00.03.E1 · precalc / equations-lines

Functions, trigonometry, and coordinate-geometry exercise pack (Lang Basic Mathematics Part III-V supplement)

shippedIntermediate-onlyLean: nonepending prereqs

Anchor (Master):

Formal definition of the pack Intermediate

Lang's Basic Mathematics Parts III-V build coordinate geometry, functions, and trigonometry as a single derived structure: lines and conics come from their geometric loci, the trigonometric functions from the unit circle, and the addition formulas from the rotation matrix acting on $R^{2}$ . This pack exercises that block. The problems test function composition and inverses 00.02.05, the equation of a line and slope conditions 00.03.01, right-triangle trigonometry and the Pythagorean identity 00.06.01, the unit-circle extension to all angles 00.07.01, the addition and double-angle formulas 00.08.01, the distance formula in the plane 00.09.01, and the standard-form equations of the conics 00.10.01.

The pack collects ten problems: three easy, four medium, three hard. Each carries a hint and a full solution. Angles are measured in radians by default, matching Lang's Part V convention and the unit-circle unit; where a degree value is more natural for a special triangle it is given explicitly. Lines are written in slope-intercept or point-slope form, conics in the centre-translated standard form, and functions in Lang's $f : A \to B$ notation.

The problems are meant to be worked alongside the prerequisite concept units. The recurring theme is Lang's reverse order: rather than declaring an equation and graphing it, the reader derives the equation from a geometric or analytic definition — a line from two points, an identity from a rotation, a conic from a focus condition — and then verifies the algebra. The trigonometry problems in particular lean on deriving identities rather than recalling them.

Key theorem with full solution Intermediate

We work one exercise in full as an exemplar of the format. The remaining nine follow the same structure: problem, hint, full answer in <details> blocks.

Lead exercise. Derive the addition formula $cos (x + y) = cos x cos y - sin x sin y$ from the rotation matrix, then deduce the double-angle formula for $cos 2 x$ .

Solution. A rotation of the plane by angle $θ$ is the linear map with matrix

R_{θ} = (cos θ sin θ - sin θ cos θ),

read off by tracking where the basis vectors $(1, 0)$ and $(0, 1)$ land on the unit circle 00.07.01. Rotating by $x$ and then by $y$ is the same as rotating by $x + y$ , so $R_{y} R_{x} = R_{x + y}$ . Multiply the matrices on the left:

R_{y} R_{x} = (cos y sin y - sin y cos y) (cos x sin x - sin x cos x) = (cos y cos x - sin y sin x \dots \dots \dots) .

The top-left entry of $R_{x + y}$ is $cos (x + y)$ by definition. Equating the top-left entries of the two sides gives

cos (x + y) = cos x cos y - sin x sin y .

The same comparison on the bottom-left entry yields $sin (x + y) = sin x cos y + cos x sin y$ .

Double angle. Set $y = x$ in the cosine formula:

cos 2 x = cos x cos x - sin x sin x = cos^{2} x - sin^{2} x .

Using the Pythagorean identity $sin^{2} x + cos^{2} x = 1$ 00.06.01, this also reads $cos 2 x = 2 cos^{2} x - 1 = 1 - 2 sin^{2} x$ . $□$

The rotation-matrix derivation is Lang's signature move: the addition formulas feel inevitable because they are just the statement that composing two rotations adds the angles. Every other identity in the pack — half-angle, product-to-sum, $cos 3 x$ — is a corollary.

Exercises Intermediate

Exercise 8 (hard, symbolic). Ellipse from the focus-sum definition.

Derive the standard equation of the ellipse defined as the locus of points whose distances to the two foci $(\pm c, 0)$ sum to the constant $2 a$ (with $a > c > 0$ ). Identify the semi-minor axis $b$ and the eccentricity.

Hint

Write the distance-sum condition, isolate one radical, square, simplify, isolate the remaining radical, and square again. The combination $a^{2} - c^{2}$ will appear; name it $b^{2}$ .

Answer

A point $(x, y)$ on the ellipse satisfies $(x + c)^{2} + y^{2} + (x - c)^{2} + y^{2} = 2 a$ .

Isolate and square the first radical: $(x + c)^{2} + y^{2} = 2 a - (x - c)^{2} + y^{2}$ . Squaring both sides,

(x + c)^{2} + y^{2} = 4 a^{2} - 4 a (x - c)^{2} + y^{2} + (x - c)^{2} + y^{2} .

The $(x \pm c)^{2}$ terms expand to $x^{2} \pm 2 c x + c^{2}$ ; subtracting, the left minus the corresponding right gives $4 c x$ . So $4 c x = 4 a^{2} - 4 a (x - c)^{2} + y^{2}$ , i.e. $a (x - c)^{2} + y^{2} = a^{2} - c x$ . Square again:

a^{2} [(x - c)^{2} + y^{2}] = a^{4} - 2 a^{2} c x + c^{2} x^{2} .

Expand the left side and cancel the $- 2 a^{2} c x$ terms that appear on both sides:

a^{2} x^{2} - a^{2} c^{2} + a^{2} c^{2} + a^{2} y^{2} = a^{4} + c^{2} x^{2} - a^{2} c^{2} \Rightarrow (a^{2} - c^{2}) x^{2} + a^{2} y^{2} = a^{2} (a^{2} - c^{2}) .

