Analytic trigonometry: synthesis and challenges

Summary

This lesson closes the unit on analytic trigonometry. It gathers into a single deductive chain what was proved in its six lessons: the fundamental identities, the formula for the cosine of a difference and all those deduced from it, the sum-to-product transformations and the harmonic form, the solution of equations and inequalities, the angle between lines, and the rotation of axes that classifies the conics. It explains how that chain rests on the preceding unit and on the course Algebra and Geometry I, in particular on the unit circle, polar coordinates, polynomials and conics, and how it prepares the following unit, in which the same formulas reappear as the rule for multiplying complex numbers. It ends with an exit assessment, parallel to the entry assessment of the introduction, and with ten challenges, harder than the problems proposed in the unit, which demand combining several lessons.

Learning objectives

Reconstruct the deductive chain of the unit, from the formula for the cosine of a difference to the classification of conics by their discriminant, and identify at each stage the result on which it depends.
Relate the results of the unit to those of the preceding unit and of Course I on the unit circle, polar coordinates, polynomials and conics.
Anticipate the role of the sum and difference formulas in the multiplication of complex numbers in polar form.
Solve problems that combine identities, equations, angles between lines and rotations, choosing in each case the transformation that makes the problem tractable.

What the unit achieved

The unit began by fixing the language. In the lesson on the fundamental trigonometric identities, an identity was defined as an equality that holds on the whole common domain of its two sides; the reciprocal, quotient, Pythagorean, parity and cofunction identities were gathered; and the method of proof was established: transform one side into the other, or both into the same expression, without starting from what is to be proved; a single counterexample suffices to refute a false identity. With that language each function was expressed in terms of another, roots were transformed by substitutions, and the ellipse and the hyperbola were recognized in the parametrizations $\left( a \cos t, b \sin t \right)$ and $\left( a \sec t, b \tan t \right)$ .

The decisive advance was the lesson on the sum and difference formulas. By computing in two ways the length of a chord of the unit circle, in Cartesian coordinates and in polar coordinates, it was proved that for all real numbers $\alpha$ and $\beta$ the following equality holds:

\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

Everything else was deduced from it. With parity came the cosine of a sum; with the cofunction, the sine of a sum and of a difference; with the quotient, the tangent of a sum. If this is so, setting $\alpha = \beta$ gives the double-angle formulas; solving for the squares, the power-reducing and half-angle formulas; adding once more, the triple-angle formulas. The substitution $t = \tan \frac{\theta}{2}$ turned the sine and the cosine into rational fractions of $t$ , and from there followed the rational parametrization of the circle and the formula for all Pythagorean triples.

The lesson on sum-to-product formulas traveled the opposite road: adding and subtracting the sum and difference formulas turns products into sums and sums into products. Its main result is that every combination $a \sin x + b \cos x$ is a single wave $R \sin(x + \varphi)$ , with $R = \sqrt{a^2 + b^2}$ ; from this follow its range, $\icc{-R}{R}$ , the superposition of oscillations of equal frequency, beats and, with a factor that makes the sum telescope, the sums of sines and cosines in arithmetic progression.

With these tools, the lesson on trigonometric equations and inequalities could solve a whole family of problems. The general solutions of $\sin x = c$ , $\cos x = c$ and $\tan x = c$ extend by periodicity those given by the inverse functions; the other equations reduce to these through identities, factoring, the equality of two sines or of two cosines, and the harmonic form, with two constant precautions: dividing can lose solutions and squaring can add them. The inequalities were solved in one turn, with the unit circle or with sign charts, and extended periodically.

