Addition Angle Formulas

Addition angle formula is based on trigonometric functions. We are having  the addition angle formulas to find the value of  the trigonometric equations and values for the trigonometric angles.
                         Cos (A+B) = Cos A Cos B - Sin A Sin B
                         Cos (A-B) = Cos A Cos B + Sin A Sin B
                         Sin (A+B) = Sin A Cos B + Cos A Sin B
                         Sin (A-B) = Sin A Cos B - Cos A Sin B
                         Tan (A+B) = `(Tan A + Tan B)/(1 - Tan A Tan B)`
                         Tan (A-B) =  `(Tan A - Tan B)/(1 + Tan A Tan B)`
                         Here we will some problems based on addition angle formulas.

Addition Angle Problems:

Problem 1:
         Solve the following trigonometric function using Addition angle formula Sin 75^o
Solution:
            Sin 75^o
                 We can write sin 75^o as Sin (45^o + 30^o)
                              We have the formula for Sin (A+B) = Sin A Cos B + Cos A Sin B
                              Where A = 45^o and  B = 30^o
                              Sin 45^o = Cos 45 = `(1)/(sqrt(2))`
                              Sin 30 = `(1)/(2)`    Cos 30 = `(sqrt(3))/(2)`
                              Sin (45^o + 30^o) = Sin 45^o Cos 30^o + Cos 45^o Sin 30^o
                                                          =  `(1)/(sqrt(2))` `(sqrt(3))/(2)` `(1)/(sqrt(2))` `(1)/(2)`
                                                          = `(sqrt(3))/(2sqrt(2))` + `(1)/(2sqrt(2))`
                                                          = `(sqrt(3)+1)/(2sqrt(2))`
Problem 2:
       Solve the following trigonometric function using Addition angle formula Cos 135^o
Solution:
               Cos 135^o
                We can write Cos 135^o as Sin (90^o + 45^o)
                              We have the formula for Cos (A+B) = Cos A Cos B - Sin A Sin B
                              Where A = 90^o and  B = 45^o
                              Sin 45^o = Cos 45 = `(1)/(sqrt(2))`
                              Sin 90^o = 1  Cos 90^o = 0
                              Cos (90^o + 45^o) = Cos 90^o Cos 45^o - Sin 90^o Sin 45^o
                                                      = 0 . `(1)/(sqrt(2))` - 1 . `(1)/(sqrt(2))`
                                                      = 0 - `(1)/(sqrt(2))`
                                                      = - `(1)/(sqrt(2))`

Problem 3:

       Solve the following trigonometric function using Addition angle formula Tan 135^o
Solution:
               Tan 135^o
               We can write Tan 135^o as Tan (180^o - 45^o)
                              We have the formula Tan (A - B) = `(Tan A - Tan B)/(1 + Tan A Tan B)`
                              Where A = 180^o and  B = 45^o
                              Tan 45^o = 1 Tan 180^o = 0
                              Tan 135^o = `(Tan 180^o - Tan 45^o)/(1 + Tan 180^o Tan 45^o)`
                              Tan 135^o = `"(0 - 1)/(1 + (0) (1)) `
                              Tan 135^o = `(- 1)/(1)`
                              Tan 135^o = -1

Compute Percentages

In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred" in French). It is often denoted using the percent sign, "%", or the abbreviation "pct". For example, 45% (read as "forty-five percent") is equal to 45 / 100, or 0.45. Let us see how to do percentages.                                                                                                                                           - Source from Wikipedia

How to compute percentages:

STEP 1: While begin the percentage` x / 100 ` = `(is) / (of)` . Out of hundred x may be the percentage, "is" denotes as fraction, and "of" denotes as whole.
STEP 2: If we are having the questions 80 : 40 percentage then we will write it as X = 40, is = 40 ("80 is"), and then of = as unknown value. Now the value is possible to write like this `40 / 100 ` = `80 / x` .
STEP 3: Let us do the cross multiplication. Now we can get a Constant value on 1 side and then multiply it with another side. Now we will get a result likes this 40x = 8,000.
STEP 4: Now you have to find out the x value. Where, x = `8000 / 40` = 200, now the x value will be 200.

