Trigonometric Ratios

⚡ Key Takeaways

Trigonometric ratios are six fixed numbers — sine, cosine, tangent, cosecant, secant, cotangent — that relate the angles of a right triangle to ratios of its side lengths. Key result in words: in any right triangle, the ratio of two specified sides depends only on the acute angle, not on the triangle's size. Where it applies: surveying, navigation, physics of waves and oscillations, engineering, computer graphics. Who/when: rooted in Greek and Indian astronomy (Hipparchus ~150 BCE; Aryabhata ~500 CE), formalized into modern notation by the 16th–18th centuries.

💡 The Core Insight

Two right triangles with the same acute angle are similar, so corresponding sides are proportional. This means the ratio of any two sides is the same for all such triangles — it is a property of the angle alone. We give each of the six possible side-ratios a name. That single fact — "ratios depend only on the angle" — is the entire foundation of trigonometry.

DefinitionThe Six Ratios in a Right Triangle

Consider a right triangle with an acute angle $\theta$ . Label the sides relative to $\theta$ :

opposite ( $o$ ): the side facing $\theta$ ,
adjacent ( $a$ ): the leg touching $\theta$ (not the hypotenuse),
hypotenuse ( $h$ ): the side opposite the right angle (the longest side).

The six trigonometric ratios are:

1

\quad \sin\theta = \frac{o}{h}, \qquad \cos\theta = \frac{a}{h}, \qquad \tan\theta = \frac{o}{a}

Derivation hint: each is a direct quotient of two labeled sides; "SOH-CAH-TOA" encodes opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent.

2

\quad \csc\theta = \frac{h}{o} = \frac{1}{\sin\theta}, \qquad \sec\theta = \frac{h}{a} = \frac{1}{\cos\theta}, \qquad \cot\theta = \frac{a}{o} = \frac{1}{\tan\theta}

Derivation hint: invert each primary ratio by swapping numerator and denominator. Here $o, a, h > 0$ are side lengths and $0° < \theta < 90°$ in the triangle definition.

Figure

A right triangle showing opposite, adjacent, and hypotenuse relative to angle θ

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TheoremPythagorean Identity

For every angle $\theta$ for which the ratios are defined,

3

\quad \sin^2\theta + \cos^2\theta = 1

Derivation hint: substitute $\sin\theta = o/h$ , $\cos\theta = a/h$ and apply the Pythagorean theorem $o^2 + a^2 = h^2$ . Two companion identities follow by dividing [3] by $\cos^2\theta$ and by $\sin^2\theta$ :

4

\quad 1 + \tan^2\theta = \sec^2\theta, \qquad 1 + \cot^2\theta = \csc^2\theta

Derivation hint: divide $\sin^2\theta + \cos^2\theta = 1$ through by $\cos^2\theta$ (first) and by $\sin^2\theta$ (second), then recognize each quotient as a named ratio.

TheoremCofunction and Quotient Relations

5

\quad \tan\theta = \frac{\sin\theta}{\cos\theta}, \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Derivation hint: $\dfrac{\sin\theta}{\cos\theta} = \dfrac{o/h}{a/h} = \dfrac{o}{a} = \tan\theta$ .

6

\quad \sin(90° - \theta) = \cos\theta, \qquad \tan(90° - \theta) = \cot\theta, \qquad \sec(90° - \theta) = \csc\theta

Derivation hint: the two acute angles of a right triangle sum to $90°$ , so the side that is "opposite" one is "adjacent" to the other — swapping roles swaps each function with its co-function.

ProofRatios depend only on the angle

We prove the foundational claim that makes $\sin\theta$ , etc., well-defined functions of $\theta$ alone. 1. Setup. Let two right triangles $T_1$ and $T_2$ both contain an acute angle equal to $\theta$ , and both contain a right angle. 2. Angle sum. Each triangle's angles sum to $180°$ . With one angle $\theta$ and one angle $90°$ , the third angle is $90° - \theta$ in both triangles. 3. Similarity (AA). Two triangles with two pairs of equal angles are similar by the Angle–Angle criterion. Hence $T_1 \sim T_2$ . 4. Proportional sides. Similarity means corresponding sides are in a common proportion: $\dfrac{o_1}{o_2} = \dfrac{a_1}{a_2} = \dfrac{h_1}{h_2} = k$ for some scale factor $k$ . 5. Ratios survive scaling. Therefore $\dfrac{o_1}{h_1} = \dfrac{k\,o_2}{k\,h_2} = \dfrac{o_2}{h_2}$ , and likewise for every pair. The ratio is unchanged by the triangle's size. 6. Conclusion. Each of the six quotients is determined by $\theta$ alone, so $\sin$ , $\cos$ , $\tan$ , $\csc$ , $\sec$ , $\cot$ are genuine functions of the angle. $\blacksquare$

