Essential Concepts

Simplifying Trigonometric Expressions with Identities

Simplifying Expressions:

• Look for patterns: difference of squares [latex]a^2 - b^2 = (a-b)(a+b)[/latex], quadratic form, perfect squares
• Reduce terms to sine and cosine
• Create common denominators to add and subtract separate fractions
• Use substitution to make expressions appear simpler (let [latex]u =[/latex] trig function)

There are multiple ways to verify an identity:

• Graphing approach: Graph both sides of the identity; if the graphs are identical, the identity is verified
• Algebraic approach: Simplify one side of the equation until it equals the other side

Steps for algebraic verification:

1. Choose the more complicated side of the equation to work with
2. Look for opportunities to factor, expand, find common denominators, or use algebraic properties (difference of squares, perfect square formulas)
3. Make substitutions using known identities

Sum and Difference Identities

• Use to find exact values of sine, cosine, or tangent of angles that can be written as sums or differences of special angles
• Can be used with inverse trigonometric functions
• Useful in verifying other identities
• Application problems are often easier to solve using these formulas

Double Angle, Half Angle, and Reduction Formulas

Using Double-Angle Formulas:

• Derived from sum formulas where both angles are equal
• Multiple forms exist for [latex]\cos(2\theta)[/latex]; choose the form that best fits the problem

Using Reduction Formulas:

• Especially useful in calculus for reducing the power of trigonometric terms
• Allow rewriting even powers of sine or cosine in terms of first power of cosine
• Apply the formula multiple times for higher powers (e.g., [latex]\cos^4 x[/latex])

Using Half-Angle Formulas:

• Allow finding values of trigonometric functions involving half-angles
• Work whether the original angle is known or not
• Choose ± sign based on the quadrant where [latex]\frac{\alpha}{2}[/latex] terminates
• Three forms available for tangent; choose the most convenient

Sum-to-Product and Product-to-Sum

Using Product-to-Sum and Sum-to-Product Formulas:

• Derived from sum and difference identities
• Trigonometric expressions are often simpler to evaluate using these formulas
• Can verify identities using these formulas or by converting to sines and cosines

Solving Trigonometric Equations

General Solution Format:

For equations with period [latex]2\pi[/latex] (sine and cosine):

• [latex]\sin\theta = \sin(\theta \pm 2k\pi)[/latex] where [latex]k[/latex] is an integer

For equations with period [latex]\pi[/latex] (tangent and cotangent):

• [latex]\tan\theta = \tan(\theta \pm k\pi)[/latex] where [latex]k[/latex] is an integer

Solving Trigonometric Equations:

Use algebraic techniques just as with algebraic equations:

• Isolate the trigonometric function using algebra
• Look for patterns: difference of squares, quadratic form, expressions that lend themselves to substitution
• Use the unit circle to find solutions
• Remember that solutions repeat based on the period of the function

Using a Calculator:

• Set calculator to proper mode (degrees or radians)
• Use inverse functions ([latex]\sin^{-1}[/latex], [latex]\cos^{-1}[/latex], [latex]\tan^{-1}[/latex]) to find one solution
• Determine additional solutions based on symmetry and the function’s period

Using Identities to Solve Equations:

• Use Pythagorean identities to express the equation in terms of one function
• Apply sum, difference, double-angle, or half-angle formulas when appropriate
• Simplify using reciprocal and quotient identities
• Convert to sines and cosines if the equation involves multiple functions

Solving Equations with Multiple Angles:

A multiple-angle equation involves a compression of a standard trigonometric function:

• Example: [latex]\sin(2x)[/latex], [latex]\cos(3x)[/latex]
• Use substitution to solve (let [latex]u = 2x[/latex], solve for [latex]u[/latex], then find [latex]x[/latex])
• Verify that all solutions fall within the given interval
• When the period is compressed by factor [latex]n[/latex], there will be [latex]n[/latex] times as many solutions in [latex][0, 2\pi)[/latex]

