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Math and Economics

A secondary curriculum sequence covering advanced mathematical modeling alongside foundational economic systems and financial literacy concepts.

JTJasia Tahir

Sinusoidal Equations

A comprehensive trigonometry lesson covering strategies for solving sinusoidal equations, from fundamental unit circle ratios to multi-frequency transformations, phase shifts, and real-world periodic modeling.

Stock Market Basics

An introductory financial literacy lesson examining how businesses raise capital through shares, how investors build wealth, and how global exchanges facilitate trade.

Math and Economics

A secondary curriculum sequence covering advanced mathematical modeling alongside foundational economic systems and financial literacy concepts.

JTJasia Tahir

Wave Hunters Worksheet

Trigonometry • Solving Sinusoidal Equations • Exact Solutions

Name:

Date: Period:

Master Strategy: 4 Steps to Solve Any Sinusoidal Equation

Reference Guide

1. Isolate the Trig Ratio Rearrange algebraically so $\sin(u) = C$ or $\cos(u) = C$ is isolated on one side.

2. Find Reference & Quadrants Determine reference angle $\theta_{\text{ref}}$. Use the sign of $C$ to pick valid quadrants (ASTC).

3. General Solutions for $u$ Set inside argument $u = \theta_1 + 2\pi k$ and $u = \theta_2 + 2\pi k$ for all integers $k$.

4. Solve for $x$ & Restrict Substitute $u = B(x - C)$, solve for $x$, and generate values within the required interval.

Sign Reminder: Sine is positive in Q I & II; Cosine is positive in Q I & IV. Periodicity: $\sin(\theta)$ & $\cos(\theta)$ repeat every $2\pi$ radians.

Worked Example: Find all solutions for $2\sin\left(2x\right) + \sqrt{3} = 0$ on $[0, 2\pi)$ Follow this blueprint

Step 1 & 2: Isolate & Quadrants

$\sin(2x) = -\frac{\sqrt{3}}{2}$. Sine is negative in Q III & Q IV.

Ref. angle: $\frac{\pi}{3}$ → $u = \frac{4\pi}{3}$ or $u = \frac{5\pi}{3}$.

Step 3: General Argument

$2x = \frac{4\pi}{3} + 2\pi k \implies x = \frac{2\pi}{3} + \pi k$

$2x = \frac{5\pi}{3} + 2\pi k \implies x = \frac{5\pi}{6} + \pi k$

Step 4: Values in $[0, 2\pi)$

For $k = 0, 1$:

$x = \frac{2\pi}{3}, \frac{5\pi}{6}, \frac{5\pi}{3}, \frac{11\pi}{6}$

Part 1

Fundamental Equations — Find all exact solutions in the interval $[0, 2\pi)$

1. $2\cos(x) - \sqrt{2} = 0$ Show work

Exact Solutions:

2. $\sqrt{3}\tan(x) + 1 = 0$ Show work

Exact Solutions:

Part 2

Multiple-Angle Equations — Find all exact solutions in the interval $[0, 2\pi)$

3. $2\sin(3x) - 1 = 0$ Expected: 6 solutions

Exact Solutions:

4. $2\cos(2x) + \sqrt{3} = 0$ Expected: 4 solutions

Exact Solutions:

Trigonometry Unit: Oscillations & Periodic Modeling Page 1 of 2

Part 3 & 4

Advanced Waves & Physical Applications

Student Name:

Part 3

Equations with Phase Shifts — Solve with exact values

Pay close attention to domain restrictions

5. $\sqrt{2}\sin\left(x - \frac{\pi}{4}\right) = 1$ Domain: $[0, 2\pi)$

Tip: Let $u = x - \frac{\pi}{4}$, solve for $u$, then adjust.

Solution(s):

6. $2\cos\left(2x + \frac{\pi}{3}\right) = -\sqrt{3}$ General Formula

Express all real solutions where $k \in \mathbb{Z}$.

General $x$:

Part 4

Ocean Harbor Tides Application

The water depth $D$ (in meters) in a coastal shipping channel on October 14 is modeled by the sinusoidal function: $D(t) = 4\cos\left(\frac{\pi}{6}(t - 2)\right) + 7$ where $t$ represents hours after midnight ($0 \le t \le 24$). Cargo vessels require a minimum depth of 9 meters to safely dock.

7a. Algebraic Setup

Set up and solve the equation $D(t) = 9$ to find all times $t$ during the 24-hour day when the water is exactly 9 meters deep.

Values of $t$:

7b. Maritime Interpretation

Between what clock hours (e.g., 2:00 AM – 6:00 AM) is it safe for cargo ships to dock? Explain using your solutions above.

