How does the method for finding secondary solutions for $\sin x = k$ differ from $\cos x = k$?

For $\sin x = k$, the secondary solution is found using $180^\circ - \theta$ (horizontal symmetry). For $\cos x = k$, the secondary solution is $360^\circ - \theta$ (vertical symmetry).

When solving $\tan x = k$, how does the periodicity affect the search for multiple solutions compared to $\sin x$?

Tangent has a period of $180^\circ$ (or $\pi$), so you find additional solutions by simply adding or subtracting $180^\circ$. Sine has a period of $360^\circ$, so you must find a secondary symmetrical solution before adding multiples of $360^\circ$.

What is the difference between the 'Principal Value' and the 'General Solution' of a trig equation?

The Principal Value is the single value returned by a calculator's inverse function. The General Solution is a formula representing all infinite possible solutions across the function's entire domain.

What error occurs if you solve for $x$ in $\sin(2x) = 0.5$ before finding all solutions for the argument $2x$?

You will likely miss half of the valid solutions. You must find all values for $2x$ within the doubled range ($0$ to $720^\circ$) first, then divide each by 2 to find $x$.

Why is it a mistake to assume $\cos x = 2$ has a solution?

The range of the cosine function is restricted to $[-1, 1]$. Any linear equation $\cos x = k$ where $|k| > 1$ has no real solutions.

What happens if you use 'Degree' mode on your calculator when the interval is given as $0 \le x \le 2\pi$?

Your principal value will be in degrees (e.g., $30^\circ$) while the required answer must be in radians (e.g., $\pi/6$). This leads to incorrect boundary checks and final values.

Define the 'CAST' acronym and its purpose in trigonometry.

CAST stands for Cosine, All, Sine, Tangent. it identifies which trigonometric functions are positive in the 4th, 1st, 2nd, and 3rd quadrants respectively.

What is a 'Transformed Range' in the context of solving $\cos(x - 30^\circ) = 0.5$?

It is the adjustment of the given interval for $x$ to match the argument $(x - 30^\circ)$. If $0 \le x \le 360$, the transformed range is $-30 \le (x - 30) \le 330$.

State the relationship between degrees and radians for the four major axes.

$90^\circ = \pi/2$, $180^\circ = \pi$, $270^\circ = 3\pi/2$, and $360^\circ = 2\pi$.

How do you find all solutions for $\tan x = k$ within a range once you have the principal value $\alpha$?

Since tangent repeats every $180^\circ$, all solutions are found by the formula $x = \alpha + 180n$, where $n$ is any integer that keeps $x$ within the specified range.

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Linear Inequalities

Linear Trigonometric Equations

Summary

Linear trigonometric equations are first-degree equations involving sine, cosine, or tangent functions. Solving them requires finding all possible angles within a specified interval that satisfy the equation, utilizing the periodic and symmetrical properties of trigonometric graphs or the CAST diagram.

1. Definition & Core Concepts

2. The CAST Diagram & Symmetry

A CAST diagram showing the four quadrants (All, Sine, Tangent, Cosine) and the standard position of an angle theta.

3. Methods for Transformed Arguments

4. Key Distinctions

5. Exam Strategy & Tips

Always check the range: After finding your initial solutions, add or subtract the period ( $360^\circ$ or $180^\circ$ ) to see if other valid solutions fall within the required interval.
Transform the range first: If the question asks for solutions of $\cos(2x)$ in $0 \le x \le 180$ , you must look for solutions of $2x$ in $0 \le 2x \le 360$ .
Sketch to verify: A quick sketch of the trig graph can help you visualize how many solutions you should expect and prevent you from missing solutions near the boundaries of the interval.

6. Common Pitfalls & Misconceptions