Key Equations

Pythagorean identity (basic)	[latex]\sin^2\theta + \cos^2\theta = 1[/latex]
Pythagorean identity (tangent)	[latex]1 + \tan^2\theta = \sec^2\theta[/latex]
Pythagorean identity (cotangent)	[latex]1 + \cot^2\theta = \csc^2\theta[/latex]
Reciprocal identities	[latex]\sin\theta = \frac{1}{\csc\theta}[/latex] [latex]\cos\theta = \frac{1}{\sec\theta}[/latex] [latex]\tan\theta = \frac{1}{\cot\theta}[/latex]
Quotient identities	[latex]\tan\theta = \frac{\sin\theta}{\cos\theta}[/latex] [latex]\cot\theta = \frac{\cos\theta}{\sin\theta}[/latex]
Cosine sum formula	[latex]\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta[/latex]
Cosine difference formula	[latex]\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta[/latex]
Sine sum formula	[latex]\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta[/latex]
Sine difference formula	[latex]\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta[/latex]
Tangent sum formula	[latex]\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}[/latex]
Tangent difference formula	[latex]\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}[/latex]
Double-angle formula (sine)	[latex]\sin(2\theta) = 2\sin\theta\cos\theta[/latex]
Double-angle formula (cosine)	[latex]\cos(2\theta) = \cos^2\theta - \sin^2\theta[/latex] [latex]\cos(2\theta) =1 - 2\sin^2\theta = 2\cos^2\theta - 1[/latex]
Double-angle formula (tangent)	[latex]\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}[/latex]
Half-angle formula (sine)	[latex]\sin\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 - \cos\alpha}{2}}[/latex]
Half-angle formula (cosine)	[latex]\cos\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 + \cos\alpha}{2}}[/latex]
Half-angle formula (tangent)	[latex]\tan\left(\frac{\alpha}{2}\right) = \frac{\sin\alpha}{1 + \cos\alpha} = \frac{1 - \cos\alpha}{\sin\alpha}[/latex]
Reduction formula (sine)	[latex]\sin^2\theta = \frac{1 - \cos(2\theta)}{2}[/latex]
Reduction formula (cosine)	[latex]\cos^2\theta = \frac{1 + \cos(2\theta)}{2}[/latex]

Glossary

double-angle formulas

Formulas derived from sum formulas by setting both angles equal, used to find trigonometric values of twice an angle

even-odd identities

Identities that relate the value of a trigonometric function at an angle to its value at the negative of that angle; cosine and secant are even functions, while sine, tangent, cotangent, and cosecant are odd functions

half-angle formulas

Formulas used to find exact values of trigonometric functions when the angle is half of a special angle

period

The length of one complete cycle of a periodic function; used when finding all solutions to trigonometric equations

product-to-sum formulas

Formulas that express products of trigonometric functions as sums or differences

Pythagorean identities

Three identities based on the Pythagorean theorem and the unit circle: [latex]\sin^2\theta + \cos^2\theta = 1[/latex], [latex]1 + \tan^2\theta = \sec^2\theta[/latex], and [latex]1 + \cot^2\theta = \csc^2\theta[/latex]

quotient identities

Identities that express tangent and cotangent as quotients: [latex]\tan\theta = \frac{\sin\theta}{\cos\theta}[/latex] and [latex]\cot\theta = \frac{\cos\theta}{\sin\theta}[/latex]

reciprocal identities

Identities that relate trigonometric functions to their reciprocals, such as [latex]\sin\theta = \frac{1}{\csc\theta}[/latex]

reduction formulas

Formulas that reduce even powers of sine or cosine to expressions involving the first power of cosine; also called power-reducing formulas

sum and difference formulas

Formulas that express trigonometric functions of sums or differences of angles in terms of functions of the individual angles

sum-to-product formulas

Formulas that express sums or differences of trigonometric functions as products

trigonometric equation

An equation that contains at least one trigonometric function and may have an infinite number of solutions due to the periodic nature of trigonometric functions

trigonometric identity

An equation involving trigonometric functions that is true for all values in the domain of the variable

verifying an identity

The process of showing that both sides of an equation are equal by transforming one side (usually the more complicated side) into the other using algebraic techniques and known identities