), and show A ≥ G ≥ H when A = B.

Prove p/r plus r/p equals a/c plus c/a when a, b, c are in AP, p, q, r in HP, and a, p, b, q, c, r are in GP.

Shows that the quadratic with roots a and b, x^2 - (a+b)x + ab = 0, becomes x^2 - 2ax + g^2 = 0 using the arithmetic mean a and geometric mean g.

Show that with h as the harmonic mean h = 2ab/(a+b), the expression 1/(h-a) + 1/(h-b) equals 1/a + 1/b.

Show that p, q, r in AP with p=(a-x)/(kx), q=(a-y)/(ky), r=(a-z)/(kz) imply 1/x, 1/y, 1/z in AP. Conclude that x, y, z are in HP.

Learn the formulae for the sum of first n natural numbers, sum of squares, and sum of cubes, and apply them to natural-number series problems.

Compute the sum to n terms of the series 1×2×3, 2×3×4, 3×4×5, … by recognizing t_n = n(n+1)(n+2) and deriving S_n = n(n+1)(n+2)(n+3)/4.

Practice simplifying factorial expressions by canceling terms, compute 30! / 28! to obtain 870, and evaluate (11! − 10!)/9! to obtain 100, using factorial properties.

Compute x by solving a factorial equation; combine 1/4! and 1/5! to a common denominator and simplify to x equals 36.

Evaluate factorial expressions and compare left-hand and right-hand sides to determine the truth of statements. The example calculates 5!, 2!, and 3! and shows both statements are false.

Understand how to apply the fundamental principle of counting by multiplying options in sequential tasks and adding when alternatives arise, illustrated with routes and selections.

Calculate the total ways to answer five multiple-choice questions by applying the fundamental principle of counting, multiplying four choices per question to obtain four raised to the power of five.

Explore permutations as the number of ways to arrange r objects from n when order matters, using the formula nPr equals n factorial over n minus r factorial.

Prove the permutation theorem nPr equals factorial n over factorial n minus r, by expanding the left-hand side and right-hand side and canceling terms.

Solve a permutation ratio problem using the nPr formula. Equate (n-1)P3 to nP4 as 1/9 and simplify to find n = 9.

Apply the nPr formula to solve 2 times 5P3 equals nP4, rewrite with factorials, cancel terms, compare factors, and deduce n equals five.

Explore combinations using nCr notation, where order does not matter. Learn the factorial formula and the symmetry nCr = nC(n - r), applying this to committees and team selection.

Determine n from the ratio (2 nC3) / (nC3) = 11 by simplifying and applying cross-multiplication to obtain n = 6.

Solve for r where 20 choose r plus one equals 20 choose 2r minus three, using the symmetry nCr = nC(n−r). The acceptable solution is r = 4.

Compute the total number of committees of two, three, or four from ten people using combinations C(10,2), C(10,3), and C(10,4) to yield 375.

Compute the number of ways to fill five teacher positions by choosing two of seven reserved category applicants and three of sixteen unreserved applicants, totaling 11,760.