Showing posts from February, 2022

Volume of Cube and Cuboid

Volume of Cube and Cuboid Welcome visitors.. Cube A cube is a three-dimensional shape bounded by six congruent square sides. A cube has 6 sides, 12 edges and 8 vertices. Look at the following picture: Cube volume formula: $$V=r^3$$ where $r$ is cube edge length Cuboid A cuboid is a three-dimensional space formed by three pairs of squares or rectangles, with at least one pair of different sizes. A block has 6 sides, 12 edges and 8 corners. Look at the following picture: The formula of cuboid volume: $$V=p.l.t$$ where $p$ is length edge, $l$ is width edge and $t$ is high edge. Thanks see you in another post and hopefully useful.

Elliptical Area Formula

Elliptical Area Formula An ellipse is an oval shape that has the basic equation $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a$ and $b$ are respectively are the vertical and horizontal diameters. The following formula is given to calculate the area of ​​an ellipse: $L=\pi.a.b$ Thanks for view, see you and happy learning.

Spherical Volume Formula

Spherical Volume Formula The sphere is a perfectly round shape. The following formula is given to determine the volume of a sphere: $$V=\frac{4}{3}.\pi . r^3$$ Example: What is the volume of a sphere with a diameter of 20 cm? Solution: $$V=\frac{4}{3}\pi . r^3$$ $$V=\frac{4}{3}\pi.(10)^3$$ $$V=4188.79~\text{cm}^3$$ Thanks for view, see you and happy learning.

Limit as Derivative Basis

Limit as Derivative Basis The concept of limit is the approximate value. Logically the approximation value is a value that is not true but can be completely. The derivative of the function comes from the logical concept of a limit that approaches the value 0. The following is the definition of the derivative of the function: $$f'(x)=\text{lim}_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ From the above definition we can find $f'(x)$ from the function $f(x)=ax^n$ as follows: $$f(x)=ax^n$$ $$f'(x)=\text{lim}_{h \to 0} \frac{a(x+h)^n-ax^n}{h}$$ $$f'(x)=\text{lim}_{h \to 0} \frac{ax^n+a.n.h.x^{n-1}+... -ax^n}{h}$$ $$f'(x)=\text{lim}_{h \to 0} a.n.x^{n-1}=a.n.x^{n-1}$$ We will try to find $D_x~\text{sin }x$, yakni kita peroleh bentuk: $$\frac{\text{sin}(x+h)-\text{sin}x}{h}$$ $$=\text{sin}x.\frac{(\text{cos}h-1)}{h}+\frac{\text{sin}h}{h}.\text{cos}x$$ $$=\text{sin}x.m+n.\text{cos}x$$ We can find the limit values ​​of $m$ and $n$ by using the approximate value

Circle Equation

Circle Equation Circle equation form: $$(x-a)^2+(y-b)^2=r^2$$ where: $(a,~b)$ is circle center, and $r$ is circle radius. If we describe the above form, we get: $$x^2+y^2-2ax-2by+a^2+b^2-r^2=0$$ $$x^2+y^2+Ax+By+C=0$$ where: $A=-2a$, $B=-2b$, and $C=a^2+b^2-r^2$. Thanks for view, see you and happy learning.

Half Angle Trigonometry Formulas

Basic Derivatives The following gives the trigonometric formula for $\displaystyle \frac{1}{2}A$: 1. $\displaystyle \text{sin}\frac{1}{2}A=\pm \sqrt{\frac{1-\text{cos}A}{2}}$ 2. $\displaystyle \text{cos}\frac{1}{2}A=\pm \sqrt{\frac{1+\text{cos}A}{2}}$ 3. $\displaystyle \text{tan}\frac{1}{2}A=\frac{1-\text{cos}A}{\text{sin}A}$ Thanks for view, see you and happy learning.

Basic Derivatives

Basic Derivatives The derivative notation of the $y=f(x)$ function is: $y'$, $f'(x)$, $\displaystyle \frac{dy}{dx}$, and $D_x ~y$. The following material describes the basic derivative formula that is very easy to remember. The derivatives of all forms of the function can be found. In contrast to integrals, not all forms of functions can be integrated. The following is the basic formula for the derivative of the function. 1. $\displaystyle y=ax^n$ so $\displaystyle y'=anx^{n-1}$ where $a$ and $n$ are numbers. 2. $\displaystyle y=(f \pm g)(x)$ so $\displaystyle y'=(f' \pm g')(x)$ 3. $\displaystyle y=(f.g)(x)$ so $\displaystyle y'=(f'.g+f.g')(x)$ 4. $\displaystyle y=\left(\frac{f}{g}\right)(x)$ so $\displaystyle y'=\left(\frac{f'g-fg'}{g^2}\right)(x)$ 5. $\displaystyle y=(f^n)(x)$ so $\displaystyle y'=(n.(f^{n-1}).f')(x)$ Example: Given function $f(x)=3x^2+5$ and $g(x)=-x+1$. Define: 1. $\displaystyle D_x f^3$

