Links: Calculating altitude without an angle measuring instrument
Introduction to Spherical Trigonometry
Application of Spherical Trigonometry
By convention the sides of right-angle triangles are named as shown in the diagram below:
Also by convention, the various ratios of the sides of a right-angle triangle are given names. The table below gives the names of the ratios of the various sides in relation to angle ø in the diagram.
RATIO OF SIDES IN RELATION TO ø NAME OF RATIO
PERPENDICULAR / BASE TANGENT
PERPENDICULAR / HYPOTENUSE SINE
BASE / HYPOTENUSE COSINE
The reciprocals of these ratios also have names as shown in the next table:
RATIO OF SIDES IN RELATION TO ø NAME OF RATIO
BASE / PERPENDICULAR COTANGENT
HYPOTENUSE / PERPENDICULAR COSECANT
HYPOTENUSE / BASE SECANT
Using Trigonometry.
Example 1. In the diagram below, angle Ø = 50o, the length of the perpendicular = 4.5 m. Calculate the length of the hypotenuse
Solution. Sin(Ø) = perpendicular / hypotenuse
⇒ hypotenuse = perpendicular / Sin(Ø)
= 4.5 / Sin(50) = 4.5 / 0.77715
∴ hypotenuse = 5.79 m.
Example 2. Using the same diagram, base = 90 cm. hypotenuse = 134 cm. Calculate angle Ø
Solution. Cos(Ø) = base / hypotenuse
= 90 / 134 = 0.6716
cos-1 (0.6716) = 47.8
∴ Ø = 47.8o .
Example 3. In the diagram below, angle a = 36o, the height of the light house (h) = 94m. Calculate the boat’s distance from the light house (d)
Solution. tan(a) = h / d
⇒ d = h / tan(a) = 94 / tan(36)
= 94 / 0.7265 = 129.38m.
∴ the boat is 129.38m. from the lighthouse.
Example 4. In the diagram below, the length of the pole is 3m, the length of the shadow = 3.8m. Calculate angle Ø
Solution. tan(Ø) = length of pole / length of shadow
⇒ tan(Ø) = 3 / 3.8 = 0.7894
tan-1(0.7894) = 38.28
∴ angle Ø = 38.28o
A fuller explanation of this topic is given in the books ‘Astro Navigation Demystified’ and ‘Applying Mathematics to Astro Navigation’.
Astro Navigation blog: Astro Navigation Demystified.
Astro Navigation Demystified 
You must be logged in to post a comment.