The last two lessons carried this calculus into analytic geometry. In the lesson on the angle between lines, the tangent of a difference gave the angle between two lines from their slopes, and the dot product a second formula, valid for vertical lines as well; the bisectors and the incenter were obtained, and inaccessible heights, harmonic motion, the range of projectiles and the reflection of light were modeled. That lesson is the only one in the unit that admits something without proof, and it says so: from Euclidean geometry, the three facts that Course I already admitted (the Pythagorean theorem, the equality of corresponding angles between parallels, and the sum $\pi$ of the angles of a triangle), and from physics, the equations of simple harmonic motion and those of the projectile. Finally, in the lesson on rotation of axes, the difference formulas described the change of coordinates when the axes are rotated; the angle satisfying $\cot 2\theta = \frac{A - C}{B}$ eliminates the term in $xy$ , the sum $A + C$ and the discriminant $B^2 - 4AC$ do not change under the rotation, and therefore the sign of the discriminant classifies every second-degree equation.

The whole chain can be summarized thus: a chord computed in two ways gives the cosine of a difference; from it come all the sum and difference formulas; from these, the transformations and the harmonic form; with all of them equations are solved; and the solved equations measure angles between lines and eliminate the mixed term of conics. The following figure gathers that chain into five segments, which follow the order of the paragraphs above: each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that are admitted without proof or come from the preceding unit or from Course I, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

AUnit circle. The axiom of traversal by arc length is admitted: for each real $s$ there is a unique point $P(s)$ of the unit circle reached from $(1, 0)$ by traveling a length $\abs{s}$ , traversals can be concatenated, $P(s + 2\pi) = P(s)$ , the abscissa of $P(s)$ is strictly decreasing on $\icc{0}{\pi}$ and increasing on $\icc{\pi}{2\pi}$ , and every point of the circle is some $P(s)$ . From that axiom it was proved, in the unit on trigonometric functions and polar coordinates, that every point of the circle is $P(\theta)$ for a unique $\theta$ in $\ico{0}{2\pi}$ , that the cosine and the sine, coordinates of $P(\theta)$ , are periodic and satisfy the parity and cofunction relations, and that they are strictly monotonic on their principal intervals. Admitted without proof in this unit.
BInverse functions. The arcsine, the arccosine and the arctangent give, on their principal intervals, the unique angle that has a given sine, cosine or tangent; proved in the unit on trigonometric functions and polar coordinates. Admitted without proof in this unit.
CPolar coordinates. A point with polar coordinates $(r, \varphi)$ has Cartesian coordinates $x = r \cos \varphi$ and $y = r \sin \varphi$ , and the distance between $(r_1, \theta_1)$ and $(r_2, \theta_2)$ satisfies $d^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)$ ; proved in the unit on trigonometric functions and polar coordinates. Admitted without proof in this unit.
DAnalytic geometry of the plane. The distance formula, the distance from a point to a line, the half-plane theorem, the dot product with the Cauchy-Schwarz inequality and the angle between two vectors, and the classification of conics with axes parallel to the coordinate axes; proved in Course I, in the unit on the analytic geometry of the plane, and in the lesson on solving triangles. Admitted without proof in this unit.
EEarlier algebra and arithmetic. The factoring of polynomials, the factor theorem and synthetic division, sign charts, telescoping sums and Euclid’s lemma; proved in Course I, in the units on the natural numbers and on algebraic functions and polynomials. Admitted without proof in this unit.
FPythagorean identity. For every $\theta$ we have $\sin^2 \theta + \cos^2 \theta = 1$ , and on their domains, $1 + \tan^2 \theta = \sec^2 \theta$ and $1 + \cot^2 \theta = \csc^2 \theta$ . Established in: 2. Fundamental identities.
GSum and difference formulas. For all real $\alpha$ and $\beta$ we have $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ and $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ , and also the difference formulas, which result from replacing $\beta$ by $-\beta$ . Established in: 6. Sum and difference formulas.
HEuclidean geometry. From Euclidean geometry are admitted, as in the Course I unit on the analytic geometry of the plane, the Pythagorean theorem and its converse, the equality of the corresponding angles that a transversal forms with two parallels, and the sum $\pi$ of the angles of a triangle; from them follow the equality of vertical angles and that of alternate angles between parallels. Admitted without proof in this unit.