Safe Time Windows:

Error Analysis — Spot the Wave Flaw

Student Work for $2\sin(2x) = 1$ on $[0, 2\pi)$:

Line 1: $\sin(2x) = \frac{1}{2}$

Line 2: $2x = \frac{\pi}{6}$ or $2x = \frac{5\pi}{6}$

Line 3: $x = \frac{\pi}{12}$ or $x = \frac{5\pi}{12}$

Line 4: "Final Answer: 2 solutions found."

Identify the critical conceptual mistake the student made in their solution, and state how many solutions actually exist in the interval $[0, 2\pi)$:

Wave Hunters • Trigonometry • Sine & Cosine Models Page 2 of 2

Wave Hunters Answer Key

Teacher Reference • Full Algebraic Solutions • Scoring Rubrics

OFFICIAL TEACHER KEY

Precalculus / Trigonometry

Key Grading Checkpoints & Common Errors

Unit Circle Ratios: Verify students identify the correct reference angle prior to choosing quadrants.

Frequency Multipliers: For $B \cdot x$, students must expand the domain to $[0, 2\pi B)$ before dividing by $B$.

Domain Bounds: Ensure solutions strictly stay within $[0, 2\pi)$; watch for extraneous negative angles.

Part 1

Fundamental Equations on $[0, 2\pi)$ — Detailed Steps

1. $2\cos(x) - \sqrt{2} = 0$ 2 pts

Step 1: Isolate ratio
$2\cos(x) = \sqrt{2} \implies \cos(x) = \frac{\sqrt{2}}{2}$

Step 2: Reference angle & Quadrants
$\cos$ is positive in Quadrants I and IV.
Reference angle: $\theta_{\text{ref}} = \frac{\pi}{4}$.

Step 3: Quadrant solutions in $[0, 2\pi)$
• Quadrant I: $x = \frac{\pi}{4}$
• Quadrant IV: $x = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$

Final Answer: $x = \frac{\pi}{4},\; \frac{7\pi}{4}$

2. $\sqrt{3}\tan(x) + 1 = 0$ 2 pts

Step 1: Isolate ratio
$\sqrt{3}\tan(x) = -1 \implies \tan(x) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Step 2: Reference angle & Quadrants
$\tan$ is negative in Quadrants II and IV.
Reference angle: $\theta_{\text{ref}} = \frac{\pi}{6}$.

Step 3: Quadrant solutions in $[0, 2\pi)$
• Quadrant II: $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$
• Quadrant IV: $x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$

Final Answer: $x = \frac{5\pi}{6},\; \frac{11\pi}{6}$

Part 2

Multiple-Angle Equations on $[0, 2\pi)$ — Detailed Steps

3. $2\sin(3x) - 1 = 0$ 3 pts

Isolate: $\sin(3x) = \frac{1}{2}$. Sine is positive in Q I & II.

Base angles for $3x$: $\frac{\pi}{6}$ and $\frac{5\pi}{6}$.

Because $x \in [0, 2\pi)$, the argument $3x \in [0, 6\pi)$ (3 full revolutions):

$3x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{13\pi}{6}, \frac{17\pi}{6}, \frac{25\pi}{6}, \frac{29\pi}{6}$

Divide each term by 3:

Final Solutions (6): $x = \frac{\pi}{18},\; \frac{5\pi}{18},\; \frac{13\pi}{18},\; \frac{17\pi}{18},\; \frac{25\pi}{18},\; \frac{29\pi}{18}$

Stock Market Worksheet

Financial Literacy • Economics

THE STOCK MARKET — EXPLAINED

Name:

Date: Period:

How the System Works: Primary & Secondary Markets

Examine the diagram carefully before answering below

Company X

Stocks (Ownership)

Capital (Funds)

Investor

Investors buy stock to become part-owners. In exchange, the company gains capital to expand.

Stockholders Can Earn in 2 Ways

1. Trade Stocks

Sell shares in the market at a higher price (capital gains).

↓ Connected to Market

2. Earn Dividends

Receive regular cash payouts from net company profits.

$ Profit sharing

Buyers

Individuals & funds seeking shares

THE STOCK MARKET $ • € • £ • ¥

NYSE • NASDAQ London (LSE) Tokyo (TSE) Euronext

Regulated exchanges securely match buy and sell orders across global currencies.

Sellers

Shareholders liquidating stock

Analysis & Application Questions

Answer thoroughly in complete sentences

1. Raising Capital vs. Part-Ownership Recall & Explain

Based on the diagram, why does Company X issue and sell stock to an Investor? In turn, what does the investor receive that changes their legal status with the company?

