📊 Precalculus Tactics

Master the essential precalculus techniques that will help you solve problems involving trigonometry, logarithms, exponential functions, and advanced function analysis efficiently.

🎯 Core Precalculus Strategies

Trigonometry Identities

• Pythagorean identities: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$
• Angle sum/difference: $\sin(A \pm B)$, $\cos(A \pm B)$, $\tan(A \pm B)$
• Double angle: $\sin(2x)$, $\cos(2x)$, $\tan(2x)$
• Half angle: $\sin(\frac{x}{2})$, $\cos(\frac{x}{2})$, $\tan(\frac{x}{2})$

Logarithm Properties

• Product rule: $\log_a(xy) = \log_a x + \log_a y$
• Quotient rule: $\log_a(\frac{x}{y}) = \log_a x - \log_a y$
• Power rule: $\log_a(x^n) = n \log_a x$
• Change of base: $\log_a x = \frac{\log_b x}{\log_b a}$

Exponential Functions

• Exponential properties: $a^x \cdot a^y = a^{x+y}$, $\frac{a^x}{a^y} = a^{x-y}$, $(a^x)^y = a^{xy}$
• Natural exponential: $e^x$ and its properties
• Exponential growth/decay: $A = A_0 e^{kt}$
• Compound interest: $A = P(1 + \frac{r}{n})^{nt}$

🔄 Trigonometry Mastery

Pythagorean Identities

Basic identity: $\sin^2 x + \cos^2 x = 1$ Derived identities:

• $1 + \tan^2 x = \sec^2 x$
• $1 + \cot^2 x = \csc^2 x$

When to use: Simplifying trigonometric expressions, solving equations

Example: Simplify $\sin^2 x + \cos^2 x + \tan^2 x$

Solution:

• $\sin^2 x + \cos^2 x + \tan^2 x = 1 + \tan^2 x = \sec^2 x$

Angle Sum and Difference

Sine formulas:

• $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• $\sin(A - B) = \sin A \cos B - \cos A \sin B$

Cosine formulas:

• $\cos(A + B) = \cos A \cos B - \sin A \sin B$
• $\cos(A - B) = \cos A \cos B + \sin A \sin B$

Tangent formulas:

• $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
• $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

Double Angle Formulas

Sine: $\sin(2x) = 2\sin x \cos x$ Cosine: $\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$ Tangent: $\tan(2x) = \frac{2\tan x}{1 - \tan^2 x}$

When to use: Simplifying expressions with double angles, solving equations

Half Angle Formulas

Sine: $\sin(\frac{x}{2}) = \pm\sqrt{\frac{1 - \cos x}{2}}$ Cosine: $\cos(\frac{x}{2}) = \pm\sqrt{\frac{1 + \cos x}{2}}$ Tangent: $\tan(\frac{x}{2}) = \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x}$

When to use: Simplifying expressions with half angles, solving equations

⚡ Logarithm Mastery

Basic Properties

Product rule: $\log_a(xy) = \log_a x + \log_a y$ Quotient rule: $\log_a(\frac{x}{y}) = \log_a x - \log_a y$ Power rule: $\log_a(x^n) = n \log_a x$ Change of base: $\log_a x = \frac{\log_b x}{\log_b a}$

When to use: Simplifying logarithmic expressions, solving equations

Logarithm Example

Problem: Simplify $\log_2(8x^3) - \log_2(2x)$

Solution:

• $\log_2(8x^3) - \log_2(2x) = \log_2(\frac{8x^3}{2x}) = \log_2(4x^2)$
• $= \log_2(4) + \log_2(x^2) = 2 + 2\log_2(x)$

Natural Logarithms

Definition: $\ln x = \log_e x$ Properties: Same as general logarithms Special values: $\ln 1 = 0$, $\ln e = 1$ Derivative: $\frac{d}{dx}[\ln x] = \frac{1}{x}$

Logarithmic Equations

Process:

1. Isolate logarithm: Get logarithm on one side
2. Apply exponential: Use $a^{\log_a x} = x$
3. Solve for variable: Use standard techniques
4. Check domain: Ensure argument is positive

Example: Solve $\log_2(x + 3) = 4$

Solution:

• $x + 3 = 2^4 = 16$
• $x = 16 - 3 = 13$
• Check: $\log_2(13 + 3) = \log_2(16) = 4$ ✓

🔢 Exponential Function Mastery

Basic Properties

Product rule: $a^x \cdot a^y = a^{x+y}$ Quotient rule: $\frac{a^x}{a^y} = a^{x-y}$ Power rule: $(a^x)^y = a^{xy}$ Zero rule: $a^0 = 1$ (for $a \neq 0$) Negative rule: $a^{-x} = \frac{1}{a^x}$

When to use: Simplifying exponential expressions, solving equations

Exponential Example

Problem: Simplify $\frac{2^{x+3} \cdot 4^{x-1}}{8^{x-2}}$

Solution:

