Algebra Tutorials!
Thursday 28th of September
Calculations with Negative Numbers
Solving Linear Equations
Systems of Linear Equations
Solving Linear Equations Graphically
Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Solving Trigonometric Equations

How would you solve the equation sin θ = 0? You know that θ = 0 is one solution, but this is not the only solution. Any one of the following values of is also a solution.

..., -3π, -2π, -π, 0, π, 2π, 3π,...

You can write this infinite solution set as {nπ: n is an integer}


Example 1

Solving a Trigonometric Equation

Solve the equation


To solve the equation, you should consider that the sine is negative in Quadrants III and IV and that

Thus, you are seeking values of θ in the third and fourth quadrants that have a reference angle of π/3. In the interval [0, 2π], the two angles fitting these criteria are

By adding integer multiples of to each of these solutions, you obtain the following general solution.

See the figure below.

Solution points of


Example 2

Solving a Trigonometric Equation

Solve cos 2θ = 2 - 3 sin θ, where 0 ≤ θ ≤ 2π.


Using the double-angle identity cos 2θ = 1 - 2 sin2θ, you can rewrite the equation as follows.

cos 2θ = 2 - 3 sin θ Given equation
1 - 2 sin2θ = 2 - 3 sin θ Trigonometric identity
0 = 2sin2θ - 3sin θ + 1 Quadratic form
0 = (2 sin θ)(sin θ - 1) Factor.

If 2 sin θ = 0, then sin θ = 1/2 and θ = π/6 or θ = 5π/6. If sin θ - 1, then sin θ = 1 and θ = π/2. Thus, for 0 ≤ θ ≤ 2π, there are three solutions.

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