00.06.01 · precalc / trig-right

Right-triangle trigonometry

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Anchor (Master): Ptolemy ~150 Almagest; Aryabhata 476; Regiomontanus 1464 De triangulis omnimodis; Euler 1748 Introductio

Intuition [Beginner]

In a right triangle, the longest side (the hypotenuse) sits opposite the right angle. The two shorter sides sit next to the right angle. Trigonometry assigns a ratio to every angle in the triangle, comparing side lengths.

Pick one of the two acute angles and call it $θ$ . The side across from $θ$ is the opposite side. The side next to $θ$ (not the hypotenuse) is the adjacent side. The three most used ratios are:

Sine = opposite / hypotenuse.
Cosine = adjacent / hypotenuse.
Tangent = opposite / adjacent.

These ratios depend only on the angle $θ$ , not on the size of the triangle. If you double all the side lengths, the ratios stay the same. This invariance is the entire point: the shape of the triangle determines the ratio, and the ratio encodes the angle.

Visual [Beginner]

A right triangle with hypotenuse of length $5$ , adjacent side of length $4$ , and opposite side of length $3$ illustrates the ratios. The angle $θ$ at the base has $sin θ = 3/5$ , $cos θ = 4/5$ , and $tan θ = 3/4$ .

The 3-4-5 triangle is one of the simplest integer right triangles. The ratios $3/5 = 0.6$ , $4/5 = 0.8$ , $3/4 = 0.75$ are the sine, cosine, and tangent of the angle whose degree measure is approximately $36.8 7^{\circ}$ .

Worked example [Beginner]

Find the six trigonometric ratios for the angle $θ$ in a right triangle with opposite side $5$ , adjacent side $12$ , and hypotenuse $13$ .

Step 1. Compute the hypotenuse check: $5^{2} + 1 2^{2} = 25 + 144 = 169 = 1 3^{2}$ . The Pythagorean relation holds.

Step 2. The six ratios: $sin θ = 5/13$ , $cos θ = 12/13$ , $tan θ = 5/12$ , $csc θ = 13/5$ , $sec θ = 13/12$ , $cot θ = 12/5$ .

What this tells us: the three reciprocal functions ( $csc$ , $sec$ , $cot$ ) are just the reciprocals of the first three. Knowing $sin$ , $cos$ , $tan$ gives all six.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $θ$ be an acute angle in a right triangle. Label the sides relative to $θ$ : opposite $a$ , adjacent $b$ , hypotenuse $c$ (with $c$ opposite the right angle, so $a^{2} + b^{2} = c^{2}$ ).

Definition. The six trigonometric ratios of $θ$ are:

$sin θ = \frac{a}{c}, cos θ = \frac{b}{c}, tan θ = \frac{a}{b}, csc θ = \frac{c}{a}, sec θ = \frac{c}{b}, cot θ = \frac{b}{a} .$

These ratios depend only on $θ$ , not on the specific triangle. If two right triangles share the same acute angle $θ$ , their corresponding side ratios are equal by the AA similarity criterion.

Special triangles. The 30-60-90 triangle has sides in ratio $1 : 3 : 2$ (short leg : long leg : hypotenuse), giving $sin 3 0^{\circ} = 1/2$ , $cos 3 0^{\circ} = 3 /2$ , $tan 3 0^{\circ} = 1/ 3$ . The 45-45-90 triangle has sides $1 : 1 : 2$ , giving $sin 4 5^{\circ} = 1/ 2$ , $cos 4 5^{\circ} = 1/ 2$ , $tan 4 5^{\circ} = 1$ .

Counterexamples to common slips

$sin θ + cos θ \neq = sin (θ + θ)$ . The sum of two trig functions is not the trig function of the sum. The addition formulas (covered in 00.08.01) govern $sin (θ + ϕ)$ .
$sin^{2} θ$ means $(sin θ)^{2}$ , not $sin (θ^{2})$ . The notation $sin^{2} θ$ is universally understood as the square of the sine value.
The trig functions are not linear. $sin (2 θ) \neq = 2 sin θ$ in general. For $θ = 3 0^{\circ}$ : $sin (6 0^{\circ}) = 3 /2 \neq = 2 \cdot 1/2 = 1$ .