How do you solve percentages some examples here:

Problem 1:
           In a question paper there is 80 questions. Laura took that test. If she gets 75% correct, how many questions did Laura missed?
Solution:
            Therefore total correct answers are 75% of 80 or else `75 / 100` × 80
            ` 75 / 100` × 80 = 60%
            So the question paper contains 80 questions and Laura got 60 exact answers, the number of questions Laura left is 80 − 60 = 20.
            Therefore Laura missed 20 questions.
Problem 2:
            Compute this, what is 85% of 15?
Solution:
Step 1: Compute the percent.
           The percent value is 85.
            P = 85
Step 2: Find out the base.
           The base is the number next the word OF, 15
            b = 15
Step 3: Identify the quantity.
           The calculation is the unknown.
            a =?
Step 4: Enter the value in the percent proportion formula.
         `a/ 15` = `85 / 100`
Step 5: Exercises the equation for the unknown.
           The Least Common Divisible of 15 and 100 is 100
     `100 / 1` * `a / 15` = `100 / 1` * `85 / 100`
            6.6a = 85
          ` (6.6a) / (6.6)` = `85 / (6.6)`
            a = 12.8
            12.8 is 85% of 15.


Practice problem for compute percentages:
Problem 1:
            What is the percentage of 67%?
Solution:
             = 0. 67.
Problem 2:
            What is percentage of 87% of 18?
Solution:
             = 15.66
            Therefore 15.66 is 87% of 18.

Taylor Polynomial Series

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials.  In the article we shall discuss about Taylor polynomial series.(Source: Wikipedia)

Taylor polynomial series:

Find the Taylor series for sin x.

    Sin x = `sum_(0)^oo` `((-1)^k)/((2k + 1)!)` x^2k+1

The remainder term is not expressible in any simple way but can be estimated by using the Lagrange's form of the remainder. The coefficients

   `((-1)^k)/((2k + 1)!)`

are easily verified by calculating successive derivatives of f(x) = sin x and using the formulas

 a_k=`(f^(k)(0))/(k!)`

To check convergence of the series, apply Lagrange's form for R_a(x); For each x`in` R. there exists Z such that

R_n(x) = `(f^(n + 1)(z))/((n + 1)!)` xⁿ⁺¹

Now |fⁿ⁺¹(z)| equals either |cos z| or |sin z| So, in either case,|fⁿ⁺¹ (z)|`<=`1 ,and

       |R_n(x) |`<=` | x |ⁿ⁺¹ /(n+1)!

Since | x |ⁿ⁺¹ /(n+1)!`->`0 as n `|->` `oo`  for all x`in` R, we can see that the remainder term |R_n(x)|`|->` 0 as n `|->` `oo`

 for all x`in` R. Thus the series representation is completely justified for all real x.

  Observe that our estimate for |R_n (x)|,

                                                    |R_n (x)|`<=`|x|^_n+1/(n+1)!

gives also a sense of the rate of convergence of the series for fixed x, for example, for | x|`<=` 1, we find

                                                    |R_n (x)|`<=`1/(n+1)!

Thus, if we want to calculate sin x on (-1, 1) to within .01, we need take only the first five terms of the series (n = 4) to achieve that degree of accuracy.

 Had we used the integral form for R_n (x) we would have obtained a similar estimate.

Sample problem for Taylor polynomial series:

Pro:  Evaluate the definite integral `int_0^1` Sin (x) dx

Sol:  The integrand has no anti derivative expressible in the terms of familiar functions. Howebver, we know how to find its Taylor series. we know that

      Sin t = t -`(t^3)/(3!)` + `(t^5)/(5!)` - `(t^7)/(7!)` + ----

Now if we substitute t = x, we have

     Sin (x) = X - `(x^3)/(3!)` + `(x^5)/(5!)` - `(x^7)/(7!)` + ----

In spite of the fact that we cannot antidifferentiate the function, we can antidifferentiate the Taylor series:

    `int_0^1` Sin (x) dx = `int_0^1` (X - `(x^3)/(3!)` + `(x^5)/(5!)` - `(x^7)/(7!)` + ----) dx

                            =(`(x^2)/(2)` -`(x^)/(4*3!)` + `(x^6)/(6*5!)`-`(x ^ 7)/(8*7!)`+ -----) |₀¹

                            = (`(x^2)/(2)` -`(x^)/(4*3!)` + `(x^6)/(6*5!)`+`(x^7)/(8*7)` + ---)

Notice that this is an alternating series so we know that it converges. if we add up the first four terms, the pattern becomes ckear: the series converges to 0.2871

Negative Integer Exponents

The exponents are which integer is placed in the power, of base numbers. It can be easily represent as, “a small number to the right side and above of base number”. It is called as exponents. These exponents have some of important rules and laws. Power with negative integer exponents is also one of the rules. Here we are going to explain about this negative integer exponent rule.

