Revision of Basic Trigonometry.

Links:  Calculating altitude without an angle measuring instrument

Introduction to Spherical Trigonometry

Application of Spherical Trigonometry

Trig Tables

By convention the sides of right-angle triangles are named as shown in the diagram below:

trig3

 

 

 

 

 

 

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.

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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.

trig4

 

 

 

 

 

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

shadow stick

 

 

 

 

 

 

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.

web: www.astronavigationdemystified.com

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