Trig Ratios

Part of Mathematics

Sine, cosine, and tangent — the three fundamental ratios that connect angles to side lengths in triangles.

Why This Matters

Trigonometric ratios are what you reach for when you know an angle and need to find a length, or when you know a length and need to find an angle. The Pythagorean theorem handles right triangles when you know two side lengths — but construction, navigation, and surveying constantly present situations where you know an angle and only one side. A cannon positioned at 45 degrees on a hill 30 m above the plain: how far does the shot travel horizontally? A roof with a 35-degree pitch and a 6-meter run: how long are the rafters? A ship traveling at bearing 310 degrees for 47 km: how far north and west has it moved?

All of these are trigonometry problems. Specifically, they require the sine, cosine, and tangent ratios — three simple relationships between the angles and sides of a right triangle that unlock an entire category of practical calculation.

Trigonometry is often taught abstractly, and students memorize definitions without understanding what they are for. This article takes the opposite approach: it defines the ratios in terms of what they do, provides tables of their values at common angles, and works through practical examples that builders, navigators, and surveyors actually face.

The Setup: Right Triangle Terminology

Every trig calculation begins with a right triangle. Identify:

• Hypotenuse (H): The longest side, opposite the right angle
• Opposite (O): The side across from the angle you are working with
• Adjacent (A): The side next to the angle you are working with (not the hypotenuse)

These change depending on which angle you are considering. If you label your angle θ (theta), then “opposite” is always the side that does not touch θ, and “adjacent” is the side that touches θ (other than the hypotenuse).

The Three Ratios

Sine (sin): sin(θ) = Opposite / Hypotenuse

Cosine (cos): cos(θ) = Adjacent / Hypotenuse

Tangent (tan): tan(θ) = Opposite / Adjacent

Memory aid: SOH-CAH-TOA

• Sin = Opposite / Hypotenuse
• Cos = Adjacent / Hypotenuse
• Tan = Opposite / Adjacent

These are the same ratio regardless of the size of the triangle — scale the triangle up or down, and the ratios remain constant for the same angle. This is the deep insight: the ratio is a property of the angle, not the triangle.

Table of Common Values

Memorize the key angles: 30°, 45°, 60°. These appear constantly.

Angle (°)	sin	cos	tan
0	0	1.000	0
15	0.259	0.966	0.268
20	0.342	0.940	0.364
25	0.423	0.906	0.466
30	0.500	0.866	0.577
35	0.574	0.819	0.700
40	0.643	0.766	0.839
45	0.707	0.707	1.000
50	0.766	0.643	1.192
55	0.819	0.574	1.428
60	0.866	0.500	1.732
65	0.906	0.423	2.145
70	0.940	0.342	2.747
75	0.966	0.259	3.732
90	1.000	0	undefined

Key values to memorize:

• sin(30°) = 0.5 exactly (a 30° angle has the opposite side exactly half the hypotenuse)
• sin(90°) = 1 (the “opposite” is the hypotenuse when the angle is 90°)
• cos(60°) = 0.5 (same relationship, different angle)
• tan(45°) = 1 (opposite = adjacent at 45°)

The complementary relationship: sin(θ) = cos(90°−θ). The sine of any angle equals the cosine of its complement. So sin(30°) = cos(60°) = 0.5. This halves the table you need to remember.

Using Trig Ratios: Finding Side Lengths

Given an angle and one side, find another side.

Example 1: Rafter length A roof has a 35° pitch and a horizontal run of 6 m. How long is the rafter?

• The 35° angle is at the base (eave)
• The run (6 m) is adjacent to this angle
• The rafter is the hypotenuse
• cos(35°) = Adjacent / Hypotenuse = 6 / H
• H = 6 / cos(35°) = 6 / 0.819 = 7.33 m

Example 2: Height from angle and distance You stand 40 m from the base of a cliff and measure the angle of elevation to the top as 50°.

• tan(50°) = Opposite / Adjacent = Height / 40
• Height = 40 × tan(50°) = 40 × 1.192 = 47.7 m

Example 3: Horizontal component of travel You travel 25 km on a bearing of 40° east of north. How far east have you moved?

• The 40° angle is from north (vertical)
• Distance traveled is the hypotenuse (25 km)
• East displacement is opposite to the 40° angle from north
• sin(40°) = Opposite / Hypotenuse = East / 25
• East = 25 × sin(40°) = 25 × 0.643 = 16.1 km
• North = 25 × cos(40°) = 25 × 0.766 = 19.2 km

Using Trig Ratios: Finding Angles

Given two side lengths, find the angle.

You need the inverse trig functions for this — arcsin, arccos, arctan. Without a calculator, use the table in reverse: find the ratio value, read off the angle.

Example: Angle of an existing roof slope A roof rises 2.5 m over a horizontal run of 4 m. What is the pitch angle?

• tan(θ) = Opposite / Adjacent = 2.5 / 4 = 0.625
• From the table: tan(30°) = 0.577, tan(35°) = 0.700
• 0.625 is between these: θ ≈ 32°
• (More precisely, interpolate: 0.625−0.577 = 0.048; range 0.700−0.577 = 0.123; fraction = 0.048/0.123 = 39%; angle ≈ 30° + 39% of 5° ≈ 32°)

Building a Sine Table from Scratch

If you have no pre-existing trig table, you can construct one using a few geometric facts and drawing.

Method using a unit circle (radius = 1 unit):

1. Draw a circle with a carefully measured radius of 10 cm (your “unit”)
2. Mark the center O
3. Draw a horizontal diameter
4. To find sin(30°): construct a 30° angle at the center; the perpendicular height from the horizontal to where the radius hits the circle = sin(30°) × 10 cm = 5 cm exactly
5. Measure this height in centimeters, divide by 10: this gives you sin(30°)
6. Repeat for 15°, 20°, 25°, etc., by constructing the angles geometrically

For 45°: draw the diagonal of a square inscribed in the circle. For 60°: construct an equilateral triangle inscribed in the circle. From 30° and 45°, you can derive 15° (difference), 75° (sum), and eventually fill in the entire table.

Practical Reference Cards

Copy these relationships onto durable material (bark, clay tablet, wood) for field use:

To find a missing side:

• O = H × sin(θ)
• A = H × cos(θ)
• O = A × tan(θ)
• H = O / sin(θ)
• H = A / cos(θ)
• A = O / tan(θ)

Common construction angles and their tangents:

• 15° roof pitch: tan = 0.268 (gentle slope, good for thatch)
• 30° roof pitch: tan = 0.577 (standard for timber-frame)
• 45° slope: tan = 1.000 (45° stairs, steep embankment)
• 60° slope: tan = 1.732 (very steep, unstable soil)

Connecting to the Pythagorean Theorem

The trig ratios and the Pythagorean theorem are complementary tools:

• Use Pythagoras when you know two sides and want the third
• Use trig ratios when you know an angle and one side

They are consistent: if you find H using trig ratios, then verify with a² + b² = c². Any discrepancy reveals a measurement or calculation error.

The trig ratios are the key that unlocks all of angle-and-distance calculation. With a table and the SOH-CAH-TOA framework, a single measurement of an angle turns into a complete solution for any triangle — powering construction, navigation, and surveying for a rebuilding civilization.