Solution of Triangles

§1Angles and their functions

Trigonometry earns its keep in engineering by converting angles into ratios of lengths — quantities a rule, micrometer or coordinate table can actually deal with. Six such ratios cover every case.

Fig. 1. The three sides named relative to the angle of interest θ. Swap to the other acute angle and “opposite” and “adjacent” swap with it.

For an acute angle θ in a right-angled triangle, each function is a ratio of two sides. Because the ratio is fixed by the angle alone, a triangle of any size gives the same values — which is precisely what makes tables and calculators possible.

The six trigonometric functions
Function	Written	Ratio	Related to
sine	sin θ	opposite ÷ hypotenuse	—
cosine	cos θ	adjacent ÷ hypotenuse	—
tangent	tan θ	opposite ÷ adjacent	sin θ ÷ cos θ
cotangent	cot θ	adjacent ÷ opposite	1 ÷ tan θ
secant	sec θ	hypotenuse ÷ adjacent	1 ÷ cos θ
cosecant	cosec θ	hypotenuse ÷ opposite	1 ÷ sin θ

The unit circle — functions for any angle

The ratio definitions stop at 90°, but engineering angles do not. Placing the angle at the centre of a circle of radius 1 extends every function to any angle whatever: the point where the rotating radius meets the circle has coordinates (cos θ, sin θ), and the vertical intercept on the tangent line at the circle’s right edge has length tan θ.

Fig. 2. At θ = 35°: cos θ = 0.8192 (horizontal), sin θ = 0.5736 (vertical), tan θ = 0.7002 (intercept on the tangent line). As the radius sweeps into other quadrants the segments change sign with direction.

Signs in the four quadrants

Beyond 90° the coordinates — and therefore the functions — take negative values according to quadrant. The workshop mnemonic All Stations To Central gives the function that stays positive in quadrants Ⅰ–Ⅳ in order.

Signs of the functions by quadrant
Quadrant	Angle range	sin	cos	tan	Positive there
Ⅰ	0° – 90°	+	+	+	All
Ⅱ	90° – 180°	+	−	−	Sine
Ⅲ	180° – 270°	−	−	+	Tangent
Ⅳ	270° – 360°	−	+	−	Cosine

Complementary angles and exact values

The two acute angles of a right triangle sum to 90°, so each function of an angle equals the co-function of its complement: sin θ = cos (90° − θ), tan θ = cot (90° − θ), sec θ = cosec (90° − θ). This is why a table running 0°–90° serves both columns at once. A handful of angles have exact closed-form values worth memorising:

Exact values at the standard angles
θ	sin θ	value	cos θ	value	tan θ	value
0°	0	0.0000	1	1.0000	0	0.0000
30°	1/2	0.5000	3/2	0.8660	1/3	0.5774
45°	2/2	0.7071	2/2	0.7071	1	1.0000
60°	3/2	0.8660	1/2	0.5000	3	1.7321
90°	1	1.0000	0	0.0000	undefined	—

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§2Degrees, minutes and radians

Two angle systems live side by side in engineering: degrees–minutes–seconds on drawings and machine dials, radians inside every formula that mixes angles with lengths.

A full circle is 360 degrees; each degree divides into 60 minutes (′) and each minute into 60 seconds (″). The radian instead measures an angle by the arc it cuts on a unit circle, so a full circle is 2π radians. The bridge between the systems:

Conversion π rad = 180° 1 rad = 57.295 78° = 57°17′45″ 1° = 0.017 453 3 rad 1′ = 0.000 290 89 rad

Decimal degrees convert to and from minutes and seconds by base-60 arithmetic: θ = D + M/60 + S/3600. For instance 37°42′ = 37.7000°, and 37.7° × π/180 = 0.658 0 rad. Going the other way, 1.2 rad = 68.7549° = 68°45′18″.

Why radians matter in the workshop

With the angle in radians, arc length and sector area collapse to their simplest possible forms — no conversion constants:

Arc & sector s = r θ A = ½ r² θ (θ in radians)

Example — arc length

A slotted link is bent to a radius of 150 mm through 38°. Developed length of the arc: θ = 38 × π/180 = 0.663 2 rad, so s = 150 × 0.6632 = 99.48 mm.

