Linear Trigonometric Equations

The Principal Value is the first solution obtained by applying the inverse trigonometric function (e.g., $x = \arcsin(k)$) on a calculator.

It is critical to ensure the calculator is in the correct mode (Degrees or Radians) before performing this calculation, as the units are not interchangeable.

While the calculator provides one value, it may not always fall within the requested interval; however, it serves as the essential starting point for finding all other valid solutions.

The CAST diagram is a coordinate tool used to identify which quadrants contain positive values for each trigonometric function: All (1st), Sine (2nd), Tangent (3rd), and Cosine (4th).

For any given value $k$, there are generally two quadrants where the trigonometric function will have that specific sign (positive or negative).

Symmetry Rules: If $\theta$ is the acute angle (related angle) to the horizontal axis, other solutions are found at $180^\circ - \theta$ (Sine), $180^\circ + \theta$ (Tangent), and $360^\circ - \theta$ (Cosine).

When the argument of the function is modified (e.g., $\sin(2x) = k$ or $\cos(x + 30^\circ) = k$), you must transform the interval before solving.

If the original range is $0 \le x < 360^\circ$, for an equation involving $2x$, you must look for all solutions in the range $0 \le 2x < 720^\circ$.

After finding all possible values for the transformed argument (e.g., $2x = 60, 300, 420, 660$), the final step is to perform the inverse operation (e.g., dividing by 2) to solve for $x$.

Degrees vs. Radians: Always check the interval units. If the interval contains $\pi$, use Radian mode; if it uses the $^\circ$ symbol, use Degree mode.

Principal vs. General Solutions: The principal value is just the start; the general solution accounts for the infinite repetitions across the entire number line.

Check the Interval: Always verify if the boundaries are inclusive ($\le$) or exclusive ($<$). Missing a solution at $0$ or $360$ is a common way to lose marks.

The 'Add/Subtract Period' Rule: Once you have your primary solutions from the CAST diagram, you can find more by adding or subtracting the period ($360^\circ$ or $180^\circ$) until you fall outside the range.

Final Verification: Always substitute your final values of $x$ back into the original equation to ensure they produce the correct constant $k$.