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 = 50^{o}, 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 = 47.8^{o} .

**Example 3.** In the diagram below, angle a = 36^{o}, 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 = 38.28^{o}

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.