Pre-Calculus · Self-Study Edition

Master the Core
Concepts

20 carefully selected problems covering Vectors, Matrices, Polar Graphs, and Trigonometry. Each question links to the big idea.

Vectors Matrices Polar Graphs Trig Basics Unit Circle Transformations
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01 Trigonometry Basics
⚡ Quick Memory Key
SOH · CAH · TOA
Sin = Opposite / Hypotenuse  |  Cos = Adjacent / Hypotenuse  |  Tan = Opposite / Adjacent
Q 01 Trig Ratios ★☆☆
In a right triangle, the side opposite to angle \(\theta\) is 3, and the hypotenuse is 5. What is \(\sin\theta\)?
💡 Recall: sin = Opposite / Hypotenuse
Q 02 Unit Circle ★★☆
Classic Trap: What is the exact value of \(\cos\!\left(\dfrac{2\pi}{3}\right)\)?
📐 Example · Reference Angle Method \(\dfrac{2\pi}{3}\) is in Quadrant II. Reference angle = \(\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}\).
In Q2, cosine is negative. So \(\cos\!\left(\dfrac{2\pi}{3}\right) = -\cos\!\left(\dfrac{\pi}{3}\right)\).
Q 03 Pythagorean Identity ★★☆
If \(\sin\theta = \dfrac{1}{3}\) and \(\theta\) is in Quadrant II, find \(\cos\theta\).
💡 KEY IDENTITY: \(\sin^2\theta + \cos^2\theta = 1\). In Q2, cos is negative!
Q 04 Trig Graphs ★★☆
The function \(y = 3\sin(2x - \pi)\) has amplitude and period equal to:
📐 Example · Reading Parameters For \(y = A\sin(Bx - C)\):
  • Amplitude = \(|A|\)   • Period = \(\dfrac{2\pi}{|B|}\)   • Phase Shift = \(\dfrac{C}{B}\)
Q 05 Double Angle ★★★
Tricky! Simplify: \(\dfrac{\sin 2\theta}{2\sin\theta}\)
💡 DOUBLE ANGLE: \(\sin 2\theta = 2\sin\theta\cos\theta\)
02 Vectors
⚡ Quick Memory Key
DOT = PROJECTION · CROSS = PERPENDICULAR
Dot Product: \(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta\) → measures how much they point together
Magnitude: \(|\vec{v}| = \sqrt{v_x^2 + v_y^2}\) · Zero dot product = vectors are perpendicular
Q 06 Vector Magnitude ★☆☆
Find the magnitude of vector \(\vec{v} = \langle -3,\, 4 \rangle\).
💡 \(|\vec{v}| = \sqrt{(-3)^2 + 4^2}\). Don't forget to square the negative!
Q 07 Dot Product ★★☆
If \(\vec{a} = \langle 2, -1 \rangle\) and \(\vec{b} = \langle 3, 6 \rangle\), what is \(\vec{a} \cdot \vec{b}\)?
📐 Example · Dot Product Formula \(\vec{a}\cdot\vec{b} = a_x \cdot b_x + a_y \cdot b_y\)
e.g., \(\langle 1,2\rangle \cdot \langle 3,4\rangle = 1\cdot3 + 2\cdot4 = 11\)
Q 08 Orthogonal Vectors ★★☆
Common Mistake: Which pair of vectors is perpendicular (orthogonal)?
💡 ORTHOGONAL KEY: \(\vec{a}\perp\vec{b} \Leftrightarrow \vec{a}\cdot\vec{b}=0\)
Q 09 Vector Direction Angle ★★☆
Find the direction angle \(\theta\) of \(\vec{v} = \langle 1, \sqrt{3} \rangle\) (measured from positive x-axis).
💡 \(\tan\theta = \dfrac{v_y}{v_x} = \dfrac{\sqrt{3}}{1} = \sqrt{3}\). Which angle gives \(\tan = \sqrt{3}\)?
Q 10 Vector Components ★★★
Link to Trig: A force of 10 N acts at an angle of 60° above the horizontal. What are its horizontal and vertical components?
📐 Example · Component Form \(\vec{F} = \langle F\cos\theta,\; F\sin\theta \rangle\)
For magnitude 8 at 30°: \(\langle 8\cos30°,\; 8\sin30°\rangle = \langle 4\sqrt{3},\; 4\rangle\)
