Analytic Trigonometry: Fundamental Identities and Sum/Difference Formulas

Chapter 5: Analytic Trigonometry

Fundamental Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which both sides are defined. These identities are essential tools for simplifying expressions and solving equations in precalculus.

• Reciprocal Identities: These relate each trigonometric function to its reciprocal.

Function	Reciprocal Identity
$an heta$	$\frac{1}{ an heta} = ext{cot} heta$
$ext{cot} heta$	$\frac{1}{ ext{cot} heta} = an heta$
$ext{sin} heta$	$\frac{1}{ ext{sin} heta} = ext{csc} heta$
$ext{csc} heta$	$\frac{1}{ ext{csc} heta} = ext{sin} heta$
$ext{cos} heta$	$\frac{1}{ ext{cos} heta} = ext{sec} heta$
$ext{sec} heta$	$\frac{1}{ ext{sec} heta} = ext{cos} heta$

• Even and Odd Identities: These describe how trigonometric functions behave under negation of their arguments.

Function	Even/Odd Identity
$ext{sin}(- heta)$	$-\text{sin} heta$ (odd)
$ext{tan}(- heta)$	$-\text{tan} heta$ (odd)
$ext{csc}(- heta)$	$-\text{csc} heta$ (odd)
$ext{cot}(- heta)$	$-\text{cot} heta$ (odd)
$ext{cos}(- heta)$	$\text{cos} heta$ (even)
$ext{sec}(- heta)$	$\text{sec} heta$ (even)

• Reciprocal Identities (Alternate Form): These express one function in terms of another.

Identity
$\tan heta = \frac{\sin heta}{\cos heta}$
$\cot heta = \frac{\cos heta}{\sin heta}$

Sum and Difference Formulas

Sum and difference formulas allow us to find the sine, cosine, or tangent of the sum or difference of two angles. These are useful for evaluating trigonometric functions at non-standard angles and for verifying identities.

• Sine and Cosine Sum/Difference Formulas:

Formula	Expression
$\sin(\alpha + \beta)$	$\sin \alpha \cos \beta + \cos \alpha \sin \beta$
$\sin(\alpha - \beta)$	$\sin \alpha \cos \beta - \cos \alpha \sin \beta$
$\cos(\alpha + \beta)$	$\cos \alpha \cos \beta - \sin \alpha \sin \beta$
$\cos(\alpha - \beta)$	$\cos \alpha \cos \beta + \sin \alpha \sin \beta$

• Tangent Sum/Difference Formulas:

Formula	Expression
$\tan(\alpha + \beta)$	$\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$
$\tan(\alpha - \beta)$	$\frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

• Applications: These formulas are used to:

  • Find exact values of trigonometric expressions for non-standard angles.

  • Verify trigonometric identities.

  • Simplify complex trigonometric expressions.

Example: Find $\sin(75^\circ)$ using sum formula: $\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

Using known values: $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 30^\circ = \frac{1}{2}$ $\sin(75^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Using Identities to Verify and Simplify Expressions

Trigonometric identities are often used to verify the equality of two expressions or to simplify complex trigonometric expressions. This process involves substituting identities and manipulating algebraic expressions.

• Steps to Verify an Identity:

  1. Simplify one or both sides using fundamental identities.

  2. Apply sum and difference formulas if needed.

  3. Combine like terms and factor where possible.

  4. Show that both sides are equal.

• Example: Verify $\tan \theta = \frac{\sin \theta}{\cos \theta}$ Substitute definitions: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\frac{\sin \theta}{\cos \theta} = \frac{\frac{\text{opposite}}{\text{hypotenuse}}}{\frac{\text{adjacent}}{\text{hypotenuse}}} = \frac{\text{opposite}}{\text{adjacent}} = \tan \theta$

Quiz Instructions (Administrative)

• You will take the quiz on MyMathLab on your own computer.

• The quiz must be completed in one attempt.

• Submit your work to Canvas within 10 minutes after submitting to Pearson.