Common angles in both systems
Degrees	Radians (exact)	Radians (decimal)	Degrees	Radians (exact)	Radians (decimal)
15°	π/12	0.26180	120°	2π/3	2.09440
30°	π/6	0.52360	135°	3π/4	2.35619
45°	π/4	0.78540	150°	5π/6	2.61799
60°	π/3	1.04720	180°	π	3.14159
90°	π/2	1.57080	360°	2π	6.28319

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§3Right-angled triangles

The right triangle is the machinist’s triangle: sine bars, tapers, dovetails and coordinate work all resolve into one. Knowing any two independent parts — two sides, or a side and an acute angle — fixes everything else.

Two relations govern the whole section. Pythagoras ties the sides together, and the angle sum ties the angles:

Governing relations c² = a² + b² A + B = 90° (right angle at C, hypotenuse c)

Solution of right triangles — sides a, b, hypotenuse c, angles A (opp. a), B (opp. b)
Given	Third side / first angle	Remaining parts
a, b	c = a² + b²	tan A = a / b, B = 90° − A
a, c	b = (c − a)(c + a)	sin A = a / c, B = 90° − A
b, c	a = (c − b)(c + b)	cos A = b / c, B = 90° − A
a, A	b = a cot A = a / tan A	c = a / sin A, B = 90° − A
b, A	a = b tan A	c = b / cos A, B = 90° − A
c, A	a = c sin A	b = c cos A, B = 90° − A
The factored forms (c − a)(c + a) are kinder to significant figures than c² − a² when a is close to c.

Example 1 — setting a sine bar

A 250 mm sine bar is to be set to 14°15′. The gauge-block stack under one roller is the opposite side of a right triangle whose hypotenuse is the bar’s roller centre distance:

H = L sin θ = 250 × sin 14.25° = 250 × 0.246 15 = 61.538 mm — built from the block set as 61.54 mm in practice.

Example 2 — included angle of a taper

A taper plug measures Ø32.00 mm at the large end and Ø25.40 mm at the small end, 120 mm apart. On a diametral section, half the diameter difference over the length gives the half-angle:

tan α = (D − d) / 2L = 6.60 / 240 = 0.027 50, so α = 1°35′ and the included angle 2α = 3°9′.

Example 3 — coordinates of holes on a pitch circle

Seven holes sit on a Ø96 mm pitch circle, hole 1 at top dead centre, measured from the circle centre. Each hole lies at x = R cos θ, y = R sin θ with R = 48 mm and θ stepping 360°/7 = 51.4286° from 90°:

Ordinate table — 7 holes, Ø96 pitch circle (mm)
Hole	θ	x	y	Hole	θ	x	y
1	90.000°	0.000	+48.000	5	295.714°	+20.826	−43.247
2	141.429°	−37.528	+29.928	6	347.143°	+46.797	−10.681
3	192.857°	−46.797	−10.681	7	38.571°	+37.528	+29.928
4	244.286°	−20.826	−43.247

The mirror symmetry of the values about the vertical axis is a built-in check on the arithmetic.

Checking a solution

Every solved right triangle should close: the two computed legs must satisfy Pythagoras, and the two acute angles must total 90°. A residual beyond the last retained figure means a rounding or transcription slip.

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§4Oblique triangles

When no right angle is present — the general, obtuse or acute “oblique” triangle — two laws carry the whole load. The rest is knowing which one to reach for.

Law of sines asin A = bsin B = csin C = 2R (R = radius of the circumscribed circle)

Law of cosines a² = b² + c² − 2bc cos A or, solved for the angle, cos A = b² + c² − a²2bc

A third relation, the law of tangents, suits logarithmic or long-hand work and remains a tidy check today:

Law of tangents a − ba + b = tan ½(A − B)tan ½(A + B)

The four cases

Choosing the attack by what is given
Given	Strategy	Solutions
ASA / AAS	Third angle from the 180° sum, then both remaining sides by the sine law.	Always one.
SAS	Third side by the cosine law, then a second angle by the cosine law (or the sine law for the angle opposite the shorter side), last angle by the sum.	Always one.
SSS	Largest angle first by the cosine law, remaining angles by the sine law; the three must total 180°.	One, provided each side is shorter than the other two combined.
SSA	Sine law gives sin B for the angle opposite the second side — but a sine fixes an angle only up to its supplement.	None, one or two — the ambiguous case, Fig. 3.

