Lesson 6.3

General Solutions vs. Specific Intervals

Trig functions repeat every period. When the problem says “find all solutions,” you need to account for that infinite periodicity using the notation.

Introduction

In Lesson 6.2, we found solutions on . But since trig functions are periodic, you can add any full period and still have a solution. This lesson teaches you to express the complete, infinite set of solutions — and to pull specific solutions from an interval when asked.

Past Knowledge

Period of sin/cos (), period of tan (), and basic equation solving (6.2).

Today's Goal

Write general solutions using and extract solutions on specific intervals.

Future Success

Every remaining lesson (6.4–6.8) may ask for general or interval-specific solutions.

Key Concepts

General Solution Format

Function	General Solution	Period Added
sin / cos	,
tan	,

Tangent Shortcut

Since tangent has period (not ), a tangent equation with two solutions per cycle can often be collapsed into one formula: . This automatically generates both quadrant solutions.

From General to Specific

To find solutions on an interval like :

Write the general solution.
Plug in (and negative if needed) until the angles leave the interval.
List all angles that fall inside the interval.

Worked Examples

Basic

General Solution for Sine

Find all solutions:

Step 1: On :

Step 2: Add to each:

General Solution:

Intermediate

General Solution for Tangent

Find all solutions:

Step 1: is one solution.

Step 2: Since tangent has period , we only need one formula:

This generates and

General Solution:

Advanced

Extracting from a Wider Interval

Find all solutions of on .

Step 1: General solution: or

Step 2: Plug in : ✓

Plug in : , ✓ (both )

Plug in : ✗ (exceeds )

Solution:

Common Pitfalls

Using for Tangent

Tangent has period , not . The general solution for tangent uses .

Stopping Too Early on Wider Intervals

On , you need to keep plugging in values of until you exceed the upper bound. Many students stop at and miss the additional solutions.

Real-Life Applications

Tide Prediction

Ocean tides follow repeating sinusoidal cycles. A harbor master who needs to know every time the water reaches a certain depth — not just the first occurrence — uses the general solution approach: find the base times, then add the tidal period to get every future (and past) occurrence.