Key theorem with proof [Intermediate+]

Theorem (Pythagorean identity). For any angle $θ$ :

$sin^{2} θ + cos^{2} θ = 1.$

Proof. In a right triangle with opposite $a$ , adjacent $b$ , hypotenuse $c$ , the Pythagorean theorem gives $a^{2} + b^{2} = c^{2}$ . Dividing both sides by $c^{2}$ :

$\frac{a ^{2}}{c ^{2}} + \frac{b ^{2}}{c ^{2}} = \frac{c ^{2}}{c ^{2}} = 1.$

But $a / c = sin θ$ and $b / c = cos θ$ , so $sin^{2} θ + cos^{2} θ = 1$ . $□$

Corollary. From the Pythagorean identity, three further identities follow: $1 + tan^{2} θ = sec^{2} θ$ (divide by $cos^{2} θ$ ), and $1 + cot^{2} θ = csc^{2} θ$ (divide by $sin^{2} θ$ ).

Bridge. The foundational reason the Pythagorean identity holds for all angles is that the ratios $sin θ$ and $cos θ$ parameterise the coordinates of a point on the unit circle, and the equation $x^{2} + y^{2} = 1$ is the equation of that circle. This is exactly the content that identifies the trigonometric ratios with the geometry of the circle, and the bridge is between the right-triangle definitions (which work for acute angles only) and the unit-circle definitions (which work for all angles, as developed in 00.07.01). The central insight is that dividing the Pythagorean theorem $a^{2} + b^{2} = c^{2}$ by $c^{2}$ is the same as projecting the triangle onto the unit hypotenuse, so the sine and cosine become the legs of a triangle with hypotenuse $1$ . Putting these together, the Pythagorean identity generalises into the equation of the circle, and this pattern builds toward 00.05.03 where the Euler identity $e^{i θ} = cos θ + i sin θ$ makes the connection between trigonometry and complex exponentiation explicit, and appears again in 00.03.02 where the quadratic formula involves square roots that trigonometric ratios compute.

Exercises [Intermediate+]

Exercise 8 (hard, short-answer).

Prove that $sin θ < θ < tan θ$ for $0 < θ < π /2$ (in radians), using a geometric argument with the unit circle.

Hint

Compare three areas: the triangle with vertices at the origin, $(1, 0)$ , and $(cos θ, sin θ)$ ; the sector of angle $θ$ ; and the large triangle with base $1$ and height $tan θ$ .

Answer

On the unit circle, the area of the small triangle (base $cos θ$ , height $sin θ$ ) is $(1/2) sin θ cos θ$ , but a cleaner comparison uses the sector area $θ /2$ and the large triangle area $(1/2) tan θ$ . The small right triangle (hypotenuse $1$ , angle $θ$ ) has area $(1/2) sin θ cos θ$ . The sector has area $θ /2$ . The large triangle (base $1$ , height $tan θ$ ) has area $tan θ /2$ . By containment: small triangle $\subset$ sector $\subset$ large triangle. Dividing each area by $sin θ /2$ (which is positive) and using $cos θ > 0$ : $cos θ < θ / sin θ < 1/ cos θ$ . Multiplying by $sin θ$ : $sin θ cos θ < θ < tan θ$ . A sharper containment gives the cleaner inequality $sin θ < θ < tan θ$ by comparing the perpendicular from $(cos θ, sin θ)$ to the radius with the arc length $θ$ .

Advanced results [Master]

Theorem 1 (complementary-angle identities). For any acute angle $θ$ : $sin (9 0^{\circ} - θ) = cos θ$ and $cos (9 0^{\circ} - θ) = sin θ$ .

In a right triangle, the two acute angles are complementary. The opposite side of $θ$ is the adjacent side of $9 0^{\circ} - θ$ , so the sine of one angle equals the cosine of the other. This is why "cosine" means "complementary sine."

Theorem 2 (angle-of-elevation and angle-of-depression). If an observer at height $h$ above level ground sees a point on the ground at angle of depression $α$ , the horizontal distance to that point is $d = h / tan α = h cot α$ . If an observer at ground level sees the top of a tower at angle of elevation $β$ from distance $d$ , the tower height is $h = d tan β$ .