Obtuse angles and the sine law

An angle and its supplement share the same sine, so the sine law alone cannot say whether an angle is 83° or 97°. The cosine law can: its result is negative exactly when the angle is obtuse. Rule of thumb — find the angle opposite the longest side by the cosine law first; the other two are then guaranteed acute and safe for the sine law.

Fig. 3. Given angle A, adjacent side b and opposite side a, the compass arc of radius a from C can cut the base twice (two triangles), touch it once (right triangle, a = b sin A), or miss it (no triangle).

SSA test — angle A acute, given sides a (opposite) and b (adjacent)
Condition	Triangles
a < b sin A	none — the arc misses the base line
a = b sin A	one, right-angled at B
b sin A < a < b	two — B and its supplement both close the triangle
a ≥ b	one
If A is obtuse the picture simplifies: a solution exists only when a > b, and it is unique.

Area of any triangle

Area K = ½ b c sin A = ½ a b sin C or K = s(s − a)(s − b)(s − c), s = ½(a + b + c)

Example 4 — SAS: the triangle on the drawing sheet

The title drawing gives b = 85.0 mm, c = 60.0 mm and the included angle A = 52°. Third side by the cosine law:

a² = 85² + 60² − 2(85)(60) cos 52° = 7225 + 3600 − 10 200 × 0.615 66 = 4545.25, so a = 67.42 mm.

Angle opposite the longest side (b = 85) by the cosine law: cos B = (4545.25 + 3600 − 7225) / (2 × 67.418 × 60) = 0.113 75, giving B = 83°28′; then C = 180° − 52° − 83°28′ = 44°32′.

Check by the sine law: a / sin A = 85.56 and c / sin C = 85.56 — the triangle closes. Area = ½ × 85 × 60 × sin 52° = 2009 mm².

Example 5 — SSS: a link plate

A link plate has hole centres 70, 90 and 120 mm apart. Largest angle first — opposite the 120 mm side:

cos C = (70² + 90² − 120²) / (2 × 70 × 90) = −1400 / 12 600 = −0.111 11 → C = 96°23′ (the negative cosine flags it obtuse). Then A = 35°26′ and B = 48°11′; sum 180°00′. Area by Heron: s = 140, K = 140 × 70 × 50 × 20 = 3130.5 mm².

Example 6 — SSA: both answers are real

Given A = 30°, opposite side a = 60 mm, adjacent side b = 100 mm. Since b sin A = 50 mm and 50 < 60 < 100, two triangles exist. Sine law: sin B = b sin A / a = 0.833 33.

First solution: B = 56°27′, C = 93°33′, c = 119.77 mm. Second: B = 123°33′, C = 26°27′, c = 53.44 mm. Which one is the part on the bench? Only a further measurement — the diagonal c or a check of angle B — can say. Design drawings avoid dimensioning a joint this way for exactly that reason.

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§5Trigonometric identities

Identities are the algebra of angles — the rewriting rules that turn an unworkable expression into a solvable one. Grouped here for reference.

Fundamental relations

Group	Identities
Reciprocal	cosec θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ
Quotient	tan θ = sin θ / cos θ cot θ = cos θ / sin θ
Pythagorean	sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ
Cofunction	sin(90° − θ) = cos θ tan(90° − θ) = cot θ sec(90° − θ) = cosec θ
Odd / even	sin(−θ) = −sin θ cos(−θ) = cos θ tan(−θ) = −tan θ

Sum and difference of angles

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = tan A ± tan B1 ∓ tan A tan B

Double and half angle

sin 2A = 2 sin A cos A cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A tan 2A = 2 tan A1 − tan²A

sin ½A = (1 − cos A)/2 cos ½A = (1 + cos A)/2 tan ½A = 1 − cos Asin A = sin A1 + cos A

Products and sums

2 sin A cos B = sin(A + B) + sin(A − B) 2 cos A cos B = cos(A + B) + cos(A − B) 2 sin A sin B = cos(A − B) − cos(A + B)

sin A + sin B = 2 sin ½(A + B) cos ½(A − B) cos A + cos B = 2 cos ½(A + B) cos ½(A − B)

Relations inside a triangle

With A + B + C = 180° and sides a, b, c opposite their angles, two elegant pairs act as full-solution checks:

Mollweide’s equations a + bc = cos ½(A − B)sin ½C a − bc = sin ½(A − B)cos ½C

Projection law a = b cos C + c cos B (and cyclic permutations)

Because every part of the triangle appears in Mollweide’s equations, one substitution catches an error anywhere in a solution — the traditional final step of careful long-hand work.

