Factoring and Verifying Trigonometric Identities 5.1

Factoring Review

Greatest Common Factor (GCF)

Factoring is a fundamental algebraic skill that is also essential for simplifying trigonometric expressions. The greatest common factor (GCF) is the largest expression that divides evenly into each term of a polynomial or trigonometric expression.

• Step 1: Identify the lowest power of each variable present in all terms.

• Step 2: Find the largest number that divides all coefficients.

• Step 3: Factor out the GCF from each term.

Example: Factor $30x^3 + 42x^2$.

• GCF of $x^3$ and $x^2$ is $x^2$.

• GCF of 30 and 42 is 6.

• So, GCF is $6x^2$.

• Factored form: $30x^3 + 42x^2 = 6x^2(5x + 7)$

Example: Factor $80\sin^4 x - 30\sin x$.

• GCF of $\sin^4 x$ and $\sin x$ is $\sin x$.

• GCF of 80 and 30 is 10.

• Factored form: $80\sin^4 x - 30\sin x = 10\sin x (8\sin^3 x - 3)$

Factoring Trinomials (Leading Coefficient 1)

For trinomials of the form $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.

• Example: Factor $x^2 + 13x + 36$.

• Numbers: 4 and 9 ($4 \times 9 = 36$, $4 + 9 = 13$)

• Factored form: $(x + 4)(x + 9)$

Factoring Trigonometric Trinomials

Apply the same process as with algebraic trinomials, replacing $x$ with the trigonometric function.

• Example: Factor $\cos^2 x - 13\cos x + 42$.

• Numbers: -6 and -7 ($-6 \times -7 = 42$, $-6 + -7 = -13$)

• Factored form: $(\cos x - 6)(\cos x - 7)$

Factoring by Grouping (AC Method)

For trinomials $ax^2 + bx + c$ where $a \neq 1$, use the AC method:

1. Multiply $a \times c$.

2. Find two numbers that multiply to $ac$ and add to $b$.

3. Rewrite the middle term as the sum of these two numbers.

4. Factor by grouping.

Example: Factor $6x^2 + 23x + 20$.

• $a \times c = 6 \times 20 = 120$

• Numbers: 8 and 15 ($8 \times 15 = 120$, $8 + 15 = 23$)

• Rewrite: $6x^2 + 8x + 15x + 20$

• Group: $(6x^2 + 8x) + (15x + 20) = 2x(3x + 4) + 5(3x + 4)$

• Factored: $(3x + 4)(2x + 5)$

Example: Factor $2\tan^2 x + 9\tan x - 35$.

• $a \times c = 2 \times -35 = -70$

• Numbers: -5 and 14 ($-5 \times 14 = -70$, $-5 + 14 = 9$)

• Rewrite: $2\tan^2 x - 5\tan x + 14\tan x - 35$

• Group: $\tan x(2\tan x - 5) + 7(2\tan x - 5)$

• Factored: $(2\tan x - 5)(\tan x + 7)$

Factoring with Common Trigonometric Factors

When all terms share a trigonometric factor, factor it out first, then factor the remaining quadratic.

• Example: Factor $\sin^2 \theta \cos \theta + 6\cos \theta \sin \theta - 7\sin \theta$.

• Factor out $\sin \theta$: $\sin \theta (\sin \theta \cos \theta + 6\cos \theta - 7)$

• Let $x = \cos \theta$, factor $\sin \theta (x + 7)(x - 1)$

• Replace $x$: $\sin \theta (\cos \theta + 7)(\cos \theta - 1)$

Trigonometric Identities

Quotient Identities

• $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$

• $\cot \theta = \dfrac{\cos \theta}{\sin \theta}$

Reciprocal Identities

• $\csc \theta = \dfrac{1}{\sin \theta}$

• $\sec \theta = \dfrac{1}{\cos \theta}$

• $\cot \theta = \dfrac{1}{\tan \theta}$

Pythagorean Identities

• $\sin^2 \theta + \cos^2 \theta = 1$

• $\sin^2 \theta = 1 - \cos^2 \theta$

• $\cos^2 \theta = 1 - \sin^2 \theta$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $\tan^2 \theta = \sec^2 \theta - 1$

• $1 + \cot^2 \theta = \csc^2 \theta$

• $\cot^2 \theta = \csc^2 \theta - 1$

Strategies for Verifying Identities

• Rewrite all terms in terms of sine and cosine.