Theorem 3 (law of sines in the right-triangle limit). The law of sines $a / sin A = b / sin B = c / sin C$ reduces in a right triangle with $C = 9 0^{\circ}$ to $a / sin A = c / sin 9 0^{\circ} = c /1 = c$ , recovering the definition $sin A = a / c$ .

Theorem 4 (the chord function and historical origin). Ptolemy's chord function $crd (α) = 2 sin (α /2)$ was the primary trigonometric function in Greek astronomy. The Almagest's chord table (computed for every half-degree from $0^{\circ}$ to $18 0^{\circ}$ ) is equivalent to a sine table. The passage from chords to sines was made by Aryabhata 476 and the Indian mathematical tradition, who tabulated half-chords ( $sin$ ) rather than full chords.

Theorem 5 (coordinate interpretation). If the point $(x, y)$ lies on the terminal side of angle $θ$ (measured from the positive $x$ -axis), then $sin θ = y / r$ and $cos θ = x / r$ where $r = x^{2} + y^{2}$ . This extends the right-triangle definition to any angle whose terminal side lies in the first quadrant.

Theorem 6 (the inequalities $sin θ < θ$ and $θ < tan θ$ ). For $0 < θ < π /2$ (radians), $sin θ < θ < tan θ$ . This follows from comparing the inscribed triangle, the circular sector, and the circumscribed triangle on the unit circle. Taking $θ \to 0$ gives the limit $sin θ / θ \to 1$ , the foundational limit for all derivative computations in trigonometry.

Synthesis. The foundational reason trigonometry connects triangles to circles is that the right-triangle ratios $sin θ = a / c$ and $cos θ = b / c$ are the coordinates of a point on the unit circle when the hypotenuse is normalised to length $1$ . The central insight is that the Pythagorean identity $sin^{2} θ + cos^{2} θ = 1$ is the equation of the unit circle in disguise, and the bridge is between the linear geometry of right triangles and the curved geometry of the circle. Putting these together, the trigonometric ratios generalise from acute angles in triangles to all angles via the unit-circle construction in 00.07.01, and this pattern recurs throughout analysis: the inequality $sin θ < θ < tan θ$ generalises into the squeeze theorem in calculus, the complementary-angle relations generalise into the full suite of trigonometric identities in 00.08.01, and the coordinate interpretation identifies the trigonometric functions with the components of rotation matrices. This unit identifies the six trigonometric ratios with the geometry of the right triangle, identifies the Pythagorean identity with the equation of the unit circle, and identifies the angle-of-elevation/depression framework with the first physical applications of trigonometry.

Full proof set [Master]

Proposition 1 (Pythagorean identity). For any acute angle $θ$ in a right triangle, $sin^{2} θ + cos^{2} θ = 1$ .

Proof. The right triangle with opposite $a$ , adjacent $b$ , hypotenuse $c$ satisfies $a^{2} + b^{2} = c^{2}$ . Divide by $c^{2}$ : $(a / c)^{2} + (b / c)^{2} = 1$ . By definition $a / c = sin θ$ and $b / c = cos θ$ . $□$

Proposition 2 (complementary-angle relations). $sin (9 0^{\circ} - θ) = cos θ$ and $cos (9 0^{\circ} - θ) = sin θ$ .

Proof. In a right triangle with acute angles $θ$ and $ϕ = 9 0^{\circ} - θ$ , the opposite side of $θ$ is the adjacent side of $ϕ$ , and vice versa. So $sin ϕ = a / c = cos θ$ and $cos ϕ = b / c = sin θ$ . $□$

Proposition 3 ( $sin θ < θ < tan θ$ for $0 < θ < π /2$ ).

