Source: SYMKC Series, Calculus Prep (Texas A&M University)
Tags: pre-calculus, precalc, even odd functions, trig identities, unit circle, composite functions, law of cosines, law of sines, trig values, parametric equations, polar equations, vectors, double angle, power reduction, reciprocal identities, sequences, series, matrices, binomial theorem, conics, logarithms, exponents, factoring, completing the square
Pre-Cal ties together algebra, geometry, and trigonometry into the toolkit you need for Calculus. This covers even/odd functions, the full unit circle, all the major trig identities (double angle, power reduction, Pythagorean, reciprocal), vectors, parametric and polar equations, conic sections, logarithm rules, exponent rules, and series. These are not optional to memorise; Calculus assumes instant recall of all of them.
Even function
A function where f(–x) = f(x) for all x. Its graph is symmetric about the y-axis. Examples: x², cos x.
Odd function
A function where f(–x) = –f(x) for all x. Its graph is symmetric about the origin. Examples: x³, sin x.
Unit circle
The circle x² + y² = 1 centred at the origin. Each point (cos θ, sin θ) on the circle corresponds to an angle θ measured from the positive x-axis.
Law of cosines
Generalisation of the Pythagorean theorem for any triangle: c² = a² + b² – 2ab cos C.
Law of sines
For any triangle: a/sin A = b/sin B = c/sin C.
Parametric equations
A pair of equations x(t) and y(t) that express the coordinates of a point as functions of a parameter t.
Polar coordinates
A system where each point is described by a distance r from the origin and an angle θ from the positive x-axis.
If f(x) is even, then f(–x) = f(x). The graph is symmetric about the y-axis.
If f(x) is odd, then f(–x) = –f(x). The graph is symmetric about the origin.
Given f(x) and g(x):
f(g(x)) means "apply g first, then f"
To find f(g(x)), substitute the entire expression for g(x) wherever x appears in f
Law of cosines:
c² = a² + b² – 2ab cos C
Use when you have SAS (two sides and the included angle) or SSS (all three sides).
Law of sines:
a / sin A = b / sin B = c / sin C
Use when you have AAS, ASA, or the ambiguous case SSA.
θ | sin θ | cos θ | tan θ |
|---|---|---|---|
0° | 0 | 1 | 0 |
π/6 (30°) | 1/2 | √3/2 | √3/3 |
π/4 (45°) | √2/2 | √2/2 | 1 |
π/3 (60°) | √3/2 | 1/2 | √3 |
π/2 (90°) | 1 | 0 | undefined |
π (180°) | 0 | –1 | 0 |
You need to know both the standard trig values and their inverse trig equivalents (e.g., tan⁻¹(1) = π/4).
The unit circle maps every angle θ to coordinates (cos θ, sin θ). Key points going anticlockwise from 0:
0° (0 rad): (1, 0)
30° (π/6): (√3/2, 1/2)
45° (π/4): (√2/2, √2/2)
60° (π/3): (1/2, √3/2)
90° (π/2): (0, 1)
120° (2π/3): (–1/2, √3/2)
135° (3π/4): (–√2/2, √2/2)
150° (5π/6): (–√3/2, 1/2)
180° (π): (–1, 0)
210° (7π/6): (–√3/2, –1/2)
225° (5π/4): (–√2/2, –√2/2)
240° (4π/3): (–1/2, –√3/2)
270° (3π/2): (0, –1)
300° (5π/3): (1/2, –√3/2)
315° (7π/4): (√2/2, –√2/2)
330° (11π/6): (√3/2, –1/2)
360° (2π): (1, 0)
sin(2x) = 2 sin x cos x
cos(2x) = cos²x – sin²x = 2cos²x – 1 = 1 – 2sin²x
All three forms of cos(2x) are useful in different contexts. The last two are derived from the Pythagorean identity.
sin²x = (1 – cos 2x) / 2
cos²x = (1 + cos 2x) / 2
These are rearrangements of the double-angle formulas for cos(2x) and are essential for integration in Calculus.
sin²x + cos²x = 1
1 + tan²x = sec²x
cot²x + 1 = csc²x
sec x = 1 / cos x, so cos x · sec x = 1
csc x = 1 / sin x, so sin x · csc x = 1
sin(–x) = –sin x (odd)
cos(–x) = cos x (even)
tan(–x) = –tan x (odd)
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β
Note the sign flip in the cosine formula: the ± on the left becomes ∓ on the right.
A vector u = ⟨a, b⟩ has components a (horizontal) and b (vertical)
Magnitude: ‖u‖ = √(a² + b²)
To graph, draw from the origin to the point (a, b)
A curve is defined by x(t) and y(t) simultaneously. To eliminate the parameter, solve one equation for t and substitute into the other, or use a trig identity if x and y involve sin t and cos t.
