Category: Trigonometry, Algebra
"Published in Newark, California, USA"
Find the values of x in the range from 0º to 360º for
Sin x + Sin 2x + Sin 3x = 0
Solution:
Consider the given equation
Sin x + Sin 2x + Sin 3x = 0
Apply the Sum and Difference of Two Angles Formula and Double Angle Formula for the above equation
Sin x + 2 Sin x Cos x + Sin (x + 2x) = 0
Sin x + 2 Sin x Cos x + Sin x Cos 2x + Cos x Sin 2x = 0
Sin x + 2 Sin x Cos x + Sin x (Cos2 x - Sin2 x) + Cos x (2 Sin x Cos x) = 0
Sin x + 2 Sin x Cos x + Sin x Cos2 x - Sin3 x + 2 Sin x Cos2 x = 0
Sin x + 2 Sin x Cos x + 3 Sin x Cos2 x - Sin3 x = 0
Take out their common factor which is Sin x, we have
Sin x (1 + 2 Cos x + 3 Cos2 x - Sin2 x) = 0
Sin x [1 + 2 Cos x + 3 Cos2 x - (1 - Cos2 x)] = 0
Sin x (1 + 2 Cos x + 3 Cos2 x - 1 + Cos2 x) = 0
Sin x (2 Cos x + 4 Cos2 x) = 0
(Sin x )(2 Cos x)(1 + 2 Cos x) = 0
Equate each factor in zero
For Sin x = 0
x = Sin-1 0
x = 0º, 180º, 360º
For 2 Cos x = 0
Cos x = 0
x = Cos-1 0
x = 90º, 270º
For 1 + 2 Cos x = 0
2 Cos x = -1
Cos x = - ½
x = Cos-1 -½
x = 120º, 240º
Therefore,
x = 0º, 90º, 120º, 180º, 240º, 270º, and 360º
This website will show the principles of solving Math problems in Arithmetic, Algebra, Plane Geometry, Solid Geometry, Analytic Geometry, Trigonometry, Differential Calculus, Integral Calculus, Statistics, Differential Equations, Physics, Mechanics, Strength of Materials, and Chemical Engineering Math that we are using anywhere in everyday life. This website is also about the derivation of common formulas and equations. (Founded on September 28, 2012 in Newark, California, USA)
Monday, December 3, 2012
Sunday, December 2, 2012
Frustum - Right Circular Cone
Category: Solid Geometry, Plane Geometry
"Published in Newark, California, USA"
The frustum of a right circular cone has a slant height of 9 ft. , and the radii of the bases are 5 ft. and 7 ft. Find the lateral area and the total area. What is the altitude of this frustum? Find the altitude of the cone that was remove to leave this frustum. Find the volume of the frustum.
Solution:
To illustrate the problem, you can draw the figure as follows
Since the slant height and the radii of the bases are given, we can get the lateral area, area of the bases, and the total area of the frustum.
Let's get the circumference of the top base of the frustum as follows
For the bottom base of the frustum
Therefore, the lateral area of the frustum is
Let's get the area of the top base of the frustum as follows
For the bottom base of the frustum
Therefore, the total area of the frustum is
Since the altitude of the frustum is not given in the problem, we have to find the altitude of the frustum with the use of the vertical section of the frustum. The vertical section of the frustum is a trapezoid.
From the figure, we can get the altitude of a trapezoid which is also the altitude of a frustum. As you notice that the end portion of a trapezoid is a right triangle. We can use the Pythagorean Theorem to solve for the altitude of the frustum as follows
In order to get the altitude of the cone that was removed to leave this frustum, we have use the formula for similar triangles as follows
Finally, we can get the volume of the frustum as follows
"Published in Newark, California, USA"
The frustum of a right circular cone has a slant height of 9 ft. , and the radii of the bases are 5 ft. and 7 ft. Find the lateral area and the total area. What is the altitude of this frustum? Find the altitude of the cone that was remove to leave this frustum. Find the volume of the frustum.
Solution:
To illustrate the problem, you can draw the figure as follows
 |
| Photo by Math Principles in Everyday Life |
Since the slant height and the radii of the bases are given, we can get the lateral area, area of the bases, and the total area of the frustum.
Let's get the circumference of the top base of the frustum as follows
For the bottom base of the frustum
Therefore, the lateral area of the frustum is
Let's get the area of the top base of the frustum as follows
For the bottom base of the frustum
Therefore, the total area of the frustum is
Since the altitude of the frustum is not given in the problem, we have to find the altitude of the frustum with the use of the vertical section of the frustum. The vertical section of the frustum is a trapezoid.
 |
| Photo by Math Principles in Everyday Life |
From the figure, we can get the altitude of a trapezoid which is also the altitude of a frustum. As you notice that the end portion of a trapezoid is a right triangle. We can use the Pythagorean Theorem to solve for the altitude of the frustum as follows
In order to get the altitude of the cone that was removed to leave this frustum, we have use the formula for similar triangles as follows
Finally, we can get the volume of the frustum as follows
Saturday, December 1, 2012
Special Products - Factoring
Category: Algebra
"Published in Newark, California, USA"
Find the factor:
(x + 2y)3 - (x3 + 8y3) = 0
Solution:
If you notice that the second group is a sum of two cubes. We can factor the second group as follows
(x + 2y)3 - (x3 + 8y3) = 0
(x + 2y)3 - (x + 2y)(x2 - 2xy + 4y2) = 0
Take out (x + 2y) as their common factor in the above equation
(x + 2y)[(x + 2y)2 - (x2 - 2xy + 4y2)] = 0
If you expand and simplify the second group, x2 and y2 will be cancel
(x + 2y)[x2 + 4xy +4y2 - x2 + 2xy - 4y2] = 0
(x + 2y)(6xy) = 0
Therefore, the answer is (x + 2y)(6xy) = 0
"Published in Newark, California, USA"
Find the factor:
(x + 2y)3 - (x3 + 8y3) = 0
Solution:
If you notice that the second group is a sum of two cubes. We can factor the second group as follows
(x + 2y)3 - (x3 + 8y3) = 0
(x + 2y)3 - (x + 2y)(x2 - 2xy + 4y2) = 0
Take out (x + 2y) as their common factor in the above equation
(x + 2y)[(x + 2y)2 - (x2 - 2xy + 4y2)] = 0
If you expand and simplify the second group, x2 and y2 will be cancel
(x + 2y)[x2 + 4xy +4y2 - x2 + 2xy - 4y2] = 0
(x + 2y)(6xy) = 0
Therefore, the answer is (x + 2y)(6xy) = 0
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