∎

ProofSpecial-angle values from geometry

We derive exact values for $30°, 45°, 60°$ without a calculator. 1. $45°$ : Take an isosceles right triangle with legs $1$ and $1$ . Hypotenuse $= \sqrt{1^2+1^2} = \sqrt2$ . Then $\sin 45° = \cos 45° = \dfrac{1}{\sqrt2} = \dfrac{\sqrt2}{2}$ and $\tan 45° = 1$ . 2. $30°$ and $60°$ : Take an equilateral triangle of side $2$ ; drop an altitude. This splits it into two right triangles with hypotenuse $2$ , short leg $1$ (half the base), and long leg $\sqrt{2^2-1^2}=\sqrt3$ . 3. The altitude creates a $30°$ angle at the apex and a $60°$ angle at the base. 4. $30°$ : opposite $=1$ , so $\sin 30° = \tfrac12$ , $\cos 30° = \tfrac{\sqrt3}{2}$ , $\tan 30° = \tfrac{1}{\sqrt3}$ . 5. $60°$ : opposite $=\sqrt3$ , so $\sin 60° = \tfrac{\sqrt3}{2}$ , $\cos 60° = \tfrac12$ , $\tan 60° = \sqrt3$ . $\blacksquare$

∎

HO

Hipparchus of Nicaea

c. 190 – c. 120 BCE

Often called the founder of trigonometry, Hipparchus compiled the first known table of chords — a precursor to the sine table — to predict the positions of the Sun and Moon. A chord is the straight line subtending an arc of a circle; the half-chord of a doubled angle is essentially the sine. His astronomical models demanded converting angular measurements at the eye into linear distances on the celestial sphere, exactly the job trigonometric ratios do. Centuries later, Indian astronomers (notably Aryabhata) replaced the full chord with the half-chord, giving us the modern jya — the word that, mistranslated through Arabic, became "sine."

🧠 Intuition

Picture standing at the corner of angle $\theta$ and looking along the adjacent side. As you tilt your gaze upward by $\theta$ , sine measures how much you rise per unit of slanted travel along the hypotenuse, while cosine measures how much you advance horizontally per unit of that travel. Tangent is steepness — rise over run — which is why a $45°$ ramp has $\tan 45° = 1$ (it goes up as fast as it goes across). Because the hypotenuse is always longest, $\sin$ and $\cos$ can never exceed $1$ , but $\tan$ has no upper bound: as $\theta \to 90°$ the run shrinks to nothing and the steepness explodes to infinity. The cofunction relations are just the observation that one person's "up" is another person's "across."

💡 The Core Insight

Place the angle at the center of a circle of radius $1$ . A point on the circle at angle $\theta$ has coordinates exactly $(\cos\theta, \sin\theta)$ . This frees the ratios from triangles: $\theta$ may now be obtuse, negative, or larger than $360°$ , and $\sin/\cos$ simply become the $y/x$ coordinates — automatically picking up correct signs in each quadrant. See Unit Circle.

ExampleFinding a height by tangent

Given: You stand $50\text{ m}$ from the base of a tower. The angle of elevation to the top is $32°$ . Your eye is at ground level. Find the tower's height $H$ .

Steps: 1. The horizontal distance is adjacent; the height is opposite. Use $\tan$ :

\tan 32° = \frac{H}{50\text{ m}}

2. Solve: $H = 50 \cdot \tan 32°$ . 3. Evaluate $\tan 32° \approx 0.6249$ . 4. $H \approx 50 \times 0.6249 = 31.2\text{ m}$ .

\boxed{H \approx 31.2\text{ m}}

ExampleRecovering an angle and side

Given: A right triangle has hypotenuse $h = 13\text{ cm}$ and one leg (opposite to angle $\theta$ ) $o = 5\text{ cm}$ . Find $\theta$ and the other leg $a$ .