If we are having variables, which is containing the exponents and it have equal bases means, we can do some mathematical operations with the exponents. Those operations are called as the “laws of exponents” or “rules of exponents”. In this rule based negative integer rule of exponent is defined as following ways,

Definition for negative integer exponents:

It is otherwise called as power with negative exponent rules. This negative exponent rule is defined as, if m is a positive integer and x is a non-zero rational number, then it can be denoted as,

X-m = (1/x)^m (or)

= (1/x)^m

Which is (x)^-m is the reciprocal of (x)^m

And we adopt the same rule for rational exponents also. If p/q is a positive rational number means, and x>0 is a rational number, then

 X^ - (p/q) = (1/x)^ (p/q) = (1/x)^ (p/q) .

Which is, (x)^-(p/q) is the reciprocal of (x) ^(p/q) or the number obtained by raising the reciprocal of x to the exponent p/q.

For example: 1). 3^-2

= (1/3)^2

= (1/3)^-2

= -6 .

2). (4)^-(2/3)

= (1/4)^-(2/3)

= (1/4)^ (2/3)

This kind of exponentiation used for discovers the negative integer exponents and simplify the problems.


Example problems for negative integer exponents:

1) Solve: (8)^-(2/3)

Solution:

Given: (8)^-(2/3)

= (1/8)^ (2/3)

= [(1/8)^ (1/3)]^ 2

= (1/2)^2, since (1/2)^3 = 1/8

= 1/4 .

2) Solve: (32/243)^-(4/5)

Solution:

Given: (32/243)^-(4/5)

= (243/32)^(4/5)

= [(243/32)^(1/5)]^4

= [(3^5/2^5)^(1/5)]^4

= [((3/2)^5)^(1/5)]^4

= (3/2)^4

= 81/16.

3) Evaluate, and find the following negative integer exponent value:

Evaluate: (27/125)^-(2/3)  xx    (27/125)^-(4/3)

Solution:

 (27/125)^- (2/3) xx (27/125)^-(4/3)

= (125/25)^(2/3) xx (125/27)^(4/3)

= [(5^3/3^3)^(1/3)]^ 2 xx [(5^3/3^3)^(1/3)]^4

= [((5/3)^3)^(1/3)]^2 xx [((5/3)^3)^(1/3)]^4

= (5/3)^2 xx (5/3)^4

= (5/3)^6

= 15625/729.

These all are the explanations and example problems may clear about the negative integer exponents.

Math Absolute Value Inequalities

In math, the absolute value |x| of a real number x is x's arithmetical value lacking view to its symbol. So, for example, 86 is the absolute value of both 86 and −86. Generalization of the absolute value for real numbers occurs in a extensive diversity of math settings. Consider an absolute value is also definite for the complex numbers, the quaternions, prepared rings, fields and vector spaces. The absolute value is strictly associated to the ideas of magnitude, distance, and norm in different math and physical contexts.

Properties of the Math absolute value inequalities

The absolute value fundamental properties are:

 |x| = sqrt(x^2)                                      (1) Basic

 |x| \ge 0                                            (2)     Non-negativity

 |x| = 0 \iff x = 0                            (3)     Positive-definiteness

 |xy| = |x||y|\,                                  (4)     Multiplicativeness

 |x+y| \le |x| + |y|                            (5)     Subadditivity

Another important property of the absolute value includes these are the:

 |-x| = |x|\,                                      (6)     Symmetry

|x - y| = 0 \iff x = y                     (7)     Identity of indiscernible (equivalent to positive-definiteness)

|x - y| \le |x - z| +|z - y|                 (8)     Triangle inequality (equivalent to sub additivity)

|x/y| = |x| / |y| \mbox{ (if } y \ne 0) \,                   (9)      Preservation of division (equivalent to multiplicativeness)

|x-y| \ge ||x| - |y||                           (10)     (Equivalent to sub additivity)

Math absolute value inequalities – Examples:

Math absolute value inequalities – Example 1:

|x - 5| < 3

Set up the two times inequality -3 < x -5 < 3 and then solve.

-3 < x – 5 < 3

2 < x < 8

In interval notation, the answer is (2, 8).
Math absolute value inequalities – Example 2:

|2x + 3| <=-8

This solution will involve setting up two separate inequalities and solving each.

2x + 3 <= -8

2x <=-11

x<=-11/2

Else

2x+3 >=8

2x >= 5

x>=5/2

In interval notation, the answer is (-oo, -11/2) U (5/2, oo) .
Math absolute value inequalities – Example 3:

|x - 9| < 4

Set up the two times inequality -4 < x -9 < 4 and then solve.

-4 < x – 9 < 4

5 < x < 13

In interval notation, the answer is (5, 13).