03 Matrices
⚡ Quick Memory Key
ROW × COL · ORDER MATTERS
Matrix multiply: (m×n)(n×p) = m×p. Take ROW of first × COLUMN of second, sum up.
\(AB \neq BA\) in general!  |  det\(\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc\)
Q 11 Matrix Multiplication ★★☆
Compute \(AB\) where \(A = \begin{pmatrix}1 & 2\\0 & 3\end{pmatrix}\), \(B = \begin{pmatrix}4 & 1\\2 & 0\end{pmatrix}\).
💡 Entry (1,1) = row1 of A · col1 of B = 1·4 + 2·2 = 8. Finish the rest!
Q 12 Determinant ★☆☆
Find \(\det\!\begin{pmatrix}5 & -2\\3 & 4\end{pmatrix}\).
📐 Example · 2×2 Determinant \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc\)
Common mistake: people compute \(ab - cd\) instead! Always: main diagonal minus anti-diagonal.
Q 13 Inverse Matrix ★★★
Trap! When does a 2×2 matrix have no inverse?
💡 INVERSE EXISTS ↔ det ≠ 0. If det = 0, the matrix is "singular" (no inverse).
Q 14 System via Matrix ★★☆
The system \(\begin{cases}2x + y = 5\\x - y = 1\end{cases}\) can be written as \(A\vec{x} = \vec{b}\). What is matrix \(A\)?
04 Polar Graphs
⚡ Quick Memory Key
POLAR ↔ CARTESIAN BRIDGE
\(x = r\cos\theta\), \(y = r\sin\theta\), \(r^2 = x^2 + y^2\)
Circle centered at origin: \(r = a\)  |  Rose curves: \(r = a\cos(n\theta)\) or \(r = a\sin(n\theta)\)
If \(n\) = even → \(2n\) petals; \(n\) = odd → \(n\) petals
Q 15 Polar → Cartesian ★★☆
Convert the polar point \(\left(4,\, \dfrac{\pi}{6}\right)\) to Cartesian coordinates.
💡 \(x = r\cos\theta\), \(y = r\sin\theta\). At \(\dfrac{\pi}{6}\): \(\cos = \dfrac{\sqrt{3}}{2}\), \(\sin = \dfrac{1}{2}\)
Q 16 Rose Curve Petals ★★☆
Classic Confusion: How many petals does \(r = 3\cos(4\theta)\) have?
📐 Example · Petal Count Rule \(r = a\cos(n\theta)\) or \(r = a\sin(n\theta)\):
  • \(n\) is odd → \(n\) petals
  • \(n\) is even → \(2n\) petals
Example: \(r = \cos(3\theta)\) → 3 petals; \(r = \cos(2\theta)\) → 4 petals.
Q 17 Limaçon ★★★
The polar curve \(r = 2 + 4\cos\theta\) is a limaçon. It has an inner loop because:
💡 For \(r = a + b\cos\theta\): inner loop exists when \(|b| > |a|\)
Q 18 Cartesian → Polar ★★☆
Convert \(x^2 + y^2 = 9\) into polar form.
💡 KEY IDENTITY: \(x^2 + y^2 = r^2\). Substitute directly!
05 Mixed Challenge
Q 19 Trig + Vectors ★★★
Connecting Topics: Vectors \(\vec{u} = \langle 1, 0 \rangle\) and \(\vec{v} = \langle \cos\theta, \sin\theta \rangle\). What is \(\vec{u}\cdot\vec{v}\)?
📐 Big Idea · Why Dot Product Equals cos θ \(\vec{u}\cdot\vec{v} = 1\cdot\cos\theta + 0\cdot\sin\theta = \cos\theta\)
This is why \(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta\). The unit vector on x-axis reveals the cosine directly!
Q 20 Matrix + Trig ★★★
Final Boss: The rotation matrix \(R = \begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\) rotates the vector \(\begin{pmatrix}1\\0\end{pmatrix}\) by angle \(\theta\). What is the result?
💡 Multiply: \(R\cdot\begin{pmatrix}1\\0\end{pmatrix}\). This reveals the connection between rotation, matrices, and the unit circle!
FINAL RESULT
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