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§6Graphs of the functions

Seeing the functions as waves explains their behaviour at a glance: where they peak, where they vanish, and where the tangent runs away to infinity.

Sine and cosine are the same wave displaced 90° — cosine leads. Both repeat every 360° and never leave the band −1 to +1. The tangent instead repeats every 180° and blows up wherever the cosine underneath it passes through zero.

Fig. 4. Sine (solid, accent) and cosine (dashed, teal) over one full turn. Period 360°, amplitude 1; cosine is sine shifted 90° to the left.

Fig. 5. The tangent, period 180°, with vertical asymptotes at 90° and 270° where cos θ = 0. Values are clipped at ±3 for display.

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§7Table of natural functions

A working table of sines, cosines, tangents and cotangents from 0° to 90° in one-degree steps. Every value below was computed for this page to five decimal places.

To use the table for angles between the tabulated degrees, interpolate as shown in §8. For angles beyond 90°, reduce to the first quadrant with the cofunction and sign rules of §1 — for example sin 132° = sin 48° and cos 132° = −cos 48°.

Natural trigonometric functions, 0°–90° (computed, five decimal places)
Angle	sin	cos	tan	cot
0°	0.00000	1.00000	0.00000	—
1°	0.01745	0.99985	0.01746	57.28996
2°	0.03490	0.99939	0.03492	28.63625
3°	0.05234	0.99863	0.05241	19.08114
4°	0.06976	0.99756	0.06993	14.30067
5°	0.08716	0.99619	0.08749	11.43005
6°	0.10453	0.99452	0.10510	9.51436
7°	0.12187	0.99255	0.12278	8.14435
8°	0.13917	0.99027	0.14054	7.11537
9°	0.15643	0.98769	0.15838	6.31375
10°	0.17365	0.98481	0.17633	5.67128
11°	0.19081	0.98163	0.19438	5.14455
12°	0.20791	0.97815	0.21256	4.70463
13°	0.22495	0.97437	0.23087	4.33148
14°	0.24192	0.97030	0.24933	4.01078
15°	0.25882	0.96593	0.26795	3.73205
16°	0.27564	0.96126	0.28675	3.48741
17°	0.29237	0.95630	0.30573	3.27085
18°	0.30902	0.95106	0.32492	3.07768
19°	0.32557	0.94552	0.34433	2.90421
20°	0.34202	0.93969	0.36397	2.74748
21°	0.35837	0.93358	0.38386	2.60509
22°	0.37461	0.92718	0.40403	2.47509
23°	0.39073	0.92050	0.42447	2.35585
24°	0.40674	0.91355	0.44523	2.24604
25°	0.42262	0.90631	0.46631	2.14451
26°	0.43837	0.89879	0.48773	2.05030
27°	0.45399	0.89101	0.50953	1.96261
28°	0.46947	0.88295	0.53171	1.88073
29°	0.48481	0.87462	0.55431	1.80405
30°	0.50000	0.86603	0.57735	1.73205
31°	0.51504	0.85717	0.60086	1.66428
32°	0.52992	0.84805	0.62487	1.60033
33°	0.54464	0.83867	0.64941	1.53986
34°	0.55919	0.82904	0.67451	1.48256
35°	0.57358	0.81915	0.70021	1.42815
36°	0.58779	0.80902	0.72654	1.37638
37°	0.60182	0.79864	0.75355	1.32704
38°	0.61566	0.78801	0.78129	1.27994
39°	0.62932	0.77715	0.80978	1.23490
40°	0.64279	0.76604	0.83910	1.19175
41°	0.65606	0.75471	0.86929	1.15037
42°	0.66913	0.74314	0.90040	1.11061
43°	0.68200	0.73135	0.93252	1.07237