2. Comparing the Two Profit Channels Contrast

Stockholders can earn money through trading stocks or earning dividends. How do these two earning methods differ in how and when the investor receives cash?

3. Role of Global Stock Exchanges System Dynamics

Notice the global ring featuring currencies ($$$, $€$, $£$, $¥$) connecting Buyers and Sellers. Why are organized stock exchanges necessary for trading shares rather than buyers and sellers negotiating independently?

4. Real-World Calculation & Scenario Analysis Application

Elena buys 50 shares of Company X at $20 per share. During the year, Company X pays $1.50 per share in dividends. At year-end, Elena sells all 50 shares at $28 per share. Calculate her: (a) total dividend income, (b) trading profit (capital gain), and (c) total profit.

Financial Literacy • Understanding Equities & Financial Markets Page 1 of 1

Stock Market Answer Key

THE STOCK MARKET — ANSWER KEY

Teacher Reference • Exemplary Student Responses • 10 Total Points

TEACHER EDITION

Grades 9–12 Economics

Scoring Criteria & Core Standards 10 Points Total

Q1–Q3 Concepts (6 pts): Full credit requires accurate economic vocabulary: capital, shareholder/equity, dividends, capital gains, and liquidity.

Q4 Calculation (4 pts): 1 pt for dividend calculation, 1 pt for capital gains, 1 pt for total profit, and 1 pt for clear step-by-step setup.

Common Misconception: Students often confuse dividends (profit sharing) with capital gains (market price increase upon selling).

Detailed Solutions & Model Responses

Suggested Point Allocations

1. Raising Capital vs. Part-Ownership 2 Points

Why does Company X issue stock, and what does the investor receive in return?

Exemplary Response: Company X issues and sells stock to raise financial capital (funds) without taking on debt, using the money to build facilities, hire staff, or develop new products. In exchange, the investor receives shares representing fractional equity (part-ownership) in the company, entitling them to voting rights and a share of future profits.

• Award 1 pt for capital/expansion rationale; 1 pt for part-ownership/shareholder status.

2. Comparing the Two Profit Channels 2 Points

How do trading stocks and earning dividends differ in mechanism and timing?

Exemplary Response: Trading stocks (capital gains) generates profit only when an investor sells their shares at a higher market price than the purchase price. In contrast, dividends are periodic direct cash payments distributed from the company's net earnings while the investor continues to hold and own the shares.

• Award 1 pt for distinguishing selling (capital gains) vs. holding; 1 pt for connecting dividends to company profit distributions.

3. Role of Global Stock Exchanges 2 Points

Why are organized stock exchanges necessary rather than private negotiating?

Exemplary Response: Organized exchanges provide a centralized, regulated marketplace that ensures liquidity, transparent real-time price discovery, and legal security. Instead of buyers independently searching for sellers, exchanges instantly match millions of orders globally and standardize currency conversion ($$$, $€$, $£$, $¥$) without counterparty risk.

4. $2\cos(2x) + \sqrt{3} = 0$ 3 pts

Isolate: $\cos(2x) = -\frac{\sqrt{3}}{2}$. Cosine is negative in Q II & III.

Base angles for $2x$: $\frac{5\pi}{6}$ and $\frac{7\pi}{6}$.

Since $x \in [0, 2\pi)$, argument $2x \in [0, 4\pi)$ (2 full revolutions):

$2x = \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{17\pi}{6}, \frac{19\pi}{6}$

Divide each term by 2:

Final Solutions (4): $x = \frac{5\pi}{12},\; \frac{7\pi}{12},\; \frac{17\pi}{12},\; \frac{19\pi}{12}$

Wave Hunters Answer Key • Parts 1 & 2 Solutions Page 1 of 2

Part 3, 4, & 5

Advanced Transformations, Modeling & Analysis

Teacher Key • Continued

Part 3

Phase Shifts & General Form Solutions

5. $\sqrt{2}\sin\left(x - \frac{\pi}{4}\right) = 1$ 3 pts

1. Isolate: $\sin\left(x - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

2. Let $u = x - \frac{\pi}{4}$. On $[0, 2\pi)$, $u = \frac{\pi}{4}$ or $u = \frac{3\pi}{4}$.

3. Solve for $x$:

• $x - \frac{\pi}{4} = \frac{\pi}{4} \implies x = \frac{2\pi}{4} = \frac{\pi}{2}$
• $x - \frac{\pi}{4} = \frac{3\pi}{4} \implies x = \frac{4\pi}{4} = \pi$

Both $\frac{\pi}{2}$ and $\pi$ fall squarely inside $[0, 2\pi)$.

