Solving Equations Revision Notes for Grade 12 NSC Matric Mathematics

Solving Equations

Understanding general solutions

Trigonometric equations often have multiple solutions because trigonometric functions are periodic - they repeat their values at regular intervals. The sine and cosine functions have a period of 360°, while the tangent function has a period of 180°. This means we need to find the general solution that includes all possible answers.

When we solve a trigonometric equation, we don't just want one solution - we want to express all solutions in a general form that accounts for this repeating pattern.

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The concept of periodicity is fundamental to understanding why trigonometric equations have infinite solutions. Just like the hands of a clock return to the same position every 12 hours, trigonometric functions return to the same values at regular intervals.

General solution formulas

These are the standard forms for writing general solutions to basic trigonometric equations:

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Critical Formulas for General Solutions

For sine equations:

If $\sin θ = x$ , then $θ = \sin^{-1} x + k × 360°$ or $θ = (180° - \sin^{-1} x) + k × 360°$

For cosine equations:

If $\cos θ = x$ , then $θ = \cos^{-1} x + k × 360°$ or $θ = (360° - \cos^{-1} x) + k × 360°$

For tangent equations:

If $\tan θ = x$ , then $θ = \tan^{-1} x + k × 180°$

In all cases, k ∈ Z (k is any integer: ..., -2, -1, 0, 1, 2, ...)

Step-by-step method for solving trigonometric equations

Follow this systematic 6-step approach to solve any trigonometric equation:

Simplify the equation using algebraic methods and trigonometric identities
Determine the reference angle (use a positive value from your calculator)
Use the CAST diagram to determine which quadrants give positive or negative values
Find restricted values within a specified interval by adding/subtracting multiples of the appropriate period
Write the general solution by adding multiples of the period to each answer
Check your answers using a calculator to verify they satisfy the original equation

Using the CAST diagram

The CAST diagram helps you identify which quadrants contain solutions:

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CAST Diagram Quadrants:

First quadrant (0° to 90°): All trigonometric ratios are positive
Second quadrant (90° to 180°): Only sine is positive
Third quadrant (180° to 270°): Only tangent is positive
Fourth quadrant (270° to 360°): Only cosine is positive

Remember the mnemonic: "All Students Take Calculus"

Worked example: Finding a general solution

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Worked Example: Basic General Solution

Question: Determine the general solution for $\sin θ = 0.3$

Solution:

Step 1: Use a calculator to find the reference angle

$\sin θ = 0.3$
Reference angle = $\sin^{-1}(0.3) = 17.5°$

Step 2: Use the CAST diagram to find which quadrants have positive sine

The CAST diagram shows that $\sin θ$ is positive in the first and second quadrants

Step 3: Find the solutions in each quadrant

In the first quadrant: $θ = 17.5°$
In the second quadrant: $θ = 180° - 17.5° = 162.5°$
Using the reduction formula: $\sin(180° - θ) = \sin θ$

Step 4: Write the general solution

$θ = 17.5° + k × 360°$ or $θ = 162.5° + k × 360°$ , where $k ∈ Z$

Step 5: Check the solution

For $k = 4$ : $θ = 17.5° + 4(360°) = 1457.5°$
$\sin(1457.5°) = 0.3007...$ ✓ (This confirms our solution is correct)

Worked example: Solving complex trigonometric equations

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Worked Example: Complex Equation with Double Angles

Question: Solve $\frac{1 - \sin y - \cos 2y}{\sin 2y - \cos y} = -1$

Solution:

Step 1: Simplify using double angle formulas

We need to manipulate the equation to have the form: single trigonometric ratio = constant
Using the identity $\cos 2y = 1 - 2\sin^2 y$ , we can rewrite the equation
After simplification, this reduces to: $\tan y = -1$

Step 2: Find the reference angle

$\tan y = -1$
Reference angle = $\tan^{-1}(1) = 45°$

Step 3: Use CAST diagram to find where tangent is negative

The CAST diagram shows that $\tan y$ is negative in the second and fourth quadrants
$y = (180° - 45°) + k × 180° = 135° + k × 180°$

Step 4: Write the final answer

$y = 135° + k × 180°$ , where $k ∈ Z$

Worked example: Using the zero product law

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Worked Example: Zero Product Law Application

Question: Find the general solution for $\sin θ \cos^2 θ = \sin^3 θ$

Solution:

Step 1: Rearrange to standard form

$\sin θ \cos^2 θ - \sin^3 θ = 0$
$\sin θ(\cos^2 θ - \sin^2 θ) = 0$
$\sin θ(\cos θ - \sin θ)(\cos θ + \sin θ) = 0$

Step 2: Apply the zero product law This gives us three separate equations:

$\sin θ = 0$ , which gives $θ = 0° + k × 360°$ or $θ = 180° + k × 360°$
$\cos θ - \sin θ = 0$ , which gives $\cos θ = \sin θ$ , so $θ = 45° + k × 180°$
$\cos θ + \sin θ = 0$ , which gives $\cos θ = -\sin θ$ , so $θ = -45° + k × 180°$

Step 3: Write the complete general solution

$θ = 0° + k × 360°$ or $θ = 180° + k × 360°$ or $θ = ±45° + k × 180°$ , where $k ∈ Z$

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Critical Exam Tips

Always check your solutions by substituting back into the original equation
Remember to divide by the appropriate number when dealing with multiple angles (e.g., if solving for 4x, divide the period by 4)
Be careful with domain restrictions - some equations may have values where certain ratios are undefined
Don't divide both sides by a trigonometric expression that could equal zero, as you might lose solutions
Use exact values when possible rather than decimal approximations in your final answer

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Key Points to Remember:

General solutions account for the periodic nature of trigonometric functions - they repeat their values at regular intervals
The systematic 6-step method ensures you find all solutions: simplify, find reference angle, use CAST, restrict values, write general solution, check answers
CAST diagram is essential for determining which quadrants contain your solutions based on the sign of the trigonometric ratio
Always verify your solutions by substituting back into the original equation to confirm they work
The zero product law is powerful for solving factorable trigonometric equations - if a product equals zero, at least one factor must equal zero

Solving Equations (Grade 12 NSC Matric Mathematics): Revision Notes