• Factor expressions where possible.

• Obtain common denominators for fractions.

• Multiply by a conjugate to simplify complex fractions.

Examples of Verifying Trigonometric Identities

Example 1: $\sec \theta = \tan \theta \cdot \csc \theta$

• Rewrite $\tan \theta$ and $\csc \theta$ in terms of sine and cosine:

• $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$, $\csc \theta = \dfrac{1}{\sin \theta}$

• Multiply: $\tan \theta \cdot \csc \theta = \dfrac{\sin \theta}{\cos \theta} \cdot \dfrac{1}{\sin \theta} = \dfrac{1}{\cos \theta} = \sec \theta$

Example 2: $\dfrac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta - \cos^2 \theta} = \sin^2 \theta + \cos^2 \theta$

• Factor numerator as difference of squares: $\sin^4 \theta - \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^2 \theta - \cos^2 \theta)$

• Cancel $\sin^2 \theta - \cos^2 \theta$ from numerator and denominator.

• Result: $\sin^2 \theta + \cos^2 \theta = 1$ (by Pythagorean Identity)

Example 3: $\cot x = \dfrac{1 - \sin x}{\cos x}$

• Rewrite $\cot x = \dfrac{\cos x}{\sin x}$

• Manipulate the right side using Pythagorean Identity: $1 - \sin x = \cos^2 x / (1 + \sin x)$ (if needed)

• Alternatively, multiply numerator and denominator by $1 + \sin x$ (conjugate) to simplify.

Example 4: $\tan \theta = \dfrac{1 - \cos 2\theta}{\sin 2\theta}$

• Use double angle identities: $\sin 2\theta = 2\sin \theta \cos \theta$, $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

• Simplify numerator and denominator to show both sides are equal.

Example 5: $\sec x = \dfrac{1 + \sin x}{\cos x}$

• Multiply numerator and denominator by $1 - \sin x$ (conjugate) to rationalize denominator if needed.

• Use Pythagorean Identity to simplify.

Example 6: $1 = \csc x \sec x (\cot x + \tan x)$

• Rewrite all terms in terms of sine and cosine:

• $\csc x = \dfrac{1}{\sin x}$, $\sec x = \dfrac{1}{\cos x}$, $\cot x = \dfrac{\cos x}{\sin x}$, $\tan x = \dfrac{\sin x}{\cos x}$

• Combine and simplify to show both sides are equal to 1.

Example 7: $\dfrac{1 - \sin \theta}{\cos \theta} = \dfrac{1 + \sin \theta}{\cos \theta}$

• Multiply numerator and denominator by the conjugate if necessary.

• Use Pythagorean Identity to simplify and show equality.

Summary Table: Key Trigonometric Identities

Type	Identity
Quotient	$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$ $\cot \theta = \dfrac{\cos \theta}{\sin \theta}$
Reciprocal	$\csc \theta = \dfrac{1}{\sin \theta}$ $\sec \theta = \dfrac{1}{\cos \theta}$ $\cot \theta = \dfrac{1}{\tan \theta}$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$

Tips for Verifying Identities

• Work on one side of the equation only; do not perform operations across the equals sign.

• Show each logical step clearly for full credit.

• There may be multiple valid approaches to verifying an identity.

Additional info: The notes emphasize the importance of logical, step-by-step reasoning and the use of algebraic techniques (factoring, common denominators, conjugates) in the context of trigonometric identities. Mastery of these skills is essential for success in precalculus and calculus.