Final Answer: $x = \frac{\pi}{2},\; \pi$

6. $2\cos\left(2x + \frac{\pi}{3}\right) = -\sqrt{3}$ 3 pts

1. Isolate: $\cos\left(2x + \frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$ (Q II & Q III).

2. Set general branches with $k \in \mathbb{Z}$:

Branch A: $2x + \frac{\pi}{3} = \frac{5\pi}{6} + 2\pi k$
$\implies 2x = \frac{3\pi}{6} + 2\pi k = \frac{\pi}{2} + 2\pi k \implies x = \frac{\pi}{4} + \pi k$
Branch B: $2x + \frac{\pi}{3} = \frac{7\pi}{6} + 2\pi k$
$\implies 2x = \frac{5\pi}{6} + 2\pi k \implies x = \frac{5\pi}{12} + \pi k$

General Solutions: $x = \frac{\pi}{4} + \pi k$ and $x = \frac{5\pi}{12} + \pi k,\; k \in \mathbb{Z}$

Part 4: Ocean Harbor Modeling — Solutions & Physical Analysis

5 pts total

7a. Solve $D(t) = 9$ Algebraic setup (3 pts)

$4\cos\left(\frac{\pi}{6}(t - 2)\right) + 7 = 9 \implies \cos\left(\frac{\pi}{6}(t - 2)\right) = \frac{2}{4} = \frac{1}{2}$

Reference angle is $\frac{\pi}{3}$. Cosine is $\frac{1}{2}$ at $\pm \frac{\pi}{3} + 2\pi k$:

• $\frac{\pi}{6}(t - 2) = \frac{\pi}{3} + 2\pi k \implies t - 2 = 2 + 12k \implies t = 4 + 12k$
• $\frac{\pi}{6}(t - 2) = -\frac{\pi}{3} + 2\pi k \implies t - 2 = -2 + 12k \implies t = 0 + 12k$

For $t \in [0, 24]$:

$t = 0,\; 4,\; 12,\; 16,\; 24\text{ hours after midnight}$

7b. Maritime Interpretation Intervals & context (2 pts)

Peak depth (high tide) is $D(2) = 4(1) + 7 = 11\text{m}$ at $t = 2$ and $t = 14$. Since depth is at or above $9\text{m}$ between crests:

Window 1: $0 \le t \le 4$ → 12:00 AM – 4:00 AM

Window 2: $12 \le t \le 16$ → 12:00 PM (Noon) – 4:00 PM

(Note: Water also reaches exactly 9m at $t=24$ / Midnight).

Cargo ships can dock safely whenever depth $\ge 9\text{m}$, giving two 4-hour docking periods.

Part 5: Error Analysis — "Spot the Wave Flaw" Rubric & Solution

3 pts

Conceptual Error Identified:

The student solved for the argument $2x$ across only one standard period $[0, 2\pi)$. However, since the question requires $x \in [0, 2\pi)$, the argument $2x$ must be evaluated over the expanded interval $[0, 4\pi)$ (two complete periods).

Missing values for $2x$: $\frac{13\pi}{6}$ and $\frac{17\pi}{6}$.

Correct Count & Full Solution Set:

There are actually 4 distinct solutions on $[0, 2\pi)$:

$x = \frac{\pi}{12},\; \frac{5\pi}{12},\; \mathbf{\frac{13\pi}{12}},\; \mathbf{\frac{17\pi}{12}}$

Teaching Tip: Remind students that the coefficient $B$ multiplies the number of solutions in the standard interval.

Wave Hunters Answer Key • Complete Reference Guide Page 2 of 2

• Award 1 pt for centralized matching / liquidity; 1 pt for price transparency / standardized safety.

4. Real-World Calculation & Scenario Analysis 4 Points

Elena buys 50 shares at $20/share, gets $1.50/share dividends, and sells at $28/share.

(a) Dividend Income: $50 \times \$1.50 = \mathbf{\$75.00}$ (1 pt)

(b) Trading Profit: $50 \times (\$28 - \$20) = \mathbf{\$400.00}$ (1 pt: Capital Gain)

(c) Total Profit: $\$75 + \$400 = \mathbf{\$475.00}$ (1 pt total + 1 pt work)

Optional Extension: Elena's total return on her initial $1,000 investment is $\frac{\$475}{\$1,000} = 47.5\%$.

Classroom Discussion Prompt: Ask students why some high-growth tech companies reinvest all profits rather than paying dividends.

Total Points: 10/10

The Stock Market Explained • Teacher Solution Key & Scoring Rubric Page 1 of 1

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