Maths⏱ 5 min read
How to Calculate Sine, Cosine, and Tangent (Without a Calculator)
Trigonometry underpins construction, navigation, physics, and engineering. Here is how SOH CAH TOA works, the key angle values worth memorising, and how to find unknown sides and angles.
Trigonometry is the branch of maths that relates angles to side lengths in triangles. Once you understand the three ratios and can use a triangle to remember them, most trig problems become mechanical.
SOH CAH TOA: The Three Ratios
In a right-angled triangle:
Hypotenuse (H): the longest side, opposite the right angle
Opposite (O): the side opposite the angle you are working with
Adjacent (A): the side next to the angle (not the hypotenuse)
SOH: sin(angle) = Opposite / Hypotenuse
CAH: cos(angle) = Adjacent / Hypotenuse
TOA: tan(angle) = Opposite / Adjacent
Memory device: "Some Old Horses Can Always Hear Their Owners Approaching"
Finding an Unknown Side
Right triangle: angle = 35 degrees, hypotenuse = 12m
Find the opposite side:
sin(35) = Opposite / Hypotenuse
Opposite = sin(35) x 12
= 0.5736 x 12 = 6.88m
Find the adjacent side:
cos(35) = Adjacent / Hypotenuse
Adjacent = cos(35) x 12
= 0.8192 x 12 = 9.83m
Check: O^2 + A^2 = H^2 (Pythagoras)
6.88^2 + 9.83^2 = 47.3 + 96.6 = 143.9
H = sqrt(143.9) = 12.0 ✓
Finding an Unknown Angle
Right triangle: opposite = 8m, hypotenuse = 13m
Find the angle:
sin(angle) = 8/13 = 0.6154
angle = arcsin(0.6154) = sin^-1(0.6154) = 38.0 degrees
Different case: adjacent = 9, opposite = 6
tan(angle) = 6/9 = 0.6667
angle = arctan(0.6667) = tan^-1(0.6667) = 33.7 degrees
Key Angles to Memorise
Anglesincostan
0 degrees010
30 degrees0.50.8660.577
45 degrees0.7070.7071
60 degrees0.8660.51.732
90 degrees10undefined
Practical Applications
Roof pitch: if a roof rises 1.8m over a horizontal run of 4.5m:
tan(angle) = 1.8/4.5 = 0.4
angle = arctan(0.4) = 21.8 degrees
Ramp gradient: need a ramp at 10 degrees, horizontal distance 3m:
Vertical rise = 3 x tan(10) = 3 x 0.1763 = 0.529m = 53cm
Height of a tree using angle of elevation:
Stand 15m from tree base, angle of elevation to top = 42 degrees:
Tree height = 15 x tan(42) = 15 x 0.9004 = 13.5m