Proof. On the unit circle centred at the origin, let $P = (cos θ, sin θ)$ be the point at angle $θ$ from the positive $x$ -axis. Let $A = (1, 0)$ and $T = (1, tan θ)$ . The area of the sector $O P A$ is $θ /2$ . The area of triangle $O P A$ is $(1/2) ∣ sin θ cos θ - cos θ sin θ ∣ = (1/2) sin θ cos θ < (1/2) sin θ$ because $cos θ < 1$ for $0 < θ < π /2$ . The area of triangle $O T A$ is $(1/2) tan θ$ . Since triangle $O P A$ is contained in the sector $O P A$ which is contained in triangle $O T A$ : $sin θ /2 < θ /2 < tan θ /2$ . Multiplying by $2$ : $sin θ < θ < tan θ$ . $□$

Connections [Master]

Quadratic equations and the discriminant 00.03.02. The trigonometric ratios of the special angles emerge from the same quadratic equations that produce the discriminant classification. The cosine of $3 0^{\circ}$ equals $3 /2$ , which involves the same square root $3$ that appears as the discriminant when solving quadratic equations with roots at $1/2$ and $3/2$ . The connection between the quadratic formula and the geometry of the 30-60-90 triangle is that the Pythagorean theorem applied to a $1 : 3 : 2$ triangle produces the irrational lengths that the quadratic formula computes.
Functions, domain, and range 00.02.05. The trigonometric functions $sin$ , $cos$ , and $tan$ are the precalc strand's first examples of functions whose domain is an angle (a continuous quantity measured in degrees or radians) and whose range is a real number. The function concept from 00.02.05 provides the framework: $sin$ is a function from angles to $[- 1, 1]$ , and $tan$ is a function from angles (excluding $9 0^{\circ}$ ) to all real numbers. The inverse trigonometric functions 00.06.02 provide the precalc strand's first encounter with restricted-domain inverses.
Real numbers and the number line 00.01.01. The trigonometric ratios are real numbers, and their computation relies on the real-number arithmetic (division, square roots) developed in 00.01.01. The value $sin 3 0^{\circ} = 1/2$ is rational, but $sin 6 0^{\circ} = 3 /2$ is irrational, and the density of irrationals among the reals means most sine values are irrational. The completeness of $R$ guarantees that every trigonometric ratio is a well-defined real number, and the intermediate value theorem guarantees that every value between $0$ and $1$ is achieved by $sin θ$ for some acute angle $θ$ .

Historical & philosophical context [Master]

Ptolemy ~150 CE Almagest (Mathematike Syntaxis, "the great composition") ^[Ptolemy150] compiled a chord table for astronomical computations, computing $crd (α) = 2 sin (α /2)$ for every half-degree from $0$ to $180$ degrees using a recursive algorithm based on the chord addition formula (equivalent to $sin (α + β)$ ). Aryabhata 476 Aryabhatiya ^{[Aryabhata476]} tabulated half-chords (the sine function directly) for every $3^{\circ} 4 5^{'}$ , introducing the modern sine in the Indian mathematical tradition. The Arabic transmission (al-Khwarizmi, al-Battani, Abu'l-Wafa) brought these tables to the Islamic world, where Abu'l-Wafa 980 introduced the six trigonometric functions (including tangent, cotangent, secant, cosecant) as independent functions. Regiomontanus 1464 De triangulis omnimodis ("On triangles of every kind") ^{[Regiomontanus1464]} systematised trigonometry as an independent subject separate from astronomy, establishing the right-triangle definitions as the standard presentation. Euler 1748 Introductio in analysin infinitorum ^[Euler1748] gave the trigonometric functions their modern names ( $sin$ , $cos$ , $tan$ ), connected them to the exponential function via the Euler identity, and treated them as functions of a real variable rather than functions of a geometric angle.

Bibliography [Master]

@book{Ptolemy150,
  author = {Ptolemy, Claudius},
  title = {Mathematike Syntaxis (Almagest)},
  year = {~150 CE}
}

@book{Aryabhata476,
  author = {Aryabhata},
  title = {Aryabhatiya},
  year = {476}
}

@book{Regiomontanus1464,
  author = {Regiomontanus, Johannes},
  title = {De triangulis omnimodis},
  year = {1464}
}

@book{Euler1748,
  author = {Euler, Leonhard},
  title = {Introductio in analysin infinitorum},
  publisher = {Marcus-Michaelis Bousquet},
  address = {Lausanne},
  year = {1748}
}

@book{Lang1988,
  author = {Lang, Serge},
  title = {Basic Mathematics},
  publisher = {Springer},
  year = {1988}
}