Math absolute value inequalities – Example 4:

|3x + 3| <= - 8

This solution will involve setting up two separate inequalities and solving each.

3x + 3 <= -8

 3x <=-11

x<=-11/3

Else

3x+3 >=8

 3x >= 5

x>=5/3

In interval notation, the answer is (-oo, -11/3) U (5/3, oo).

Negative Number Calculator

In this article we are discussing about subtracting negative number by using calculator. Negative number also real number. The negative number is represented as minus symbol “- “. Negative number or elements are less than zero such as -4, -7, -/3. A negative number may be parenthesized with its symbol, For example a subtracting is clearer if written (-7) + (−5) = -13 Using calculator we can the negative number.
Negative number calculator:

Let us see how to negative number using calculator.

Negative number is the similar as subtracting the corresponding negative number:

Example: (-7) + (-7) = -14

Negative number calculator – Example problems:

Example 1:

(-3) + (-4) = ?

Solution:

First enter the negative number in the calculator.

Then press the equal to button.

The solution is displayed on the calculator.

negative number calculator

The subtract solution is -7.

Example 2:

(-5) - (-4) = ?

Solution:

First enter the negative number in the calculator.

Then press the equal to button.

The solution is displayed on the calculator.

negative number calculator

The subtract solution is -1.

Example 3:

(-4) x (-4) = ?

Solution:

First enter the negative number in the calculator.

Then press the equal to button.

The solution is displayed on the calculator.

negative number calculator

The subtract solution is 16.

Example 4:

(-9) `-:` (-3) = ?

Solution:

First enter the negative number in the calculator.

Then press the equal to button.

The solution is displayed on the calculator.

negative number calculator

The subtract solution is 3.

Example 5:

(-10) x (-20) = ?

Solution:

First enter the negative number in the calculator.

Then press the equal to button.

The solution is displayed on the calculator.

negative number calculator

The subtract solution is 200.

Negative number calculator – practice problems:

Problem 1: (-5) + (-6)

Problem 2: (-4) – (-2)

Problem 3: (-9) x (-18)

Problem 4: (-5) `-:` (-20)

Negative number calculator – answer key:

Problem 1: -11

Problem 2: -2

Problem 3: 162

Problem 4: -0.25

Solving Monotonicity and Concavity

A function is said to be rising on an interval [a, b] = I if f (z1) < f (z2) whenever z1 < z2 for z1, z2 `in` I. A function is said to be decreasing on an interval [a, b] = I if f (z1) > f (z2) whenever z1 < z2 for z1, z2 `in` I. A function which is strictly increasing or strictly decreasing on an interval is said to be monotonic on that interval. A function is said to be curved in up on an interval [a, b] if f ’’ (z) > 0 on [a, b]. A function is said to be curved in down on an interval [a, b] if f ’’ (z) < 0 on [a, b]. A point (c, f(c)) on the graph of y = f (z) is called an inflection point if the concavity changes at the point. (I.E., it is curved in up on some interval (a, c) and curved in down on some interval (c, b) or vice versa.)

Solving monotonicity and concavity - Examples:

Solving monotonicity and concavity - Example 1:

Find inflection points & determine concavity for f (x)

f ‘(x) = `(x^5)/20 + (x^4)/12 - (3x^3)/3 - 10`

f ’(x) = `(x^4)/4 + (x^3)/3 - 3(x^2)`

f ‘’ (x) = x3 + x2 – 3x

= x(x2 + x - 3)

f ‘‘(x) = 0

x = 0, -2.3, 1.3

Inflection pts: x= -2.3, 0, 1.3

Curved in up: (-2.3,0), (1.3,infinity)

Curved in down: (-oo,-2.3), (0,1.3)

Solving monotonicity and concavity

Solving monotonicity and concavity - More Examples:

Solving monotonicity and concavity - Example 1:

Find inflection points & determine concavity for f (x)

f (x) = `(x^5)/20 + (x^4)/12 - (x^3)/3 +10`

f ‘ (x) = `(x^4)/4 + (x^3)/3 - x^2`

f ‘’ (x) = x3 + x2 – 2x = x(x2 + x - 2)

= x (x+2) (x-1)

f ‘‘ (x) = 0

x = 0, -2, 1

f ’’(-5) < 0, f’’(-1) > 0, f ’’(.5) < 0, f ’’(10) > 0

Inflection pts: x=-2,0,1

Curved in up: (-2,0), (1,`oo` )

Curved in down: (`-oo` .,-2), (0,1)

Solving monotonicity and concavity