Year 12 Trigonometry: General solutions of trigonometric equations

A general solution question asks for the entire repeating family of angles, not just the solutions on a single interval. This is where periodicity matters. Sine and cosine repeat every $2 pi$ , while tangent repeats every $pi$ . The trick is to find the core solutions once and then attach the correct repeating pattern. Many otherwise correct responses lose marks because they stop with an interval answer when the question explicitly asks for every solution.

General solutions are interval solutions plus the correct repeating term, usually written with

n

where

n

is an integer.

Core patterns

If $\sin (x) = a$ , there are usually two families separated by quadrant logic, repeating every $2 pi$ .
If $\cos (x) = a$ , there are usually two symmetric families, also repeating every $2 pi$ .
If $\tan (x) = a$ , there is one main family repeating every $pi$ .

Worked example 1

Problem: Solve

\sin (x) = sqrt (3) / 2

for all real

x

The reference angle is $pi / 3$ .
Sine is positive in quadrants 1 and 2.
The basic solutions on one full turn are $x = pi / 3$ and $x = 2 pi / 3$ .
Since sine repeats every $2 pi$ , add $2 pi * n$ to each solution.

Answer:

x = pi / 3 + 2 pi * n

x = 2 pi / 3 + 2 pi * n

, where

n

is an integer.

Worked example 2

Problem: Solve

\tan (x) = - 1

for all real

x

The reference angle is $pi / 4$ .
Tangent is negative in quadrants 2 and 4, but the tangent graph repeats every $pi$ .
One valid starting solution is $x = - pi / 4$ .
Add $pi * n$ to capture the full repeating family.

Answer:

x = - pi / 4 + pi * n

, where

n

is an integer.

When the angle is transformed

General-solution questions become slightly harder when the angle is not just $x$ . If the equation is $\cos (2 x) = - 1 / 2$ , you first solve for $2 x$ using standard trig reasoning, then divide the resulting families by $2$ . This is conceptually the same as any other trig equation, but students often lose track of the periodic pattern after the substitution step. The safest habit is to name the inner angle, solve in that variable, and only then convert back.

Worked example 3

Problem: Solve

\cos (2 x) = - 1 / 2

for all real

x

Let $u = 2 x$ . Then solve $\cos (u) = - 1 / 2$ .
The reference angle is $pi / 3$ , and cosine is negative in quadrants 2 and 3.
So $u = 2 pi / 3 + 2 pi * n$ or $u = 4 pi / 3 + 2 pi * n$ .
Now divide by $2$ : $x = pi / 3 + pi * n$ or $x = 2 pi / 3 + pi * n$ .

Answer:

x = pi / 3 + pi * n

x = 2 pi / 3 + pi * n

Common traps

Stopping with interval answers instead of writing the repeating family.
Adding the wrong period to the solution.
Forgetting to divide or otherwise transform the angle back after substitution.
Writing a family that misses one of the required quadrants for sine or cosine.

Revision focus

To revise general solutions effectively, pair every interval problem with its all-real-numbers version. That makes the difference between the two formats much clearer. The interval version asks, "Which solutions lie here?" The general version asks, "What is the full repeating pattern?" They use the same trig thinking but finish in different ways.

It also helps to write the period of the relevant function before you finish the question. This single line often prevents the common error of attaching $2 pi * n$ to tangent or $pi * n$ to sine and cosine without thinking.

Practice links

General solution of trigonometric equations