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Trigonometric Identities and Equations

Simplifying Trigonometric Expressions

When Simplifying Trigonometric Identities it is important not to get frustrated and give up. The purpose of simplifying Trigonometric Identities is to simplify the expression to one single trigonometric function or number. To do this it is often helpful to follow a step by step process. While following these steps may help you simplify the expression they are not often the only way to simplify the expression. Keep an open mind try different identities and operations if the expression gets to confusing. Try another method. Her are the steps I like to follow when Simplifying Trigonometric Expressions:

  1. Apply operations such as multiplying using distributive property, combining rational expressions, or factor out common terms.
  2. Apply Trigonometric Identities to convert trigonometric functions.  Try to convert expressions so that reciprocal functions are present together, look to use Pythagorean Identities when functions are squared.
  3. Convert all trigonometric functions in terms of sines and cosines
  4. Look to apply the division property to simplify the expression to one single quantity

Simplifying Trigonometric Expressions

Verifying Trigonometric Identities

Trigonometric Identities simply state that the right side of the equation is equal to the left side of the equation.  Our goal is to simplify the identity to prove or disprove this.  We will do this in similar fashion on how we simplified trigonometric expressions.  However it is easiest when verifying to choose only one side to simplify.  I like to choose the side which is the most “difficult” meaning has operations that can be applied or the most work need to simplify. If you choose one side and get stuck, try another method or the other side.

Verifying Trigonometric Identities

Verifying Trigonometric Identities with multiple steps

Verifying Trigonometric Identities with rational expression

Verifying Trigonometric Identities using Cofunction and even/odd identities

Verifying Trigonometric Identities using double angle formulas

Verifying Trigonometric Identities using half angle formulas

Verifying Trigonometric Identities using sum and difference formulas

Solving Trigonometric Equations

When solving trigonometric equations we will bring together our understanding of simplifying trigonometric expressions to simplify the equations to solve.  Our goal with trigonometric equations is to simplify the equation down to one trigonometric function equal to a value or a factored equation equal to zero so that we can apply the zero product property.  Once we have the equation simplified we will find the solutions(angles) that make the equation true.  Then based on the problem we will either find all of the solutions or the solutions only between [0,2pi)

Solving for all the solutions of Trigonometric Equation

Solving for all the solutions Trigonometric Equations using Inverse Operations

Solving Trigonometric Equations with more than one Function

Solving Trigonometric Equations using Pythagorean Identities

Solving Trigonometric Equations using Factoring

Solving Trigonometric Equations with Multiple Angles

Solving Trigonometric Equations with Double Angle Formulas

Solving Trigonometric Equations with Sum and Difference Formulas

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