Segment 1 The language: identities and their proof
- Define an identity as an equality that holds on the whole common domain of its sides, and distinguish it from an equation.
- Gather the reciprocal, quotient, Pythagorean, parity and cofunction identities.
- Prove identities without starting from what is to be proved, and refute false ones with a counterexample.
- Transform roots by substitutions and recognize the ellipse and the hyperbola in their trigonometric parametrizations.
1. Identity and its proof. An identity is an equality that holds on the whole common domain of its two sides; it is proved by transforming one side into the other, or both into the same expression, without starting from it, and it is refuted by a single counterexample.
  Proved in: Fundamental trigonometric identities.
2. Fundamental identities. On the common domain of each one, the reciprocal, quotient and Pythagorean identities hold; the sine, the tangent, the cosecant and the cotangent are odd, the cosine and the secant are even, and each function of $\frac{\pi}{2} - \theta$ is its cofunction of $\theta$ .
  Uses: 1. Identity and its proof. Rests on: A (Unit circle). Proved in: Fundamental trigonometric identities.
3. Trigonometric substitutions. If $\theta = \arcsin x$ , then $\sqrt{1 - x^2} = \cos \theta$ ; if $\theta = \arctan x$ , then $\sqrt{1 + x^2} = \sec \theta$ ; and if $x \geq 1$ and $\theta = \arccos \frac{1}{x}$ , then $\sqrt{x^2 - 1} = \tan \theta$ .
  Uses: 2. Fundamental identities. Rests on: F (Pythagorean identity), B (Inverse functions). Proved in: Fundamental trigonometric identities.
4. Parametric ellipse and hyperbola. As $t$ runs through $\ico{0}{2\pi}$ , the point $(a \cos t, b \sin t)$ traces the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ exactly once; and the point $(a \sec t, b \tan t)$ traces the right branch of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ exactly once as $t$ runs through $\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}$ , and the left branch as it runs through $\ioo{\frac{\pi}{2}}{\frac{3\pi}{2}}$ .
  Uses: 2. Fundamental identities. Rests on: A (Unit circle), F (Pythagorean identity). Proved in: Fundamental trigonometric identities.
Segment 2 The formula for the cosine of a difference and its consequences
- Prove the cosine of a difference by computing in two ways a chord of the unit circle.
- Deduce from it the cosine and the sine of a sum and of a difference, and the tangent of a sum.
- Obtain the double-angle, power-reducing, half-angle and triple-angle formulas.
- Express the sine and the cosine as rational fractions of $t = \tan \frac{\theta}{2}$ , parametrize the circle rationally and generate all Pythagorean triples.
1. Cosine of a difference. For all real $\alpha$ and $\beta$ we have $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ , because the chord joining $P(\alpha)$ to $P(\beta)$ has the same length computed in Cartesian coordinates and in polar coordinates.
  Rests on: C (Polar coordinates), D (Analytic geometry of the plane), F (Pythagorean identity). Proved in: Sum and difference formulas, double angle and half angle.
2. Sum and difference formulas. With parity and the cofunction one obtains $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ , $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ and the sine of a difference; and, if the tangents of $\alpha$ and $\beta$ are defined and $\tan \alpha \tan \beta \neq 1$ , then $\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$ .
  Uses: 2. Fundamental identities, 5. Cosine of a difference. Proved in: Sum and difference formulas, double angle and half angle.
3. Double angle. We have $\sin 2\theta = 2 \sin \theta \cos \theta$ , $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta$ and, where it is defined, $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$ .
  Rests on: G (Sum and difference formulas), F (Pythagorean identity). Proved in: Sum and difference formulas, double angle and half angle.