Understanding Statements

If a sentence is only true or false but not both true and false, then the sentence is called a statement. Consider the following examples of statements: Example 1: a. "The sun rises from the east and sets in the west." Sentences are true.  b. "7 is an odd number". sentence is true.  c. "Water boils at 10 degrees Celsius". sentence is false. d, "2 is an odd number". sentence is false.  If the examples in Example 1 are studied more deeply, it will be seen that a statement is a sentence that explains something or a declarative sentence. With this it can also be seen that a sentence that does not explain something or is not a declarative sentence is not a statement. Example 2: a. Don't try to fight! b. When did Maradona lead the Argentine national team to the world cup soccer championship? c. No Smoking! d. Where do you come from? Each sentence in example 2 cannot be determined whether it is true or false. Thus, it is in accordance with the

Basic Integral Formula

Basic Integral Formula There are two types of basic integrals: indedeminate and certain integrals. 1. Integral Uncertainty Here is the basic formula of indede sureness integrals that are very easy to understand: $$\int ax^n~dx=\frac {a}{n+1}x^{n+1}+C$$ The result of an inde baepenal integral is a function with an uncertaintal constant $C$. Example: $\displaystyle \int -3x^2~dx=...$ Solution: $\displaystyle \int -3x^2~dx=\frac{-3}{2+1}x^{2+1}+C$ $\displaystyle =-x^3+C$ 2. Specific Integrals Given $\displaystyle \int f(x)~dx=F(x)+C$, Then the form of certain integrals is: $$\int_b^a f(x)~dx=F(a)-F(b)$$ A particular integral will produce a value. Example: $$\int_1^4(2x+3)~dx=...$$ Solution: $$\int(2x+3)~dx=x^2+3x+C$$ $$\int_1^4(2x+3)~dx=(4^2+3.(4))-(1^2+3.(1))$$ $$\int_1^4(2x+3)~dx=28-5=23$$ Thus this material, see you in other material and hopefully useful.


Scale 1. Scale Image Maps and plans include scale drawings. In this case, the scale is the ratio between the distance on the map and the actual distance. $$\text{Scale}=\frac{\text{Distance on map}}{\text{real distance}}$$ Example: The distance between two cities on the map is 6 cm, while the actual distance is 18 km. Determine the scale of the map. Solution: Actual distance = 18 km = 1,800,000 cm $$\text{Scale}=\frac{6}{1,800,000}=\frac{1}{300,000}$$ So, The scale of the map is 1 : 300,000, meaning that every 1 cm on the map represents 300,000 cm in actual distance. 2. Scale Model Models are small imitations with the exact shape of the ones being imitated. For example, airplane models and car models. The size comparison in the scaled model is the same as the actual size comparison. Let the length, width and height of the model respectively are $\displaystyle p_m$, $\displaystyle l_m$ and $\displaystyle t_m$, and the actual length, width and height are $\displ

Double Angle

Double Angle Welcome visitors.. The trigonometric formula for double angles is derived from the trigonometric formula for the sum of angles. We take $A=B$ so that $A+B=2A=2B$. The following are trigonometric formulas for the Double Angle: $\text{sin}(2A)=\text{sin}A.\text{cos}A+\text{sin}A.\text{cos}A$ $\text{sin}(2A)=2.\text{sin}A.\text{cos}A$ $\text{cos}(2A)=\text{cos}A.\text{cos}A-\text{sin}A.\text{sin}A$ $\displaystyle \text{cos}(2A)=\text{cos}^2A-\text{sin}^2A$ $\displaystyle \text{tan}2A=\frac{2.\text{tan}A}{1-\text{tan}^2A}$ Example: It is known that $\text{sin }M= \frac{3}{5}$. Determine the value of $\text{sin}2M$. Solution: First we have to find the value of cos$M$. $$\text{cos}M=\sqrt{1-\text{sin}^2M}$$ $$=\sqrt{1-\frac{9}{25}}=\frac{4}{5}$$ So $$\text{sin}2M=2.\frac{3}{5}.\frac{4}{5}=\frac{24}{25}$$ Thanks for view, see you and happy learning.