Steps: 1. $\sin\theta = \dfrac{o}{h} = \dfrac{5}{13} \approx 0.3846$ . 2. $\theta = \sin^{-1}(0.3846) \approx 22.6°$ . 3. Find $a$ by the Pythagorean theorem: $a = \sqrt{h^2 - o^2} = \sqrt{169 - 25} = \sqrt{144} = 12\text{ cm}$ . 4. Check: $\cos\theta = \dfrac{12}{13} \approx 0.923$ , and $\sin^2+\cos^2 = \left(\tfrac{5}{13}\right)^2 + \left(\tfrac{12}{13}\right)^2 = \tfrac{25+144}{169} = 1$ . ✓

\boxed{\theta \approx 22.6°, \qquad a = 12\text{ cm}}

⚠️ Common mistake

$\sin^2\theta$ does not mean $\sin(\sin\theta)$ . The notation $\sin^2\theta$ is a convention for $(\sin\theta)^2$ — the value squared. Worse, $\sin^{-1}\theta$ means the inverse function (arcsine), not $1/\sin\theta = \csc\theta$ . These two superscript conventions clash, so read carefully.

Degrees vs. radians: $\sin 30 \ne \sin 30°$ on most calculators. Always confirm the mode. $\sin 30° = 0.5$ , but $\sin(30\text{ rad}) \approx -0.988$ .

SOH-CAH-TOA only works for the acute angle inside a right triangle. For oblique (non-right) triangles you need the Law of Sines or Law of Cosines.

Counterexample

Claim (false): " $\sin(A+B) = \sin A + \sin B$ ."

Test it: Let $A = B = 30°$ . Then $\sin(A+B) = \sin 60° = \dfrac{\sqrt3}{2} \approx 0.866$ . But $\sin 30° + \sin 30° = 0.5 + 0.5 = 1$ . Since $0.866 \ne 1$ , the claim fails. Trigonometric functions are not linear; the correct expansion is $\sin(A+B) = \sin A\cos B + \cos A\sin B$ (an angle-addition identity).

connection

Trigonometric ratios sit at a crossroads of geometry and analysis:

The Unit Circle generalizes the ratios from triangles to all real angles, giving them periodicity.
Trigonometric Functions (as graphs over $\mathbb{R}$ ) emerge by sweeping $\theta$ continuously, producing the sine and cosine waves.
The Law of Sines and Law of Cosines extend the ratios to any triangle, not just right ones.
Radian Measure reveals the calculus-friendly limit $\lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1$ , which underlies the Derivatives of Trigonometric Functions.
Vectors decompose into components using $\cos\theta$ and $\sin\theta$ , linking trigonometry to physics and linear algebra.
Complex Numbers unify everything through Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ .

🔬 Where this appears

⏳ History

- c. 1800 BCE — Babylonian tablet Plimpton 322 lists Pythagorean triples, hinting at early ratio relationships between triangle sides.

- c. 150 BCE — Hipparchus compiles the first table of chords, the ancestor of the sine table, for astronomy.

- c. 100 CE — Ptolemy's Almagest refines chord tables to high precision and proves the addition theorem in chord form (Ptolemy's theorem).

- c. 500 CE — Aryabhata introduces the half-chord (jya) and tabulates sine values, a decisive step toward the modern function.

- c. 9th–12th c. — Islamic scholars (al-Battani, Abu al-Wafa) develop all six ratios, the spherical law of sines, and tangent/cotangent tables for astronomy and timekeeping.

- 1464 — Regiomontanus' De triangulis omnimodis gives Europe its first systematic trigonometry treatise.

- 1583 — Bartholomäus Pitiscus coins the word trigonometria ("triangle measuring").

- 1748 — Leonhard Euler treats sine and cosine as functions and unifies them with the exponential via

e^{i\theta}=\cos\theta+i\sin\theta

, completing the modern viewpoint.

open-problem

Beyond elementary triangles, several deep questions arise:

Transcendence of values: By the Lindemann–Weierstrass theorem, $\cos\theta$ is transcendental for any nonzero algebraic angle in radians — so $\cos 1$ (radian) cannot satisfy any polynomial with rational coefficients. Which angles give rational sine/cosine? Niven's theorem answers: among angles that are rational multiples of $90°$ , the only rational sine values are $0, \pm\tfrac12, \pm1$ .
Constructibility: Which angles can be trisected or whose ratios constructed with compass and straightedge connects directly to Galois theory; $\cos 20°$ is not constructible, which is why a general angle cannot be trisected.
Best rational/computational approximation: Efficiently evaluating $\sin$ and $\cos$ (CORDIC algorithms, minimax polynomials) remains an engineering art balancing speed and accuracy in hardware.
Generalizations: Hyperbolic functions ( $\sinh, \cosh$ ) mirror these ratios on a hyperbola rather than a circle, and elliptic functions generalize periodicity to two dimensions — rich territory beyond this level.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also