Standard conversion formulas between polar (r, θ) and rectangular (x, y):
x = r cos θ
y = r sin θ
tan θ = y / x
x² + y² = r²
Sigma notation: Σ represents summation from a lower index to an upper index
Know arithmetic and geometric series formulas (see Algebra 2 notes)
For 2×2 matrices A and B:
Subtraction: A – B is computed element by element
Multiplication: AB follows the row-by-column rule: each entry in the result is the dot product of the corresponding row of A and column of B
Matrix multiplication is not commutative: AB ≠ BA in general
(u + v)² = u² + 2uv + v²
(u – v)² = u² – 2uv + v²
(u + v)³ = u³ + 3u²v + 3uv² + v³
(u – v)³ = u³ – 3u²v + 3uv² – v³
(a + b)ⁿ = Σ from k=0 to n of C(n, k) · aⁿ⁻ᵏ · bᵏ
a⁰ = 1 (a ≠ 0)
(ab)ⁿ = aⁿbⁿ
aˣ · aʸ = aˣ⁺ʸ
√a = a^(1/2)
aˣ / aʸ = aˣ⁻ʸ
ⁿ√a = a^(1/n)
(a/b)ˣ = aˣ / bˣ
ⁿ√(aᵐ) = a^(m/n)
a⁻ˣ = 1 / aˣ
ⁿ√(ab) = ⁿ√a · ⁿ√b
(aˣ)ʸ = aˣʸ
ⁿ√(a/b) = ⁿ√a / ⁿ√b
Converting between forms:
log to exponential: if log_b(y) = x, then bˣ = y
ln y = x means eˣ = y
log y = x means 10ˣ = y
Core rules:
log_a(a) = 1
log_a(1) = 0
log_a(aⁿ) = n
log_b(mn) = log_b(m) + log_b(n)
log_b(m/n) = log_b(m) – log_b(n)
log_b(mⁿ) = n · log_b(m)
Change of base:
log_c(a) = log_b(a) / log_b(c)
Compound interest: A = P(1 + r/n)^(nt)
Continuously compounded: A = Pe^(rt)
Exponential growth/decay: y = a · bᵗ (or y = ae^(kt), where k > 0 for growth, k < 0 for decay)
To find the x-intercept of any function: set y = 0 and solve for x
To find the y-intercept of any function: set x = 0 and solve for y
Circle: (x – h)² + (y – k)² = r²
Parabola:
Vertical: (x – h)² = 4p(y – k)
Horizontal: (y – k)² = 4p(x – h)
Ellipse:
Vertical major axis: (x – h)²/b² + (y – k)²/a² = 1, where a > b
Horizontal major axis: (x – h)²/a² + (y – k)²/b² = 1, where a > b
Hyperbola:
Vertical: (y – k)²/a² – (x – h)²/b² = 1
Horizontal: (x – h)²/a² – (y – k)²/b² = 1
To rewrite ax² + bx + c in vertex form: factor out a from the x-terms, take half the coefficient of x, square it, add and subtract inside the parentheses, then simplify.
Difference of squares: a² – b² = (a + b)(a – b)
Difference of cubes: a³ – b³ = (a – b)(a² + ab + b²)
Sum of cubes: a³ + b³ = (a + b)(a² – ab + b²)
Perfect square trinomial: a² – 2ab + b² = (a – b)²; a² + 2ab + b² = (a + b)²
Grouping: ac + ad + bc + bd = (a + b)(c + d)
Used to break a rational expression into simpler fractions. The method depends on the factors in the denominator (linear, repeated linear, irreducible quadratic).
A shortcut for dividing a polynomial by a linear factor (x – c). Write only the coefficients and carry out the algorithm.
√(x²) = |x|
e ≈ 2.718
π ≈ 3.14159
x(t) = v₀ cos(θ) · t
y(t) = v₀ sin(θ) · t – ½gt²
where v₀ is initial velocity, θ is launch angle, and g is gravitational acceleration (9.8 m/s² or 32 ft/s²).
⚠️ The unit circle is not optional. You will use it in virtually every calculus problem involving trig. Memorise it.
⚠️ There are three equivalent forms of cos(2x). Choosing the right one can simplify a problem dramatically.
⚠️ Power reduction identities are the basis for integrating sin²x and cos²x in Calculus.
⚠️ In the cosine sum/difference formula, the sign flips: cos(α – β) uses a + sign between the terms. Students mix this up with sine's formula.
⚠️ log_b(m + n) ≠ log_b(m) + log_b(n). The product rule applies to log_b(mn), not to sums inside the log.
⚠️ For conics, know which variable is squared in which direction. (x – h)² = 4p(y – k) opens vertically; (y – k)² = 4p(x – h) opens horizontally.
Q: Is f(x) = x³ – x even, odd, or neither?
A: f(–x) = –x³ + x = –(x³ – x) = –f(x). It is odd.
Q: What is cos(2x) in terms of cos x only?
A: cos(2x) = 2cos²x – 1.
Q: State sin(π/6) and cos(π/6).
A: sin(π/6) = 1/2, cos(π/6) = √3/2.
Q: Simplify log₃(81).
A: 81 = 3⁴, so log₃(81) = 4.
Q: Factor a³ + 8.
A: a³ + 2³ = (a + 2)(a² – 2a + 4).
Q: Convert the polar point (4, π/3) to rectangular coordinates.
A: x = 4 cos(π/3) = 4(1/2) = 2; y = 4 sin(π/3) = 4(√3/2) = 2√3. The point is (2, 2√3).
pre-calculus review, SYMKC, even function, odd function, unit circle, trig values, double angle identity, power reduction, Pythagorean identity, reciprocal identity, sum and difference formulas, law of cosines, law of sines, vectors, magnitude, parametric equations, polar coordinates, polar to rectangular, conics, circle, ellipse, hyperbola, parabola, completing the square, factoring, difference of cubes, sum of cubes, logarithm rules, change of base, exponent rules, compound interest, exponential growth, exponential decay, binomial theorem, matrices, matrix multiplication, synthetic division, partial fractions, projectile motion