44°	0.69466	0.71934	0.96569	1.03553
45°	0.70711	0.70711	1.00000	1.00000
46°	0.71934	0.69466	1.03553	0.96569
47°	0.73135	0.68200	1.07237	0.93252
48°	0.74314	0.66913	1.11061	0.90040
49°	0.75471	0.65606	1.15037	0.86929
50°	0.76604	0.64279	1.19175	0.83910
51°	0.77715	0.62932	1.23490	0.80978
52°	0.78801	0.61566	1.27994	0.78129
53°	0.79864	0.60182	1.32704	0.75355
54°	0.80902	0.58779	1.37638	0.72654
55°	0.81915	0.57358	1.42815	0.70021
56°	0.82904	0.55919	1.48256	0.67451
57°	0.83867	0.54464	1.53986	0.64941
58°	0.84805	0.52992	1.60033	0.62487
59°	0.85717	0.51504	1.66428	0.60086
60°	0.86603	0.50000	1.73205	0.57735
61°	0.87462	0.48481	1.80405	0.55431
62°	0.88295	0.46947	1.88073	0.53171
63°	0.89101	0.45399	1.96261	0.50953
64°	0.89879	0.43837	2.05030	0.48773
65°	0.90631	0.42262	2.14451	0.46631
66°	0.91355	0.40674	2.24604	0.44523
67°	0.92050	0.39073	2.35585	0.42447
68°	0.92718	0.37461	2.47509	0.40403
69°	0.93358	0.35837	2.60509	0.38386
70°	0.93969	0.34202	2.74748	0.36397
71°	0.94552	0.32557	2.90421	0.34433
72°	0.95106	0.30902	3.07768	0.32492
73°	0.95630	0.29237	3.27085	0.30573
74°	0.96126	0.27564	3.48741	0.28675
75°	0.96593	0.25882	3.73205	0.26795
76°	0.97030	0.24192	4.01078	0.24933
77°	0.97437	0.22495	4.33148	0.23087
78°	0.97815	0.20791	4.70463	0.21256
79°	0.98163	0.19081	5.14455	0.19438
80°	0.98481	0.17365	5.67128	0.17633
81°	0.98769	0.15643	6.31375	0.15838
82°	0.99027	0.13917	7.11537	0.14054
83°	0.99255	0.12187	8.14435	0.12278
84°	0.99452	0.10453	9.51436	0.10510
85°	0.99619	0.08716	11.43005	0.08749
86°	0.99756	0.06976	14.30067	0.06993
87°	0.99863	0.05234	19.08114	0.05241
88°	0.99939	0.03490	28.63625	0.03492
89°	0.99985	0.01745	57.28996	0.01746
90°	1.00000	0.00000	—	0.00000
— : function unbounded (undefined) at this angle. For minutes between entries, interpolate as in §8; for angles beyond 90°, reduce by the sign and cofunction rules of §1.

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§8Interpolation

Tables step in fixed increments; the workshop does not. Linear interpolation bridges the gap, on the assumption that the function is straight enough between neighbouring entries — true to a few units in the last figure for one-degree steps.

Linear interpolation f(θ) ≈ f(θ₁) + θ − θ₁θ₂ − θ₁ × [ f(θ₂) − f(θ₁) ]

Example 7 — reading between the lines

Find sin 24°36′. The table gives sin 24° = 0.406 74 and sin 25° = 0.422 62; the tabular difference is 0.015 88. Since 36′ is 0.6 of the way to the next degree:

sin 24°36′ ≈ 0.406 74 + 0.6 × 0.015 88 = 0.416 27. The directly computed value is 0.416 28 — the single-unit discrepancy in the fifth figure is the interpolation error.

Example 8 — inverse interpolation

What angle has tan θ = 0.500 00? The table brackets it: tan 26° = 0.487 73 and tan 27° = 0.509 53. The fraction is (0.500 00 − 0.487 73) / (0.509 53 − 0.487 73) = 0.563, so θ ≈ 26° + 0.563° = 26.563° = 26°34′. Direct computation gives 26.565° — agreement to the minute.

