Applications of Trigonometry
     Right-angled Triangle or Right Triangle
      Trigonometric functions of an acute angle defined in a right triangle
      Solving right triangles
         Solving right triangles examples
Right-angled Triangle or Right Triangle
Trigonometric functions of an acute angle defined in a right triangle
Trigonometric functions of an acute angle are defined in a right triangle as a ratio of its sides.
Solving the right triangle
To solve a right triangle means to find all unknown sides and angles using its known parts.
While solving a right triangle we use Pythagoras’ theorem and trigonometric functions of an acute angle depending which pair of its parts is given
Note, right triangles are usually denoted as follows;  c stands for the hypotenuse,  a and b for the perpendicular sides called legs, and a and b for the angles opposite to a and b respectively.
There are four basic cases that can occur, given
             a) hypotenuse and angle,            c) hypotenuse and leg,  
             b) leg and angle,                         d) two legs.
Solving right triangles examples
Example:   Which highest level h will reach a bob pendulum of length l, starting from rest, where j is the largest angle attained by the pendulum.
Solution:  Given l and j.    h = ?
From the right triangle in the right figure
Example:   In an isosceles triangle two equal sides subtend angles a at the endpoints of the third side. 
What is the length of the third side if the difference between the altitude of the triangle to that side, and the radius of the inscribed circle, equals d?
Solution:  Given a and d.    a = ?
angle ESC = angle EAD = a  - the angles with mutually perpendicular sides
Example:   From a distance an observer sees the top of a church tower at the angle of elevation a = 18°9′34″ when comes 100 m closer, under the angle b = 28°39′50″. Find the height h of the church and the distance x.
Solution:  From the two right triangles
Example:   Find the angle a between a lateral edge and the base, and the base-to-face dihedral angle b of the regular tetrahedron shown in the figure below.
Solution:  R -the radius of circumscribed circle, r -the radius of inscribed circle, h -height of the tetrahedron and ha -height of a face (the equilateral triangle).
Geometry and use of trigonometry contents - B
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