Solving Trigonometric Equations, 9

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 all angles of the trigonometric functions are different? You can convert the multiple angles into single angles by the sum and difference of two angles formula but the equation will be more complicated. In this case, we will use 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. 

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