Where linear interpolation struggles

Interpolation error grows where a function curves hard: the tangent and secant near 90°, and the cosine near 0° (where it is flat, so the inverse problem is ill-conditioned instead). In those regions use a finer table, work from the cofunction, or compute directly.

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§9Versed sine and related functions

Four historical companions of the main six survive because they name quantities the workshop measures every week — chiefly the height of a circular segment.

Definitions vers θ = 1 − cos θ covers θ = 1 − sin θ havers θ = ½ vers θ exsec θ = sec θ − 1

Fig. 6. A chord c subtending angle θ at the centre. The segment height h is the radius times the versed sine of the half-angle.

For a chord of a circle of radius r subtending a central angle θ:

c = 2r sin ½θ h = r vers ½θ = r(1 − cos ½θ)

And the shop-floor inverse — radius from a chord and its rise, needing no angle at all:

r = c²/4 + h²2h

Measure a chord of 84 mm rising 9 mm, for instance, and the arc’s radius is (1764 + 81) / 18 = 102.5 mm.

Example 9 — depth of a flat on round stock

A flat 18 mm wide is milled on a Ø50 mm shaft. The half-chord is 9 mm, so sin ½θ = 9/25 = 0.360 and ½θ = 21°6′. Depth of cut:

h = r vers ½θ = 25 × (1 − cos 21°6′) = 25 × 0.067 05 = 1.676 mm.

The haversine — half the versed sine — earned its own tables in celestial navigation because it turns the spherical distance formula of §12 into pure addition of logarithms; the term still headlines the standard great-circle formula in software today. The exsecant gave railway and road engineers the “external distance” from a curve’s intersection point to the arc.

Versed functions, 0°–90° in 5° steps (computed)
θ	vers θ	covers θ	havers θ	exsec θ
0°	0.00000	1.00000	0.00000	0.00000
5°	0.00381	0.91284	0.00190	0.00382
10°	0.01519	0.82635	0.00760	0.01543
15°	0.03407	0.74118	0.01704	0.03528
20°	0.06031	0.65798	0.03015	0.06418
25°	0.09369	0.57738	0.04685	0.10338
30°	0.13397	0.50000	0.06699	0.15470
35°	0.18085	0.42642	0.09042	0.22077
40°	0.23396	0.35721	0.11698	0.30541
45°	0.29289	0.29289	0.14645	0.41421
50°	0.35721	0.23396	0.17861	0.55572
55°	0.42642	0.18085	0.21321	0.74345
60°	0.50000	0.13397	0.25000	1.00000
65°	0.57738	0.09369	0.28869	1.36620
70°	0.65798	0.06031	0.32899	1.92380
75°	0.74118	0.03407	0.37059	2.86370
80°	0.82635	0.01519	0.41318	4.75877
85°	0.91284	0.00381	0.45642	10.47371
90°	1.00000	0.00000	0.50000	—
— : unbounded at 90°. vers θ = 1 − cos θ; covers θ = 1 − sin θ; havers θ = ½ vers θ; exsec θ = sec θ − 1.

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§10Involute and sevolute functions

The involute function is gear mathematics distilled to one line. It converts a pressure angle into the polar angle of a point on an involute curve — the profile of nearly every metal gear tooth in service.

Definitions inv φ = tan φ − φ (φ in radians) sev φ = sec φ − inv φ

Geometrically: unwind a taut string from a base circle of radius r_b. Its free end traces the involute. When the straight, unwound portion of string is tangent to the base circle and makes the angle φ with the radius to the tracing point, that point sits at radius r = r_b / cos φ from the centre, and at a polar angle of exactly inv φ from the curve’s starting point on the base circle. One scalar function therefore locates any point of the profile.

Fig. 7. The involute traced by unwinding a string from the base circle. The unwound length TP equals the arc it left behind, TP is tangent at T, and P lies at polar angle inv φ from the starting point S, at radius r_b/cos φ.

Worked directly: for the standard 20° pressure angle, inv 20° = tan 20° − (20 × π/180) = 0.363 970 − 0.349 066 = 0.014 904. The other stock pressure angles give inv 14.5° = 0.005 545 and inv 25° = 0.029 975. The sevolute pairs with it in checking involute profiles with pins and balls: sev 20° = sec 20° − inv 20° = 1.064 178 − 0.014 904 = 1.049 27.

