PC40S
Unit 1:Unit Circle
- Relate degrees and radians for angles in standard position
- Determine exact sin, cos, and tan values using the unit circle
- Use reference angles to find trig values in all quadrants
- Connect special triangles (30°–60°–90°, 45°–45°–90°) to coordinates on the circle
- Apply periodic properties of trig functions
- Solve problems involving coterminal and related angles
Key Formulas & Concepts:
- $\sin^{2}(\theta) + \cos^{2}(\theta) = 1$
- $\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}$
- Radian–degree conversion: $\pi\ \text{rad} = 180^\circ$
- Arc length: $s = r\theta$
Unit 2:Transformations
- Describe effects of horizontal/vertical shifts, stretches, and reflections
- Use mapping rule and master form y = a f(b(x − h)) + k
- Sketch and analyze absolute value, reciprocal, and inverse functions
- Identify non-permissible values and transformation order
- Graph sinusoidal (sine/cosine) functions with amplitude, period, phase shift
- Model real-world transformations
Key Formulas & Concepts:
- Period of trig function: $P = \dfrac{2\pi}{|b|}$
- Mapping rule: $(x, y) \to \big(\tfrac{x - h}{b},\ a\,y + k\big)$
- Reciprocal: $y = \dfrac{1}{f(x)}$
- Inverse reflection: across line $y = x$
Unit 3:Radicals &
Rationals
- Simplify and combine radical/rational expressions
- Convert between radical and rational exponent forms
- Rationalize denominators properly
- Solve equations with radicals or rational exponents
- Identify domain restrictions
- Apply radicals in geometric and real-world contexts
Key Formulas & Concepts:
- $a^{m/n} = \sqrt[n]{a^{m}}$
- Product rule: $\sqrt{a}\cdot\sqrt{b} = \sqrt{a\cdot b}$
- Quotient rule: $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$
- Rationalization: multiply by conjugate if denominator has two terms
Unit 4:Exponents &
Logarithms
- Simplify using exponent laws
- Convert between exponential and logarithmic forms
- Apply log laws: product, quotient, and power
- Solve exponential/logarithmic equations algebraically
- Model exponential growth/decay situations
- Analyze exp/log graphs (domain, range, asymptotes)
Key Formulas & Concepts:
- $b^{x} = y \iff \log_{b}(y) = x$
- $\log_{b}(MN) = \log_{b}M + \log_{b}N$
- $\log_{b}\!\left(\dfrac{M}{N}\right) = \log_{b}M - \log_{b}N$
- $\log_{b}(M^{k}) = k\,\log_{b}M$
- Exponential growth: $y = a\,b^{t}$
Unit 5:Polynomials
- Identify degree, leading coefficient, and end behavior
- Factor using common, grouping, synthetic division methods
- Apply Factor and Remainder Theorems
- Sketch using zeros, multiplicities, and intercepts
- Solve polynomial equations and inequalities
Key Formulas & Concepts:
- Remainder Theorem: $f(c) =$ remainder when divided by $(x - c)$
- Factor Theorem: $(x - c)$ is a factor if $f(c) = 0$
- Sum of zeros = $-\dfrac{\text{coeff of }x^{n-1}}{\text{coeff of }x^{n}}$
- Polynomial form: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$
Unit 6:Permutations,
Combinations & Binomial Theorem
- Differentiate between permutations and combinations
- Compute arrangements and selections with or without repetition
- Apply factorial notation correctly
- Expand binomials using the Binomial Theorem
- Identify specific terms or coefficients in an expansion
- Apply counting principles to probability problems
Key Formulas & Concepts:
- $P(n,r) = \dfrac{n!}{(n-r)!}$
- $C(n,r) = \dfrac{n!}{r!(n-r)!}$
- Binomial Theorem: $(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^{r}$
- Specific term: $T_{r+1} = \binom{n}{r} a^{n-r} b^{r}$
Unit 7:Trigonometry
Identities
- Apply angle addition and subtraction identities
- Find exact values of special angles
- Simplify or prove trig identities step-by-step
- Use double- and half-angle formulas
- Solve trig equations in specified intervals
- Verify identities by converting to sin and cos
Key Formulas & Concepts:
- $\sin(A\pm B)=\sin A\cos B \pm \cos A\sin B$
- $\cos(A\pm B)=\cos A\cos B \mp \sin A\sin B$
- $\sin(2A)=2\sin A\cos A$
- $\cos(2A)=\cos^{2}A-\sin^{2}A=1-2\sin^{2}A=2\cos^{2}A-1$
- $\tan(2A)=\dfrac{2\tan A}{1-\tan^{2}A}$
Unit 8:Functions
- Identify domain, range, and notation of a function
- Determine inverse and composite functions
- Interpret intercepts, intervals, and asymptotes
- Work with piecewise-defined and restricted functions
- Model real data using linear, quadratic, and exponential relations
- Analyze transformations within function families
Key Formulas & Concepts:
- Function notation: $f(x)$ means “value of $f$ at $x$”
- Inverse: $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$
- Composition: $(f\circ g)(x) = f(g(x))$
- Average rate of change: $\dfrac{f(b)-f(a)}{b-a}$