More Spherical Triangle Problems

Category: Trigonometry

"Published in Newark, California, USA"

Find the side opposite the given angle for a spherical triangle having

(a) b = 60°, c = 30°, A = 45°
(b) a = 45°, c = 30°, B = 120°

Solution:

A spherical triangle is a triangle whose sides are the edges of a sphere. It is not the same as a plane triangle because the sides of a spherical triangles are curve and not a straight line. A spherical triangle looks like this

Photo by Math Principles in Everyday Life

As you notice that the measurements of the edges of a spherical triangle are expressed in degrees because the sides of a spherical triangle are the arcs of a sphere. The measurements of the arcs of a sphere or the edges of a spherical triangle are measured from the center of a sphere. 

In this case, we will use the formulas that are completely different from the formulas of plane triangles. The formulas that we will use are the following:

The above formulas are called the Law of Cosines. You must  remember or memorize all the above formulas because you will use these often in solving the sides and the angles of spherical triangles.

Now, let's go back to the given problem, if 

(a) b = 60°, c = 30°, A = 45°

then we have to solve for the measurement of arc a. Use the first formula as follows

Substitute the values b, c, and A, we have

or

Let's have another one, if

(b) a = 45°, c = 30°, B = 120°

then we have to solve for the measurement of arc b. Use the second formula as follows

or

Algebraic Operations - Exponents

Category: Algebra

"Published in Suisun City, California, USA"

Simplify the expression and eliminate any negative exponents for

Solution:

Consider the given equation above

The first grouped term has an exponent which is 3. If you will cube the first grouped term, then the exponents of variables inside the parenthesis will be multiplied by 3 as follows

Since the exponents of both grouped terms are now equal to 1, then we can multiply the two fractions directly. Numerator times numerator and denominator times denominator. If you  multiply the two terms with the same variable, then their exponents will be added as follows

If you divide the two terms with the same variable, then their exponents will be subtracted as follows

Any number or any variable (except zero) raised to zero power is always equal to one. Since there's no y at the numerator, then the exponent of y at the numerator is equal to zero.  

In order to eliminate the negative exponents of b and y, we have to transfer the two terms in the denominator as follows

Therefore,

Some Substitution Methods

Category: Differential Equations, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above

Since the above equation contains trigonometric functions, then it is considered as a complicated equation with differentials. To avoid the confusion in solving the equation, it is better to rewrite the equation by using the substitution method. We will substitute all trigonometric functions with another variables.

Let

so that 

Let

so that

Substitute all the above values to the given equation, as follows

Rewrite the above equation as a first order, first degree linear equation, we have

where

and

Since the above equation is already a first order, first degree linear equation in terms of v, then the integrating factor will be equal to

Therefore, the general solution for the above equation is

but

and

and the final answer for the above equation is