Neurochispas
Neurochispas is a website that offers various resources for the study of Mathematics, Physics, and English. Our goal is to help students learn in an accessible and interactive way.
Algebra
Learn about variables, equations, functions, and more advanced algebraic topics.
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10 Examples of Composite Functions with Answers
Composite functions are functions where we use the output values or results of one function as the inputs of another function. For example, if we have the functions f(x) and g(x), a composite function is formed when we write f(g(x)). Essentially, we are applying a function to the result of another function. Here, we will…
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10 Examples of Equations with Absolute Value with Answers
Absolute value equations can be solved by squaring both sides of the equation. In this way, the expression inside the absolute value sign will always be positive. Then we can expand the expressions and simplify them. Finally, the obtained equation can be solved with any applicable method. Here, we will look at 10 examples of…
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10 Examples of the Remainder Theorem with Answers
The remainder theorem tells us that when a polynomial f(x) is divided by (αx–β), the remainder is equal to f(β/α). This means that to find the remainder of the division, we have to evaluate the function using the β/α value, that is, the x value of the factor (the divisor). Below, we will look at…
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10 First Degree Equations Examples and Practice Problems
First-degree equations can be solved by applying different operations to both sides of the equal sign so that we simplify the equation and solve for the variable. In this article, we will look at a summary of solving first-degree equations. In addition, we will look at several examples with answers and practice problems. ALGEBRA Relevant…
Geometry
Learn about angles, perimeter, area and volume of different geometric figures.
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3D Pythagorean Theorem – Formula and Examples
The 3D Pythagorean theorem is an extension of the 2D Pythagorean theorem that can be used to solve problems in three dimensions such as cubes, and rectangular pyramids. In three dimensions, the general formula of the 3D Pythagorean theorem is $latex {{c}^2}={{x}^2}+{{y}^2}+{{z}^2}$, where x, y, and z are the corresponding lengths of the three dimensions. In this lesson, we…
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Acute Scalene Triangle – Formulas and Examples
Acute scalene triangles are scalene triangles in which all of their interior angles are acute. These triangles satisfy the definition of a scalene triangle and an obtuse triangle at the same time. Recall that an acute angle is an angle that is less than 90 degrees, and scalene triangles are triangles that have sides of…
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Adjacent Angles – Definition, Properties and Examples
Adjacent angles are angles that have a common vertex and a common side but do not overlap. Recall that a vertex is a point where two segments or two sides of an angle meet and the sides are simply the line segments that contain the angle. A common example of adjacent angles is the interior…
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All the Cross-Sections of a Cone
Cone cross-sections are obtained when we cut a cone with a plane. We can obtain different cross-sections depending on the orientation of the plane. It is possible to obtain circular, elliptical, parabolic, and hyperbolic cross-sections. Here, we will learn about each of the cross-sections of a cone using diagrams. GEOMETRY Relevant for… Learning about all…
Trigonometry
Learn about angles in a triangle, trigonometric functions and trigonometric identities.
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Amplitude of Trigonometric Functions with Examples
The amplitude of a function is defined as the distance from the central axis to the maximum or minimum value of the function. In the case of the sine and cosine functions, the central axis is called the sinusoidal axis. The amplitude can also be defined as half the distance between the maximum value and…
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Cartesian to Cylindrical Coordinates – Formulas and Examples
Cylindrical coordinates are an alternate three-dimensional coordinate system to the Cartesian coordinate system. Cylindrical coordinates have the form (r, θ, z), where r is the distance in the xy plane, θ is the angle of r with respect to the x-axis, and z is the component on the z-axis. This coordinate system can have advantages over the Cartesian system when graphing cylindrical figures such as…
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Cartesian to Spherical Coordinates – Formulas and Examples
Spherical coordinates are written in the form (ρ, θ, φ), where, ρ represents the distance from the origin to the point, θ represents the angle with respect to the x-axis in the xy plane and φ represents the angle formed with respect to the z-axis. Spherical coordinates can be useful when graphing spheres or other three-dimensional figures represented by angles. This coordinate system is particularly…
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Cosecant of an Angle – Formulas and Examples
The cosecant of an angle is defined with respect to the sides of a right triangle. In a right triangle, the cosecant is equal to the length of the hypotenuse divided by the side opposite the angle. The cosecant is the reciprocal function of the sine. Here, we will learn more about the cosecant and…
Precalculus
Learn about parabolas, ellipes, circumferences, and hyperbolas.
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Applications of the Ellipse
Ellipses are conic sections formed by a plane that intersects a cone. Ellipses are characterized by the fact that the sum of the distances from any point on the ellipse to two fixed points is equal to a constant. The fixed points are called the foci of the ellipse. The shape of the ellipse and…
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Applications of the Hyperbola
Hyperbolas are conic sections formed when a plane intersects a pair of cones. For the hyperbola to be formed, the plane has to intersect both bases of the cones. Hyperbolas are made up of two branches that are shaped like a parabola. We have a vertex and a focus in each branch, which serve to…
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Arc of a Circumference – Formulas and Examples
An arc of a circumference or of a circle is a portion of the circumference. The length of an arc is simply the length of this portion of the circumference. The circumference itself can be considered an arc that goes around the circle. Here, we will learn about the different formulas and methods that we…
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Asymptotes of a Hyperbola – Formulas and Examples
The asymptotes of a hyperbola are straight lines that the curve approaches as the values of the independent variable (x) increase. The branches of the hyperbola approach the asymptotes but never touch them. All hyperbolas have two asymptotes, which intersect at the center of the hyperbola. The equations of the asymptotes can have four different…
English
Learn about English grammar with sentences.
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50 Sentences in Past Simple – With Pronunciation
Do you need the audio? 1. James completed his project a week before the deadline. 2. Sarah and Rachel went to the museum last Saturday. 3. Michael wrote a compelling novel that captivated many readers. 4. Emma traveled to Japan for her summer vacation last year. 5. John and his friends played football at the…
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50 Sentences in Present Continuous – With Pronunciation
Do you need the audio? 1. Sarah is studying for her math exam. 2. Are you watching the latest episode of your favorite TV show, Michael? 3. They are discussing the details of the project. 4. Emma and Liam are going on a hike this weekend. 5. I am reading an interesting book about space…
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50 Sentences in Simple Present – With Pronunciation
Do you need the audio? 1. Alice reads her favorite book every evening before going to bed. 2. John and Sarah walk their dog in the park every morning. 3. Maria listens to classical music while she studies for her exams. 4. Tom exercises at the gym three times a week to stay healthy. 5.…
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50 Sentences with Adjectives – With Pronunciation
Do you need the audio? 1. The garden was full of colorful flowers. 2. She wore a gorgeous evening gown. 3. He served me a cup of steaming coffee. 4. My grandmother has a very delicate health. 5. The meal was delicious. 6. The box is empty. 7. The event was a resounding success. 8.…