4. Power reduction and half angle. We have $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$ and $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$ ; from them, the sine and the cosine of $\frac{\theta}{2}$ are the square roots of $\frac{1 - \cos \theta}{2}$ and of $\frac{1 + \cos \theta}{2}$ with the sign of their quadrant, and $\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}$ if $\cos \theta \neq -1$ .
  Uses: 7. Double angle. Rests on: F (Pythagorean identity). Proved in: Sum and difference formulas, double angle and half angle.
5. Triple angle. For every real $\theta$ we have $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$ and $\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$ .
  Uses: 7. Double angle. Rests on: F (Pythagorean identity), G (Sum and difference formulas). Proved in: Sum and difference formulas, double angle and half angle.
6. Half-angle tangent. If $\theta$ is not an odd multiple of $\pi$ and $t = \tan \frac{\theta}{2}$ , then $\sin \theta = \frac{2t}{1 + t^2}$ and $\cos \theta = \frac{1 - t^2}{1 + t^2}$ .
  Uses: 7. Double angle. Rests on: F (Pythagorean identity). Proved in: Sum and difference formulas, double angle and half angle.
7. Rational parametrization. The line of slope $t$ through $(-1, 0)$ meets the unit circle again at $\left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right)$ ; every other point of the circle is obtained in this way for a unique $t$ , and it is rational if and only if $t$ is.
  Uses: 10. Half-angle tangent. Rests on: E (Earlier algebra and arithmetic). Proved in: Sum and difference formulas, double angle and half angle.
8. Pythagorean triples. The primitive Pythagorean triples with $b$ even are exactly $a = m^2 - n^2$ , $b = 2mn$ and $c = m^2 + n^2$ , with $m > n$ , $\operatorname{gcd}(m, n) = 1$ and $m - n$ odd.
  Uses: 11. Rational parametrization. Rests on: E (Earlier algebra and arithmetic). Proved in: Sum and difference formulas, double angle and half angle.
Segment 3 The transformations and the harmonic form
- Convert products into sums and sums into products by adding and subtracting the sum and difference formulas.
- Write every combination $a \sin x + b \cos x$ as a single wave $R \sin(x + \varphi)$ and obtain its range.
- Compose oscillations of equal frequency and explain beats.
- Add sines and cosines in arithmetic progression by means of a telescoping sum.
1. Product to sum. We have $\sin \alpha \cos \beta = \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)$ , and likewise $\cos \alpha \sin \beta$ , $\cos \alpha \cos \beta$ and $\sin \alpha \sin \beta$ are half-sums or half-differences of the sines or of the cosines of $\alpha + \beta$ and of $\alpha - \beta$ .
  Rests on: G (Sum and difference formulas). Proved in: Sum-to-product formulas and the harmonic form.
2. Sum to product. For all real $p$ and $q$ we have $\sin p + \sin q = 2 \sin \frac{p + q}{2} \cos \frac{p - q}{2}$ , with analogous formulas for $\sin p - \sin q$ , $\cos p + \cos q$ and $\cos p - \cos q$ ; from this follow the beats of $\sin(ax) + \sin(bx)$ .
  Uses: 13. Product to sum. Proved in: Sum-to-product formulas and the harmonic form.
3. Harmonic form. If $a$ and $b$ are not both zero and $R = \sqrt{a^2 + b^2}$ , there is a unique $\varphi$ in $\ico{0}{2\pi}$ such that $a \sin x + b \cos x = R \sin(x + \varphi)$ for every $x$ ; that form is unique, and the function has range $\icc{-R}{R}$ .
  Rests on: A (Unit circle), D (Analytic geometry of the plane), F (Pythagorean identity), G (Sum and difference formulas). Proved in: Sum-to-product formulas and the harmonic form.
4. Superposition of oscillations. Unless they cancel, the oscillations $A_1 \sin(\omega t + \varphi_1)$ and $A_2 \sin(\omega t + \varphi_2)$ add up to an oscillation of the same frequency, whose amplitude satisfies $A^2 = A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\varphi_1 - \varphi_2)$ .
  Uses: 5. Cosine of a difference, 15. Harmonic form. Rests on: F (Pythagorean identity), G (Sum and difference formulas). Proved in: Sum-to-product formulas and the harmonic form.
