TRIG 8.1

8.1 Trigonometric Identities

OBJECTIVE 1: Reviewing the Fundamental Identities

The Quotient Identities

  • Definition of Quotient Identities:

    • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

    • cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}

The Reciprocal Identities

  • Definition of Reciprocal Identities:

    • csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}

    • sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}

    • cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}

Additional Reciprocal Identities

  • Further elucidation of Reciprocal Identities:

    • sin⁡(θ)=1csc⁡(θ)\sin(\theta) = \frac{1}{\csc(\theta)}

    • cos⁡(θ)=1sec⁡(θ)\cos(\theta) = \frac{1}{\sec(\theta)}

    • tan⁡(θ)=1cot⁡(θ)\tan(\theta) = \frac{1}{\cot(\theta)}

The Pythagorean Identities

  • Definition of Pythagorean Identities:

    • sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

    • 1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)

    • 1+cot⁡2(θ)=csc⁡2(θ)1 + \cot^2(\theta) = \csc^2(\theta)

The Odd Properties

  • Definitions and properties of Odd functions:

    • sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)

    • tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta)

    • csc⁡(−θ)=−csc⁡(θ)\csc(-\theta) = -\csc(\theta)

    • cot⁡(−θ)=−cot⁡(θ)\cot(-\theta) = -\cot(\theta)

The Even Properties

  • Definitions and properties of Even functions:

    • cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta)

    • sec⁡(−θ)=sec⁡(θ)\sec(-\theta) = \sec(\theta)

OBJECTIVE 2: Substituting Known Identities to Verify an Identity

  • Utilizing previously established identities to confirm the validity of new identities.

OBJECTIVE 3: Changing to Sines and Cosines to Verify an Identity

  • Converting all trigonometric functions into sine and cosine to simplify verification processes.

OBJECTIVE 4: Factoring to Verify an Identity

  • Techniques for simplifying or proving identities using algebraic factoring methods:

    • Difference of Two Squares Formula:

    • a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

    • Perfect Square Formulas:

    • a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2

    • a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2

OBJECTIVE 5: Separating a Single Quotient into Multiple Quotients to Verify an Identity

  • Process of expressing a single fraction into a sum of smaller fractions to ease verification:

    • AB+C=AB+AC\frac{A}{B + C} = \frac{A}{B} + \frac{A}{C} if applicable.

OBJECTIVE 6: Combining Fractional Expressions to Verify an Identity

  • Steps to bring together multiple fractions to form single expressions:

    • AB+CD=AD+BCBD\frac{A}{B} + \frac{C}{D} = \frac{AD + BC}{BD}

OBJECTIVE 7: Multiplying by Conjugates to Verify Identities

  • Technique involving multiplication of expressions by their conjugates to simplify or verify:

    • For expressions of the form $A + B$ and $A - B$, the identity states:

    • AB⋅CD=A⋅CB⋅D\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D}

OBJECTIVE 8: Summarizing the Techniques for Verifying Identities

  • Approach for identity verification:

    • Start with the more complex side of the identity you wish to verify. Transform it towards matching the simpler side.

    • If stuck, switch to starting over with the other side of the identity.

    • Follow listed techniques in the following order, ensuring a thorough exploration:

    1. Apply known identities, including reciprocal identities and the even/odd properties.

    2. Convert trigonometric expressions to sines and cosines.

    3. Factor out the greatest common factor and use algebraic techniques such as the difference of two squares or perfect squares.

    4. Separate terms in complicated fractions if necessary.

    5. Combine multiple fractions into one using a common denominator.

    6. If needed, multiply fractions by the conjugates of their numerators or denominators.