Example 10 — angle from an involute value

A gear calculation ends with inv φ = 0.020 000; the pressure angle at that point is needed. The table below brackets the value: inv 21° = 0.017 345 and inv 22° = 0.020 054. Inverse interpolation (§8):

fraction = (0.020 000 − 0.017 345) / (0.020 054 − 0.017 345) = 0.980, so φ ≈ 21° + 0.980° = 21°59′.

Tooth thickness, backlash and measurement-over-pins calculations all run on this function; a dedicated Gears and Gearing page in this Library will carry it forward into those methods.

Involute and sevolute functions, 10°–40° (computed; inv φ to six decimal places)
φ	inv φ	sev φ	φ	inv φ	sev φ
10°	0.001794	1.01363	26°	0.033947	1.07865
11°	0.002394	1.01632	27°	0.038287	1.08404
12°	0.003117	1.01922	28°	0.043017	1.08955
13°	0.003975	1.02233	29°	0.048164	1.09519
14°	0.004982	1.02563	30°	0.053751	1.10095
15°	0.006150	1.02913	31°	0.059809	1.10682
16°	0.007493	1.03281	32°	0.066364	1.11281
17°	0.009025	1.03667	33°	0.073449	1.11891
18°	0.010760	1.04070	34°	0.081097	1.12512
19°	0.012715	1.04491	35°	0.089342	1.13143
20°	0.014904	1.04927	36°	0.098224	1.13784
21°	0.017345	1.05380	37°	0.107782	1.14435
22°	0.020054	1.05848	38°	0.118061	1.15096
23°	0.023049	1.06331	39°	0.129106	1.15765
24°	0.026350	1.06829	40°	0.140968	1.16444
25°	0.029975	1.07340
inv φ = tan φ − φ (φ in radians); sev φ = sec φ − inv φ. Interpolate between entries as in §8 — see Example 10.

Contents

§11Compound angles

A hole drilled on a slant in two directions, a surface tipped twice on a sine plate — compound angles arise whenever one inclination is stacked on another. Two distinct situations share the name, and they do not give the same answer.

Case 1 — angles given as projections

A drawing shows the axis of a hole inclined at α to the vertical in the front view and at β in the side view. Both are projections of one line. Drop a unit height along the vertical: the horizontal offsets seen in the two views are tan α and tan β, at right angles to each other, so the true offset is their hypotenuse:

Projections → true angle tan θ = tan²α + tan²β tan δ = tan βtan α

θ is the true angle from the vertical; δ is the compass direction — the angle the vertical plane containing the axis makes with the front-view plane. Set the work by rotating δ in plan, then tilting θ.

Case 2 — angles applied as successive tilts

A compound sine plate tilts the work α about one horizontal axis, then β about the now-rotated perpendicular axis. Each tilt scales the vertical component of the surface normal by its cosine, so the resultant inclination of the surface obeys:

Successive rotations cos θ = cos α cos β

Fig. 8. Case 1 in pictures: the two drawing views each show a projection of the inclined axis; the true angle θ exceeds both α and β.

Example 11 — hole given by two views

α = 15° in the front view, β = 10° in the side view. tan θ = 0.26795² + 0.17633² = 0.102 89 = 0.320 76, so θ = 17°47′, and tan δ = 0.17633 / 0.26795 = 0.658 08, giving δ = 33°21′.

Example 12 — the same numbers on a sine plate

Tilt 15° about the first axis, then 10° about the second: cos θ = cos 15° cos 10° = 0.965 93 × 0.984 81 = 0.951 25, so θ = 17°58′ — eleven minutes steeper than Case 1.

Establish the convention first

The same pair “15° and 10°” yielded 17°47′ under one reading and 17°58′ under the other. Before machining, confirm whether a drawing’s compound callout means projected angles or successive tilts.

Contents

§12Spherical trigonometry

On a sphere the triangle’s sides are themselves arcs, measured in degrees, and the three angles sum to more than 180°. Navigation, astronomy, and the geometry of angular machine motions live here.