5. Sums in arithmetic progression. If $x$ is not an integer multiple of $2\pi$ , then $\sum_{k=1}^{n} \sin(kx) = \frac{\sin \frac{nx}{2} \sin \frac{(n + 1)x}{2}}{\sin \frac{x}{2}}$ , and the sum of the cosines is obtained with $\cos \frac{(n + 1)x}{2}$ in place of $\sin \frac{(n + 1)x}{2}$ .
  Uses: 13. Product to sum, 14. Sum to product. Rests on: E (Earlier algebra and arithmetic). Proved in: Sum-to-product formulas and the harmonic form.
Segment 4 Equations and inequalities
- Obtain the general solutions of $\sin x = c$ , $\cos x = c$ and $\tan x = c$ from the inverse functions and periodicity.
- Reduce the other equations to the elementary ones through identities, factoring, the equality of two sines or of two cosines, and the harmonic form.
- Neither lose solutions when dividing nor add them when squaring.
- Solve inequalities in one turn, with the unit circle or with sign charts, and extend them periodically.
1. General solutions. If $\abs{c} \leq 1$ , the solutions of $\sin x = c$ are $\arcsin c + 2k\pi$ and $\pi - \arcsin c + 2k\pi$ , and those of $\cos x = c$ are $\arccos c + 2k\pi$ and $-\arccos c + 2k\pi$ ; for every real $c$ , those of $\tan x = c$ are $\arctan c + k\pi$ , with $k$ an integer.
  Uses: 2. Fundamental identities. Rests on: A (Unit circle), B (Inverse functions). Proved in: Trigonometric equations and inequalities.
2. Equality of sines and of cosines. We have $\sin A = \sin B$ if and only if $A = B + 2k\pi$ or $A = \pi - B + 2k\pi$ , and $\cos A = \cos B$ if and only if $A = B + 2k\pi$ or $A = -B + 2k\pi$ , for some integer $k$ ; two defined tangents are equal if and only if $A = B + k\pi$ .
  Uses: 14. Sum to product. Rests on: A (Unit circle), G (Sum and difference formulas). Proved in: Trigonometric equations and inequalities.
3. Polynomial and homogeneous equations. The equation $P(\sin x) = 0$ is equivalent to the equations $\sin x = r$ for the real roots $r$ of $P$ ; a homogeneous equation in $\sin x$ and $\cos x$ reduces to a polynomial one in $\tan x$ by examining the zeros of the cosine separately; and every solution obtained by squaring must be checked.
  Uses: 18. General solutions. Rests on: E (Earlier algebra and arithmetic). Proved in: Trigonometric equations and inequalities.
4. Equation linear in sine and cosine. The equation $a \sin x + b \cos x = c$ has a solution if and only if $c^2 \leq a^2 + b^2$ , and then its solutions are $-\varphi + \arcsin \frac{c}{R} + 2k\pi$ and $\pi - \varphi - \arcsin \frac{c}{R} + 2k\pi$ , where $R$ and $\varphi$ are those of the harmonic form.
  Uses: 15. Harmonic form, 18. General solutions. Proved in: Trigonometric equations and inequalities.
5. Inequalities. If $-1 < c < 1$ and $\beta = \arccos c$ , then $\cos x > c$ if and only if $-\beta + 2k\pi < x < \beta + 2k\pi$ for some integer $k$ , and the analogous statement holds for the sine; products and quotients of trigonometric factors are solved with a sign chart in one turn.
  Uses: 18. General solutions. Rests on: A (Unit circle), B (Inverse functions), E (Earlier algebra and arithmetic). Proved in: Trigonometric equations and inequalities.
Segment 5 Analytic geometry: lines and conics
- Measure the angle between two lines with the tangent of a difference and with the dot product.
- Obtain the bisectors of two lines and the incenter of a triangle.
- Model inaccessible heights, harmonic motion, the range of a projectile and the reflection of light.
- Deduce the rotation formulas, eliminate the term in $xy$ and classify every second-degree equation by the sign of the discriminant.
1. Angle between lines. The slope of a non-vertical line is the tangent of its angle of inclination; two lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $1 + m_1 m_2 = 0$ , and if they are not, the angle $\theta$ between them satisfies $\tan \theta = \abs{\frac{m_2 - m_1}{1 + m_1 m_2}}$ , in agreement with the dot product formula for their direction vectors.
