The Timpson Lyceum · Calculus 1 (2024)

Trig Identities, Graphs, & Theorems - Lecture 3 (Calculus 1)

Overview

This lecture is a thorough second pass through trigonometry, building every tool from the ground up rather than asking you to memorize it. It starts with radians and arc length, defines the six trig functions from a point on a circle, and then derives (not just states) the standard family of trig identities: the Pythagorean identities, the angle-sum/difference formulas, the double-angle and half-angle formulas, and the Law of Cosines and Law of Sines. It closes with the small-angle inequality $\sin\theta \le \theta$, the graphs of all six trig functions, and how to transform a sinusoid (amplitude, period, and shifts). The payoff is being able to evaluate "off-the-unit-circle" angles exactly and to read or build any sine/cosine wave — skills you lean on constantly in calculus, where these functions and their identities appear everywhere.

Key Concepts

Radians and arc length

• A radian measures an angle by how many radii of arc you travel: walk an arc length equal to one radius and you have swept exactly one radian.
• Arc length formula: $s = r\theta$, equivalently $\theta = s/r$, with $\theta$ in radians.
• A full circle is $2\pi r$ around, so $2\pi$ radians is once around and $\pi$ radians is halfway. From here on, every angle is assumed to be in radians unless degrees are stated.
• On the unit circle ($r = 1$) the arc length equals the angle, $s = \theta$.

The six trig functions (circle definition)

• For a point at angle $\theta$ on a circle of radius $r$, drop to the $x$-axis to get coordinates $(x, y)$ (either coordinate may be negative): $\cos\theta = \tfrac{x}{r},\quad \sin\theta = \tfrac{y}{r},\quad \tan\theta = \tfrac{y}{x}$
• The reciprocals: $\sec\theta = \tfrac{r}{x}$, $\csc\theta = \tfrac{r}{y}$, $\cot\theta = \tfrac{x}{y}$.
• On the unit circle, $\cos\theta$ is just the $x$-coordinate and $\sin\theta$ is the $y$-coordinate. A useful consequence for any radius: $x = r\cos\theta$ and $y = r\sin\theta$.

Pythagorean identities (from $x^2 + y^2 = 1$ on the unit circle)

• $\sin^2\theta + \cos^2\theta = 1$
• Dividing through by $\cos^2\theta$: $\tan^2\theta + 1 = \sec^2\theta$
• Dividing through by $\sin^2\theta$: $1 + \cot^2\theta = \csc^2\theta$

Angle-sum and difference formulas (the hardest derivations, built from a geometric construction)

• $\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$
• $\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$
• The $\mp/\pm$ pairing matters: the top signs go together, the bottom signs go together.

Double-angle formulas (special case $\alpha = \beta = \theta$)

• $\sin 2\theta = 2\sin\theta\cos\theta$
• $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$ (the last two use $\sin^2 + \cos^2 = 1$).

Half-angle formulas (solve the $\cos 2\theta$ forms for $\cos^2$ / $\sin^2$, then take roots)

• $\cos\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}$
• $\sin\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}$
• Pick the sign from the quadrant of the half-angle.

Law of Cosines (generalized Pythagorean theorem for any triangle, sides $a,b,c$ opposite angles $A,B,C$)

• $c^2 = a^2 + b^2 - 2ab\cos C$ (and the cyclic versions for $a^2$ and $b^2$).
• When the included angle is $90^\circ$, $\cos C = 0$ and it collapses to $c^2 = a^2 + b^2$ — the ordinary Pythagorean theorem.

Law of Sines

• $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ — these three ratios are always equal in any triangle (proved by dropping an altitude and matching its two expressions).

Small-angle inequality

• For $\theta$ in radians, $\sin\theta \le \theta$ (and the geometry gives $\sin^2\theta \le \theta^2$ and $(1-\cos\theta)^2 \le \theta^2$). On the unit circle the chord/height can never exceed the arc, since a straight line is the shortest path. This depends on using radians.

Graphs and their features

• $\sin x$ and $\cos x$ oscillate between $-1$ and $1$ (they are the $y$- and $x$-coordinates of a spinning point); $\cos x$ is $\sin x$ shifted left.
• $\csc x$, $\sec x$ have vertical asymptotes wherever $\sin x$, $\cos x$ equal $0$ (reciprocal of $0$ is undefined), so their domains exclude those points.
• $\tan x$ rises with slope, with vertical asymptotes at odd multiples of $\tfrac{\pi}{2}$; $\cot x$ is its decreasing counterpart.
• Parity: $\sin$, $\csc$, $\tan$, $\cot$ are odd ($f(-x) = -f(x)$); $\cos$ and $\sec$ are even ($f(-x) = f(x)$).

Transformations of a sinusoid

• General form $y = a\,\sin\!\big(b(x - c)\big) + d$:
  • $a$ = amplitude (vertical stretch),
  • $d$ = vertical shift, $c$ = horizontal shift,
  • $b$ controls the period: a large $b$ compresses, a small $b$ stretches.
• Periodic function: $f(x + p) = f(x)$ for some positive $p$; the smallest such $p$ is the period. The period of a sinusoid is $\tfrac{2\pi}{b}$, so $\sin x$ (with $b=1$) repeats every $2\pi$; $\tan x$ has period $\pi$.

Worked Example

Evaluate $\cos\dfrac{7\pi}{12}$ exactly using the cosine angle-sum formula. First write the angle as a sum of two known angles, $\tfrac{7\pi}{12} = \tfrac{\pi}{3} + \tfrac{\pi}{4}$, then apply $\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$:

The negative result is consistent with $\tfrac{7\pi}{12}$ landing just past $\tfrac{\pi}{2}$ in the second quadrant, where the $x$-coordinate (the cosine) is slightly negative.

Summary

• A radian is "one radius of arc," giving the workhorse formula $s = r\theta$; on the unit circle the angle equals the arc length.
• The six trig functions are ratios of $(x, y)$ to $r$; on the unit circle $\cos\theta = x$ and $\sin\theta = y$, and in general $x = r\cos\theta$, $y = r\sin\theta$.
• The three Pythagorean identities all fall out of $\sin^2\theta + \cos^2\theta = 1$ by dividing through by $\cos^2\theta$ or $\sin^2\theta$.
• The angle-sum/difference formulas are the parents of the double-angle and half-angle formulas, and they let you compute exact values for angles outside the unit circle, like $\cos\tfrac{7\pi}{12}$.
• The Law of Cosines generalizes the Pythagorean theorem to any triangle (recovering it when the included angle is $90^\circ$), and the Law of Sines equates the side-to-sine ratios.
• Each trig function has a characteristic graph — bounded waves for $\sin/\cos$, asymptotes for the reciprocals and $\tan/\cot$, with definite parity — and any sinusoid is captured by $y = a\sin(b(x-c)) + d$, where $a$ is amplitude and the period is $\tfrac{2\pi}{b}$.