Fundamental Trigonometric Identities

Why This Matters

Trigonometric identities aren't just formulas to memorize—they're the algebraic backbone of everything you'll do with trig functions. On the AP Pre-Calculus exam, you're being tested on your ability to recognize equivalent forms, simplify complex expressions, and verify relationships between functions. These identities connect directly to Unit 3's focus on periodic phenomena and will reappear constantly when you work with polar functions, solve equations, and analyze graphs.

Think of identities as a toolkit for transformation. Just like you learned to convert between factored and standard forms of polynomials in Unit 1, trig identities let you rewrite expressions in whatever form is most useful for a given problem. The key categories you need to master are reciprocal relationships, Pythagorean connections, symmetry properties, and angle manipulation formulas. Don't just memorize these—understand which identity solves which type of problem and why each one works.

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Foundational Definitions: Reciprocal and Quotient Identities

These identities establish the basic relationships between the six trigonometric functions. Every trig function can be expressed in terms of sine and cosine, which is why these two functions are considered fundamental.

Reciprocal Identities

• Reciprocal pairs—$\sin\theta = \frac{1}{\csc\theta}$, $\cos\theta = \frac{1}{\sec\theta}$, $\tan\theta = \frac{1}{\cot\theta}$
• Domain restrictions arise when denominators equal zero; secant is undefined where cosine equals zero
• Simplification strategy: replace unfamiliar functions (sec, csc, cot) with their reciprocals to work entirely in sine and cosine

Quotient Identities

• Tangent as a ratio—$\tan\theta = \frac{\sin\theta}{\cos\theta}$, explaining why tangent has vertical asymptotes where $\cos\theta = 0$
• Cotangent as the flip—$\cot\theta = \frac{\cos\theta}{\sin\theta}$, with asymptotes where $\sin\theta = 0$
• Conversion power: these identities let you rewrite any expression using only sine and cosine, the universal language of trig simplification

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Pythagorean Relationships: The Unit Circle Connection

These identities come directly from the Pythagorean theorem applied to the unit circle. The point $(\cos\theta, \sin\theta)$ lies on a circle of radius 1, so the sum of squares equals 1.

Primary Pythagorean Identity

• The master identity—$\sin^2\theta + \cos^2\theta = 1$, the foundation from which all other Pythagorean identities derive
• Rearranged forms: $\sin^2\theta = 1 - \cos^2\theta$ and $\cos^2\theta = 1 - \sin^2\theta$ are equally important for substitution
• Geometric meaning: represents the equation of the unit circle, connecting algebra to the circular definition of trig functions

Derived Pythagorean Identities

• Secant-tangent form—$1 + \tan^2\theta = \sec^2\theta$, obtained by dividing the primary identity by $\cos^2\theta$
• Cosecant-cotangent form—$1 + \cot^2\theta = \csc^2\theta$, obtained by dividing by $\sin^2\theta$
• Recognition tip: if you see $\tan^2$ or $\sec^2$ in an expression, think Pythagorean substitution

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Symmetry Properties: Even-Odd Identities

These identities describe how trig functions behave when you negate the input. They reflect the geometric symmetry of the unit circle across the x-axis and y-axis.

Even and Odd Functions

• Cosine is even—$\cos(-\theta) = \cos\theta)$, meaning the graph is symmetric about the y-axis
• Sine and tangent are odd—$\sin(-\theta) = -\sin\theta$ and $\tan(-\theta) = -\tan\theta$, giving origin symmetry
• Reciprocal inheritance: secant is even (like cosine), while cosecant and cotangent are odd (like sine and tangent)

Cofunction Identities

• Complementary relationship—$\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$ and $\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$
• Extended pairs: $\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$, $\sec\left(\frac{\pi}{2} - \theta\right) = \csc\theta$
• Why "cofunction": the "co" in cosine, cotangent, and cosecant means complement—these functions equal their partners at complementary angles

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Angle Combination Formulas: Sum, Difference, and Multiples

These identities let you break down or build up angles, converting between single-angle and multi-angle expressions. They're essential for evaluating trig functions at non-standard angles.

Sum and Difference Identities

• Sine of a sum/difference—$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$, with matching signs on both sides
• Cosine of a sum/difference—$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$, with opposite signs (note the $\mp$)
• Memory trick: sine keeps the same sign pattern ($\pm$ stays $\pm$), cosine flips it ($\pm$ becomes $\mp$)

Double Angle Identities

• Sine double angle—$\sin(2\theta) = 2\sin\theta\cos\theta$, useful for converting products to single functions
• Cosine double angle has three forms—$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
• Strategic choice: pick the cosine form based on what's in your expression; the $1 - 2\sin^2\theta$ form is ideal when you only have sine

Half Angle Identities

• Sine half angle—$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$, with sign determined by the quadrant of $\frac{\theta}{2}$
• Cosine half angle—$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$, also requiring quadrant analysis
• Derivation connection: these come from solving the double angle cosine formulas for $\sin\theta$ and $\cos\theta$, then replacing $\theta$ with $\frac{\theta}{2}$

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Product and Sum Conversions

These identities transform between products and sums of trig functions. They're particularly useful for simplifying expressions and solving equations where one form is easier to work with than the other.

Product-to-Sum Identities

• Sine times sine—$\sin a \sin b = \frac{1}{2}[\cos(a-b) - \cos(a+b)]$, converting a product into a difference of cosines
• Cosine times cosine—$\cos a \cos b = \frac{1}{2}[\cos(a-b) + \cos(a+b)]$, converting to a sum of cosines
• Mixed product—$\sin a \cos b = \frac{1}{2}[\sin(a+b) + \sin(a-b)]$, useful when sine and cosine are multiplied

Sum-to-Product Identities

• Sum of sines—$\sin a + \sin b = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$, factoring a sum into a product
• Sum of cosines—$\cos a + \cos b = 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$
• Equation solving: when you need to find where $\sin a + \sin b = 0$, converting to a product lets you set each factor to zero separately

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Quick Reference Table

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Self-Check Questions

1. Which two Pythagorean identities would you use to simplify an expression containing both $\tan^2\theta$ and $\sec^2\theta$? How are they related?

2. If you're given $\cos(75°)$ and need to find its exact value, which identity would you apply, and how would you break down the angle?

3. Compare and contrast the even-odd identities with the cofunction identities: what type of angle transformation does each handle, and how do their applications differ?

4. An FRQ asks you to verify that $\frac{\sin(2\theta)}{1 + \cos(2\theta)} = \tan\theta$. Which identities would you need, and in what order would you apply them?

5. When would you choose the sum-to-product identity over leaving an expression as $\sin a + \sin b$? Give a specific problem type where this conversion is advantageous.