Fig. 9. A spherical triangle. Side a is the arc BC expressed as the angle it subtends at the sphere’s centre; angle A is the dihedral angle between the planes of arcs b and c.

Working rules that replace their plane cousins:

Law of sines sin asin A = sin bsin B = sin csin C

Law of cosines — sides cos a = cos b cos c + sin b sin c cos A

Law of cosines — angles cos A = −cos B cos C + sin B sin C cos a

The angle sum exceeds 180° by the spherical excess E = A + B + C − 180°, and the triangle’s area is proportional to it: area = π r² E / 180°.

Right spherical triangles — Napier’s rules

With the right angle at C, the remaining five parts, written as a, b, co-A, co-c, co-B (“co-” meaning the complement, 90° minus the part), arrange in a circle in that order. Napier’s two rules then generate every formula:

Napier’s rules

The sine of any middle part equals the product of the tangents of the two adjacent parts — and also the product of the cosines of the two opposite parts.

Fig. 10. The five circular parts in their working order; the right angle at C does not appear. Pick any part as “middle” and read its neighbours and opposites straight off the wheel.

The ten relations for a right spherical triangle (C = 90°)
Relation	Relation
sin a = sin c sin A	sin b = sin c sin B
tan a = tan c cos B	tan b = tan c cos A
tan a = tan A sin b	tan b = tan B sin a
cos A = cos a sin B	cos B = cos b sin A
cos c = cos a cos b	cos c = cot A cot B

Example 13 — solving a right spherical triangle

Given hypotenuse c = 70° and angle A = 50° (right angle at C). By Napier’s rules:

sin a = sin c sin A = 0.939 69 × 0.766 04 → a = 46°3′; tan b = tan c cos A = 2.747 48 × 0.642 79 → b = 60°29′; cos B = cos b sin A = 0.492 62 × 0.766 04 → B = 67°49′.

Check: cos a cos b = 0.694 08 × 0.492 76 = 0.342 02 = cos 70° — the triangle closes.

Example 14 — great-circle distance

The shortest route between two points on a sphere follows a great circle, and its length falls straight out of the law of cosines applied to the triangle formed with the pole (sides are the two colatitudes, included angle the longitude difference):

cos d = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ

Sydney (33.869° S, 151.209° E) to Auckland (36.849° S, 174.763° E): Δλ = 23.554°, cos d = 0.943 3, so the central angle is d = 19°23′. On a 6371 km mean-radius Earth that is 6371 × 0.338 30 rad = 2156 km (1164 nautical miles).

Contents

§13Quick reference

The working core of the page on one card rack.

Right triangle

c² = a² + b² A + B = 90°

a = c sin A = b tan A

Law of sines

a/sin A = b/sin B = c/sin C = 2R

Law of cosines

a² = b² + c² − 2bc cos A

cos A = (b² + c² − a²)/2bc

Area

K = ½ bc sin A

K = √(s(s−a)(s−b)(s−c))

Radians

π rad = 180° s = rθ A = ½r²θ

Circular segment

c = 2r sin ½θ h = r vers ½θ

r = (c²/4 + h²)/2h

Involute

inv φ = tan φ − φ (rad)

sev φ = sec φ − inv φ

Compound angles

tan θ = √(tan²α + tan²β)

cos θ = cos α cos β (tilts)

Spherical

cos a = cos b cos c + sin b sin c cos A

Contents

§1Angles and their functions

The unit circle — functions for any angle

Signs in the four quadrants

Complementary angles and exact values

§2Degrees, minutes and radians

Why radians matter in the workshop

§3Right-angled triangles

§4Oblique triangles

The four cases

Area of any triangle

§5Trigonometric identities

Fundamental relations

Sum and difference of angles

Double and half angle

Products and sums

Relations inside a triangle

§6Graphs of the functions

§7Table of natural functions

§8Interpolation

§9Versed sine and related functions

§10Involute and sevolute functions

§11Compound angles

Case 1 — angles given as projections

Case 2 — angles applied as successive tilts

§12Spherical trigonometry

Right spherical triangles — Napier’s rules

§13Quick reference

Right triangle

Law of sines

Law of cosines

Area

Radians

Circular segment

Involute

Compound angles

Spherical

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