Set $b^{2} = a^{2} - c^{2} > 0$ and divide by $a^{2} b^{2}$ :

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1.

The eccentricity is $e = c / a \in (0, 1)$ , and $b = a^{2} - c^{2}$ is the semi-minor axis. The derive-then-verify order is Lang's 00.10.01: the geometric locus comes first, the algebra confirms it.

Exercise 9 (hard, proof). Perpendicular slopes multiply to $- 1$ .

Prove that two non-vertical lines with slopes $m_{1}$ and $m_{2}$ are perpendicular if and only if $m_{1} m_{2} = - 1$ . Use the distance formula and the converse of the Pythagorean theorem.

Hint

Translate so the lines cross at the origin. Take the points $(1, m_{1})$ and $(1, m_{2})$ on them; the lines are perpendicular iff the triangle with vertices $O, (1, m_{1}), (1, m_{2})$ has its right angle at $O$ , i.e. iff the Pythagorean relation holds on the three squared distances.

Answer

Place the intersection at the origin, so the lines are $y = m_{1} x$ and $y = m_{2} x$ . Pick $P_{1} = (1, m_{1})$ and $P_{2} = (1, m_{2})$ , one on each line. The lines are perpendicular exactly when the angle at $O$ in triangle $O P_{1} P_{2}$ is a right angle, which by the converse of the Pythagorean theorem holds iff

∣ O P_{1} ∣^{2} + ∣ O P_{2} ∣^{2} = ∣ P_{1} P_{2} ∣^{2} .

Compute the squared distances 00.09.01: $∣ O P_{1} ∣^{2} = 1 + m_{1}^{2}$ , $∣ O P_{2} ∣^{2} = 1 + m_{2}^{2}$ , and $∣ P_{1} P_{2} ∣^{2} = (1 - 1)^{2} + (m_{1} - m_{2})^{2} = m_{1}^{2} - 2 m_{1} m_{2} + m_{2}^{2}$ .

The Pythagorean condition becomes

(1 + m_{1}^{2}) + (1 + m_{2}^{2}) = m_{1}^{2} - 2 m_{1} m_{2} + m_{2}^{2},

i.e. $2 = - 2 m_{1} m_{2}$ , i.e. $m_{1} m_{2} = - 1$ . Each step is an equivalence, so perpendicularity holds iff $m_{1} m_{2} = - 1$ . $□$

Exercise 10 (hard, symbolic). Sum-to-product from the addition formulas.

Derive the identity $sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})$ , and use it to express $sin 75° + sin 15°$ in closed form.

Hint

Set $A = s + t$ and $B = s - t$ , so $s = (A + B) /2$ and $t = (A - B) /2$ . Add the sine addition formulas for $sin (s + t)$ and $sin (s - t)$ .

Answer

Let $s = (A + B) /2$ and $t = (A - B) /2$ , so $A = s + t$ and $B = s - t$ . By the sine addition formula 00.08.01,

sin (s + t) = sin s cos t + cos s sin t, sin (s - t) = sin s cos t - cos s sin t .

Adding, the $cos s sin t$ terms cancel:

sin A + sin B = sin (s + t) + sin (s - t) = 2 sin s cos t = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2}) .

Application. With $A = 75°$ , $B = 15°$ : $(A + B) /2 = 45°$ and $(A - B) /2 = 30°$ , so

sin 75° + sin 15° = 2 sin 45° cos 30° = 2 \cdot \frac{2}{2} \cdot \frac{3}{2} = \frac{6}{2} .

The sum-to-product identities are corollaries of the addition formulas, obtained by adding or subtracting the $\pm t$ versions — Lang's derived-not-memorised treatment.

Exercise pack for Lang, Basic Mathematics, Part III-V. Functions, trigonometry, and coordinate geometry: composition and inverses, lines and conics, right-triangle and unit-circle trigonometry, the addition formulas. Distribution: 3 easy / 4 medium / 3 hard.

Prerequisites

00.02.05 pending
00.03.01 pending
00.06.01 pending
00.07.01 pending
00.08.01 pending
00.09.01 pending
00.10.01 pending

Tier anchors

beginner
intermediate: Lang, Basic Mathematics, Part III (Coordinate Geometry, Ch. 9-13), Part IV (Functions, Ch. 14), and Part V (Trigonometry, Ch. 16-19) exercise sets: functions and graphs, lines and conics, right-triangle and unit-circle trigonometry, and the addition formulas
master

References

Lang — Basic Mathematics (Springer, 1988) · Part III (Ch. 9 lines, Ch. 10-12 conics, Ch. 13 polar/parametric), Part IV (Ch. 14 functions), Part V (Ch. 16 right-triangle trig, Ch. 17 unit-circle and addition formulas, Ch. 18 inverse trig and laws); chapter-end exercise sets
Axler — Precalculus: A Prelude to Calculus · Ch. 1-2 (functions, transformations, composition, inverses), Ch. 4-6 (trigonometric functions, unit circle, identities)
Stillwell — The Four Pillars of Geometry · Ch. 3-4 (coordinates, the distance formula, lines, and conics from the Cartesian viewpoint)

Estimated time

intermediate: 4h