Sum and Difference of Angles

Sum and Difference of Angles Welcome visitors.. The following are trigonometric formulas for the sum and difference of angles: $\text{sin}(A+B)=\text{sin}A.\text{cos}B+\text{sin}B.\text{cos}A$ $\text{cos}(A+B)=\text{cos}A.\text{cos}B-\text{sin}A.\text{sin}B$ $\text{sin}(A-B)=\text{sin}A.\text{cos}B-\text{sin}B.\text{cos}A$ $\text{cos}(A-B)=\text{cos}A.\text{cos}B+\text{sin}A.\text{sin}B$ $\displaystyle \text{tan}(A+B)=\frac{\text{tan}A+\text{tan}B}{1-\text{tan}A.\text{tan}B}$ $\displaystyle \text{tan}(A-B)=\frac{\text{tan}A-\text{tan}B}{1+\text{tan}A.\text{tan}B}$ Example: It is known that $\text{sin }A= \frac{2}{5}$and $\text{sin }B = \frac{3}{5}$. Determine the value of: 1. $\text{sin}(A+B)$ 2. $\text{cos}(A-B)$ 3. $\text{tan}(A+B)$ Solution: First we have to find the value of cos$A$, cos$B$, tan$A$, and tan$B$. By using the trigonometric identity: $\displaystyle \text{cos}A=\sqrt{1-\text{sin}^2A}$ $\displaystyle \text{cos}B=\sqrt

Relating Angle

Relating Angle Welcome visitors.. In this post, we will explain about Relating Angle. First, the reader must understand the quadrant. From the two-dimensional Cartesian coordinates, we know that there are 4 regions (quadrants) namely on the top right (K I), top left (K II), bottom left (K III) and bottom right (K IV), K stands for quadrant. Consider the following proposition of related angles. The value of trigonometric ratios (sin, cos, tan, csc, sec and cot) in that quadrant depends on the ratio formula. For example, the value of tan in quadrant II is $\displaystyle \frac{y(+)}{x(-)}$ or negative (-). Another example, the value of sin in quadrant III is $\displaystyle \frac{y(-)}{r(+)}$ or negative (-). Note that the value of $r$ is always +. The following is a formula for trigonometric ratios for related angles: $$\text{sin}(90^\text{o}-\alpha)=\text{cos }\alpha$$ $$\text{sin}(270^\text{o}-\alpha)=-\text{cos }\alpha$$ $$\text{cos}(90^\text{o}-\alpha)=\text{sin

Special Angle

Special Angle Welcome visitors.. In this post, we will explain about Special Angle. Take a look at the following table of special angles: $0^\text{o}$ $30^\text{o}$ $45^\text{o}$ $60^\text{o}$ $90^\text{o}$ sin 0 $\displaystyle \frac{1}{2}$ $\displaystyle \frac{1}{2}\sqrt{2}$ $\displaystyle \frac{1}{2}\sqrt{3}$ 1 cos 1 $\displaystyle \frac{1}{2}\sqrt{3}$ $\displaystyle \frac{1}{2}\sqrt{2}$ $\displaystyle \frac{1}{2}$ 0 tan 0 $\displaystyle \frac{1}{3}\sqrt{3}$ 1 $\displaystyle \sqrt{3}$ $\infty$ csc $\infty$ 2 $\displaystyle \sqrt{2}$ $\displaystyle \frac{2}{3}\sqrt{3}$ 1 sec 1 $\displaystyle \frac{2}{3}\sqrt{3}$ $\displaystyle \sqrt{2}$ 2 $\infty$ cot $\infty$ $\displaystyle \sqrt{3}$ 1 $\displaystyle \frac{1}{3}\sqrt{3}$ 0 Example Given a right angled triangle ABC at B. Length AC = 2 and angle C = $30^\text{o}$. Length AB = .... Answer: From the special corner table, we get: sin

Trigonometry (Basic Formulas and Trigonometric Identity)

Trigonometry (Basic Formulas and Trigonometric Identity) Welcome visitors.. In this post, we will explain about trigonometry. A right triangle with hypotenuse $r$, vertical side $y$ and horizontal side $x$. Given the angle $\alpha$ formed from the sides $x$ and $r$, then there are 6 basic formulas in trigonometry, namely: 1. sin $\displaystyle \alpha =\frac{y}{r}$ 2. cos $\displaystyle \alpha =\frac{x}{r}$ 3. tan $\displaystyle \alpha =\frac{y}{x}$ 4. csc $\displaystyle \alpha =\frac{r}{y}$ 5. sec $\displaystyle \alpha =\frac{r}{x}$ 6. cot $\displaystyle \alpha =\frac{x}{y}$ Description: sin be read "sine" cos be read "cosine" tan be read "tangent" csc be read "cosecant" sec be read "secant" cot be read "cotangent" Trigonometry Identity The trigonometric identity comes from the Pythagorean formula. From the triangle we have defined above, the Pythagorean formula is $x^2+y^2=r^2