Solving Trigonometric Equations, 8

Category: Trigonometry

"Published in Vacaville, California, USA"

Solve for the value of x for the equation:

Solution:

Consider the given equation above

Did you notice that the given equation consists of the product of trigonometric functions? Well, we have to split the product of trigonometric functions first into single trigonometric functions by using the sum and product formula as follows

Take the inverse cosine on both sides of the equation, we have

                            or

Therefore, the values of x are

where n is the number of revolutions.

Regular Polygon Problems, 3

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of a regular hexagon with perimeter 12 cm.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

There are six triangles of a regular hexagon if you draw the three largest diagonal that passes through its center which is the vertex of each triangles. The first thing that we need to do is to get the vertex angle of a triangle in a regular hexagon as follows

If the given figure is a regular hexagon, then all vertex angles of the triangles are congruent. Also, the two adjacent sides of each triangle are congruent. If that's the case, the other two angles of a triangle are also congruent. The other two equal angles of a triangle are

Since all angles of a triangle are congruent, then the triangle is equilateral or equiangular triangle. 

If the perimeter of a regular hexagon is given, then we can solve for the length of a base of each equilateral triangle as follows

If you draw a perpendicular line segment from the vertex to the base of a triangle, then that line segment is the altitude of each triangle or the apothem of a regular hexagon. The altitude of a triangle bisects the base. There are two 30° - 60° right triangles of an equilateral triangle. By Pythagorean Theorem, the altitude of a triangle is

Therefore, the area of a regular hexagon is

                            or

 

Equilateral Triangle Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of an equilateral triangle inscribed in a circle with radius 2√3.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

If an equilateral triangle is inscribed in a circle, then the radius of a circle bisects the angles of an equilateral triangle. Each angle of an equilateral triangle is 60°. In this case if you analyze further the figure, there are six 30° - 60° right triangles. In this problem, we need the length of the base and the altitude of an equilateral triangle.

The base of an equilateral triangle is
 

 

The altitude of an equilateral triangle is
 

 

Therefore, the area of an equilateral triangle is
 

 

 

There's another way in getting the area of an equilateral triangle. If you know the sides of an equilateral triangle, then we can use the Heron's Formula as follows
 

The semi-perimeter of an equilateral triangle is
 

 

 

Therefore, the area of an equilateral triangle is