• $\frac{2^{x+3} \cdot 4^{x-1}}{8^{x-2}} = \frac{2^{x+3} \cdot (2^2)^{x-1}}{(2^3)^{x-2}}$
• $= \frac{2^{x+3} \cdot 2^{2x-2}}{2^{3x-6}} = \frac{2^{3x+1}}{2^{3x-6}} = 2^{7} = 128$

Natural Exponential

Definition: $e^x$ where $e \approx 2.718$ Properties: Same as general exponentials Special values: $e^0 = 1$, $e^1 = e$ Derivative: $\frac{d}{dx}[e^x] = e^x$

Exponential Equations

Process:

1. Isolate exponential: Get exponential on one side
2. Apply logarithm: Take logarithm of both sides
3. Solve for variable: Use standard techniques
4. Check answer: Verify the solution

Example: Solve $3^{2x-1} = 27$

Solution:

• $3^{2x-1} = 3^3$, so $2x - 1 = 3$
• $2x = 4$, so $x = 2$
• Check: $3^{2(2)-1} = 3^3 = 27$ ✓

🎯 Advanced Function Techniques

Function Composition

Definition: $(f \circ g)(x) = f(g(x))$ Properties: Not commutative, associative Inverse functions: $(f \circ f^{-1})(x) = x$ Domain: Must consider domain of inner function

Function Inverses

Definition: $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$ Finding inverse: Swap $x$ and $y$, solve for $y$ Graph: Reflect across $y = x$ Domain/range: Swap domain and range

Function Transformations

Vertical shift: $f(x) + c$ (up if $c > 0$) Horizontal shift: $f(x - c)$ (right if $c > 0$) Vertical stretch: $a \cdot f(x)$ (stretch if $|a| > 1$) Horizontal stretch: $f(bx)$ (compress if $|b| > 1$) Reflection: $-f(x)$ (across $x$-axis), $f(-x)$ (across $y$-axis)

⚡ Quick Precalculus Tricks

Special Angles

Degrees: 0°, 30°, 45°, 60°, 90° Radians: 0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$ Values: Use unit circle or special triangles

Common Values

$\sin 30° = \frac{1}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$ $\sin 45° = \frac{\sqrt{2}}{2}$, $\cos 45° = \frac{\sqrt{2}}{2}$ $\sin 60° = \frac{\sqrt{3}}{2}$, $\cos 60° = \frac{1}{2}$

Logarithmic Identities

$\log_a a = 1$, $\log_a 1 = 0$ $a^{\log_a x} = x$, $\log_a(a^x) = x$ $\log_a x = \frac{\ln x}{\ln a}$ (change of base)

Exponential Identities

$e^{\ln x} = x$, $\ln(e^x) = x$ $e^{x+y} = e^x \cdot e^y$, $e^{x-y} = \frac{e^x}{e^y}$ $(e^x)^y = e^{xy}$

🎯 Problem-Specific Strategies

Trigonometry Problems

• Use identities: Apply appropriate trigonometric identities
• Use special angles: Apply known values
• Use unit circle: Visualize trigonometric values
• Use graphs: Understand trigonometric functions

Logarithm Problems

• Use properties: Apply logarithmic properties
• Use change of base: Convert to common logarithms
• Use exponentials: Convert to exponential form
• Check domain: Ensure arguments are positive

Exponential Problems

• Use properties: Apply exponential properties
• Use logarithms: Take logarithms to solve
• Use special values: Apply known exponential values
• Check answers: Verify solutions

Function Problems

• Use composition: Apply function composition
• Use inverses: Find and use inverse functions
• Use transformations: Apply function transformations
• Use graphs: Visualize function behavior

🚨 Common Precalculus Mistakes

Avoid These Errors:

• Trig identity errors: Check your trigonometric identities
• Logarithm errors: Check your logarithmic properties
• Exponential errors: Check your exponential properties
• Domain errors: Check domains of functions
• Sign errors: Be careful with signs

Red Flags:

• Answer doesn’t work: Check your calculations
• Negative under square root: Check your work
• Division by zero: Check your work
• Impossible result: Check your work

📊 Quick Reference

Trigonometry Checklist:

• Identify the function: What trigonometric function is involved?
• Apply identities: Use appropriate trigonometric identities
• Use special angles: Apply known values
• Simplify: Reduce to simplest form
• Check answer: Verify the result

Logarithm Checklist:

• Identify the base: What is the logarithm base?
• Apply properties: Use logarithmic properties
• Use change of base: Convert if needed
• Solve for variable: Use standard techniques
• Check domain: Ensure argument is positive

Exponential Checklist:

• Identify the base: What is the exponential base?
• Apply properties: Use exponential properties
• Use logarithms: Take logarithms if needed
• Solve for variable: Use standard techniques
• Check answer: Verify the solution

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