Calculus

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How to Prove Trigonometric Identities When You Start Off with Fractions

When the trig expression you're given begins with fractions, most of the time you have to add (or subtract) them to get things to simplify. Here's one example of a proof where doing just that gets the [more…]

How to Simplify Trigonometric Expressions with a Binomial in a Fraction's Denominator

When a trigonometric expression is a fraction with a binomial in its denominator, always consider multiplying by the conjugate before you do anything else. Most of the time, this technique allows you to [more…]

How to Prove Complex Identities by Working Individual Sides of a Trig Proof

Sometimes doing work on both sides of a trig proof, one side at a time, leads to a quicker solution. This is because in order to prove a very complicated identity, you may need to complicate the expression [more…]

How to Calculate the Sine of Special Angles in Degrees

Measuring angles in degrees for the sum and difference formulas for sine is easier than measuring in radians, because adding and subtracting degrees is much easier than adding and subtracting radians. [more…]

How to Calculate the Sine of Special Angles in Radians

You can use the concept of sum and difference formulas to calculate the sine of special angles in radians. This process is different than solving equations because here you're asked to find the trig value [more…]

How to Apply the Sine Sum and Difference Formulas to Trig Proofs

When dealing with sine sum and difference formulas in a trig proof, you need to make one side of the given equation look like the other. You can work on both sides to get a little further if need be, but [more…]

Trigonometry Proofs and Pythagorean Identities

The Pythagorean identities pop up frequently in trig proofs. Pay attention and look for trig functions being squared. Try changing them to a Pythagorean identity and see whether anything interesting happens [more…]

How to Find Trigonometric Functions of an Angle by Using Pythagorean Identities

You can use Pythagorean identities to find the trig function of an angle if you know one trig function of the angle and are looking for another. For example, if you know the sine of an angle, you can use [more…]

Understanding the Binomial Theorem

A binomial is a polynomial with exactly two terms. Multiplying out a binomial raised to a power is called binomial expansion. Your pre-calculus teacher may ask you to use the binomial theorem to find the [more…]

How to Solve a Triangle When You Know Two Consecutive Side Lengths (SSA)

In some trig problems, you may be given two sides of a triangle and an angle that isn't between them, which is the classic case of SSA, or Side-Side-Angle [more…]

How to Identify the Four Conic Sections in Graph Form

Each conic section has its own standard form of an equation with x- and y-variables that you can graph on the coordinate plane. You can write the equation of a conic section if you are given key points [more…]

How to Identify the Four Conic Sections in Equation Form

Each conic section has its own standard form of an equation with x- and y-variables that you can graph on the coordinate plane. You can write the equation of a conic section if you are given key points [more…]

2 Ways to Graph a Circle

Circles are simple to work with in pre-calculus. A circle has one center, one radius, and a whole lot of points, but you follow slightly different steps, depending on whether you are graphing a circle [more…]

How to Identify the Min and Max on Vertical Parabolas

Vertical parabolas give an important piece of information: When the parabola opens up, the vertex is the lowest point on the graph — called the minimum [more…]

Calculus: How to Graph an Ellipse

An ellipse is basically a circle that has been squished either horizontally or vertically. From a pre-calculus perspective, an ellipse is a set of points on a plane, creating an oval, curved shape such [more…]

How to Graph Conic Sections in Polar Form Based on Eccentricity

When you graph conic sections on the polar plane, you use equations that depend on a special value known as eccentricity, which describes the overall shape of a conic section. The value of a conic's eccentricity [more…]

Understanding Systems of Inequalities

In a system of inequalities, you see more than one inequality with more than one variable. Before pre-calculus, teachers tend to focus mostly on systems of linear inequalities. The graphs of those inequalities [more…]

How to Apply Basic Operations to Matrices

When you apply basic operations to matrices, it works a lot like operating on multiple terms within parentheses; you just have more terms in the "parentheses [more…]

How to Write a System in Matrix Form

In a system of linear equations, where each equation is in the form Ax + By + Cz + . . . = K, you can represent the coefficients of this system in matrix, called the [more…]

Writing a Matrix in Augmented Form

An alternative to writing a system of equations as the product of a coefficient matrix and variable matrix equaling an answer matrix is what's known as [more…]

How to Solve a System of Equations Using the Inverse of a Matrix

If you have a coefficient tied to a variable on one side of a matrix equation, you can multiply by the coefficient's inverse to make that coefficient go away and leave you with just the variable. For example [more…]

How to Calculate a Sequence's Terms by Using the Sequence Expression

If you're given a few terms of a sequence, you can often use these terms to find the general formula for the sequence. If you're given the general formula [more…]

How to Find the nth Term Using the First Terms of an Arithmetic Sequence

If you know the first few terms of an arithmetic sequence, you can write a general expression for the sequence to find the nth term. To write the general expression, you must look for a pattern in the [more…]

How to Recognize Recursive Arithmetic Sequences

A recursive sequence is an arithmetic sequence in which each term depends on the term(s) before it; the Fibonacci sequence is a well-known example. When your pre-calculus teacher asks you to find any term [more…]

How to Find the General Formula for the nth Term of an Arithmetic Sequence Using Any Two Terms

At some point, your pre-calculus teacher will ask you to find the general formula for the nth term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will [more…]

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