Equivalent representations of trigonometric functions — AP Precalculus
1. Pythagorean Identities and Simplification ★★☆☆☆ ⏱ 3 min
Pythagorean identities are the most commonly used tools for rewriting trigonometric expressions, derived directly from the unit circle equation. For any angle $\theta$ corresponding to a point $(x,y)$ on the unit circle, $x^2 + y^2 = 1$, so substituting $x = \cos\theta$ and $y = \sin\theta$ gives the core identity:
cos^2 \theta + \sin^2 \theta = 1
Dividing both sides by $\cos^2 \theta$ (for $\cos\theta \neq 0$) gives the tangent-secant form: $1 + \tan^2 \theta = \sec^2 \theta$, and dividing by $\sin^2 \theta$ (for $\sin\theta \neq 0$) gives the cotangent-cosecant form: $1 + \cot^2 \theta = \csc^2 \theta$. These identities are used to replace quadratic terms, cancel common factors, and simplify to a single basic trigonometric term.
Exam tip: Always confirm that your simplified expression has the same domain as the original. If the original excludes input values allowed in the simplified form, explicitly note the excluded values for full credit on FRQs.
2. Double-Angle and Power-Reduction Identities ★★★☆☆ ⏱ 3 min
Double-angle and power-reduction identities convert between trigonometric functions of $2\theta$ and functions of $\theta$, and convert quadratic powers of sine/cosine into linear functions of double angles. These are derived from sum identities for sine and cosine. Setting $A = B = \theta$ in the sine sum identity gives:
\sin 2\theta = 2\sin \theta \cos \theta
For cosine, setting $A = B = \theta$ gives three equivalent forms:
\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta
Rearranging the last two forms gives the power-reduction identities:
\sin^2 \theta = \frac{1 - \cos 2\theta}{2}, \quad \cos^2 \theta = \frac{1 + \cos 2\theta}{2}
Exam tip: Memorize the sign pattern for power-reduction: 'sine is minus, cosine is plus' to avoid sign errors when simplifying.
3. Amplitude-Phase Form of Combined Sinusoids ★★★★☆ ⏱ 3 min
Any linear combination of sine and cosine with the same period can be rewritten as a single equivalent sinusoidal function, which is much easier to analyze for amplitude, maximum/minimum values, and phase shift. This equivalent representation is called amplitude-phase form, and follows the formula:
a\cos Bx + b\sin Bx = C\cos(Bx - \phi) + D
Where $C = \sqrt{a^2 + b^2}$ (the amplitude, positive by convention), $\cos \phi = \frac{a}{C}$, $\sin \phi = \frac{b}{C}$, and $D$ is any vertical shift.
Exam tip: Always confirm the quadrant of $\phi$ using the signs of $\cos \phi$ and $\sin \phi$, do not just use the output of arctangent directly, as arctangent only gives values between $-\pi/2$ and $\pi/2$ and will miss angles in other quadrants.
4. Product-to-Sum and Sum-to-Product Identities ★★★★☆ ⏱ 3 min
These identities let you rewrite products of sines/cosines as sums, or sums of sines/cosines as products, which is useful for factoring trig expressions and solving equations with multiple trigonometric terms. The most commonly used sum-to-product identity for sine is:
\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)
This identity converts a sum of two sines into a product, which can then be set equal to zero and solved using the zero product property.
Exam tip: Only use sum-to-product for two trigonometric functions of the same type (two sines or two cosines) with the same period; for mixed sums of sine and cosine, use amplitude-phase form instead.
5. AP-Style Worked Practice ★★★☆☆ ⏱ 5 min
Common Pitfalls
Why: The original expression excludes all $\theta$ where $\cos \theta = 0$, while $\cos \theta$ is defined for all $\theta$, so they are not fully equivalent without noting exclusions
Why: Students mix up the sign pattern for sine and cosine power-reduction
Why: Students remember the factor of 2 but drop the cosine term when simplifying quickly
Why: Students misplace the sign of $\phi$ inside the argument, adding instead of subtracting
Why: Students forget that sum-to-product only works for two trigonometric functions of the same type with matching frequencies