  Uses: 5. Cosine of a difference, 6. Sum and difference formulas. Rests on: B (Inverse functions), D (Analytic geometry of the plane), F (Pythagorean identity), H (Euclidean geometry). Proved in: The angle between two lines and applications of trigonometry.
2. Bisectors and incenter. The points equidistant from two intersecting lines form two mutually perpendicular lines, the bisectors, each of which bisects a different pair of vertical angles; and the three internal bisectors of a triangle, each of which meets the opposite side, meet at an interior point, the incenter, center of the inscribed circle.
  Uses: 8. Power reduction and half angle. Rests on: D (Analytic geometry of the plane), H (Euclidean geometry). Proved in: The angle between two lines and applications of trigonometry.
3. Range of a projectile. Admitting from physics the equations of motion, which combine a uniform horizontal advance with a uniformly accelerated fall, the range of a projectile launched with speed $v$ and elevation $\theta$ is $\frac{v^2 \sin 2\theta}{g}$ ; it is maximal for $\theta = \frac{\pi}{4}$ , and each smaller range is obtained with exactly two angles, which are complementary.
  Uses: 18. General solutions, 7. Double angle. Proved in: The angle between two lines and applications of trigonometry.
4. Rotation formulas. If the axes are rotated through the angle $\theta$ , the new coordinates are $X = x \cos \theta + y \sin \theta$ and $Y = -x \sin \theta + y \cos \theta$ , and the distance between two points does not change.
  Rests on: A (Unit circle), C (Polar coordinates), F (Pythagorean identity), G (Sum and difference formulas). Proved in: Rotation of axes and the general second-degree equation.
5. Coefficients after the rotation. When the axes are rotated, the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ keeps its constant term, and its new coefficient of $XY$ is $B_1 = B \cos 2\theta - (A - C) \sin 2\theta$ .
  Uses: 26. Rotation formulas, 7. Double angle. Proved in: Rotation of axes and the general second-degree equation.
6. Elimination of the xy term. If $B \neq 0$ , there is a unique angle $\theta$ in $\ioo{0}{\frac{\pi}{2}}$ whose rotation makes $B_1$ vanish: the one that satisfies $\cot 2\theta = \frac{A - C}{B}$ .
  Uses: 27. Coefficients after the rotation. Rests on: B (Inverse functions). Proved in: Rotation of axes and the general second-degree equation.
7. Invariants of rotation. For every angle of rotation we have $A_1 + C_1 = A + C$ and $B_1^2 - 4A_1C_1 = B^2 - 4AC$ .
  Uses: 27. Coefficients after the rotation, 7. Double angle. Rests on: F (Pythagorean identity). Proved in: Rotation of axes and the general second-degree equation.
8. Classification by the discriminant. If $A$ , $B$ and $C$ are not all zero and $\Delta = B^2 - 4AC$ , the equation represents an ellipse, a point or no point if $\Delta < 0$ ; a parabola, two parallel lines, one line or no point if $\Delta = 0$ ; and a hyperbola or two intersecting lines if $\Delta > 0$ .
  Uses: 26. Rotation formulas, 28. Elimination of the xy term, 29. Invariants of rotation. Rests on: D (Analytic geometry of the plane). Proved in: Rotation of axes and the general second-degree equation.

Toward complex numbers in polar form. When $\cos \alpha + i \sin \alpha$ is multiplied by $\cos \beta + i \sin \beta$ , the cosine and the sine of $\alpha + \beta$ appear: the sum and difference formulas are the multiplication rule of complex numbers, and the following unit gathers them into a single one, De Moivre’s formula.

In the rest of this lesson

The connection with what came before
The link with the following unit
Exit assessment
Challenges
The tangents of the multiples of pi over seven
A telescoping sum of arctangents
The largest sum of cosines in a triangle
A product of cosines equal to one
The nearest and farthest points of a rotated ellipse
The tangents to a parabola that form forty-five degrees
A tower seen at angles in progression
A positive sum of four sines
The largest rectangle in a circular sector
A family of conics with foci at distance two