Question

Without solving, determine the number of solutions for each trigonometric equation in the specified domain. Explain your reasoning. a) \(\sin \theta=\frac{\sqrt{3}}{2}, 0 \leq \theta < 2 \pi\) b) \(\cos \theta=\frac{1}{\sqrt{2}},-2 \pi \leq \theta < 2 \pi\) c) \(\tan \theta=-1,-360^{\circ} \leq \theta \leq 180^{\circ}\) d) \(\sec \theta=\frac{2 \sqrt{3}}{3},-180^{\circ} \leq \theta < 180^{\circ}\)

Short answer

a) 2 solutions, b) 4 solutions, c) 3 solutions, d) 2 solutions.

Step-by-step solution

Analyze the equation \(\sin \theta=\frac{\sqrt{3}}{2}, 0 \leq \theta < 2 \pi\)

The sine function equals \(\frac{\sqrt{3}}{2}\) at two angles within one period \(0 \leq \theta < 2 \pi\). These angles are \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\). Therefore, there are 2 solutions.

Analyze the equation \(\cos \theta=\frac{1}{\sqrt{2}},-2 \pi \leq \theta < 2 \pi\)

The cosine function equals \(\frac{1}{\sqrt{2}}\) at two angles within one period \(0 \leq \theta < 2 \pi\). These angles are \(\frac{\pi}{4}\) and \(\frac{7\pi}{4}\). The given domain extends over two periods, thus there are 4 solutions.

Analyze the equation \(\tan \theta=-1,-360^{\circ} \leq \theta \leq 180^{\circ}\)

The tangent function equals \(-1\) at two angles within one period \(0^{\circ} \leq \theta < 360^{\circ}\). These angles are \(135^{\circ}\) and \(315^{\circ}\). Given the domain includes one full circle plus an additional half-circle in the negative direction, there are 3 solutions: \(135^{\circ}, 315^{\circ}, -45^{\circ}\).

Analyze the equation \(\sec \theta=\frac{2 \sqrt{3}}{3},-180^{\circ} \leq \theta < 180^{\circ}\)

The secant function equals \(\frac{2\sqrt{3}}{3}\) when the cosine function equals the reciprocal of this value, \(\frac{1}{2}\). The cosine function equals \(\frac{1}{2}\) at 60° and 300° within one period. Considering the domain extends from -180° to 180°, these angles correspond to \(60^{\bullet}\) and \(-60^{\circ}\). Therefore, there are 2 solutions.

Key concepts

Number of Solutions

To determine how many solutions an equation has within a specified domain, you first need to understand how many times the trigonometric function equals the given value in one period. Trigonometric functions like \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \) each have unique periods and properties. Once you identify these values in one period, you can extrapolate to find out how they appear across the entire domain.
For example, the sine function typically has two solutions in one period for a given value. The cosine function also usually has two, while the tangent function has one per period but repeats every 180 degrees or \( \pi \) radians.

Trigonometric Functions

Different trigonometric functions behave differently within one period:

• Sine and Cosine: These functions repeat their values every \( 2 \pi \) radians (or every 360 degrees). Therefore, for any specific value, like \( \frac{\sqrt{3}}{2} \) for sine or \( \frac{1}{\sqrt{2}} \) for cosine, there are usually two solutions within one period.
• Tangent: The tangent function repeats every \( \pi \) radians (or 180 degrees). This means that for values like -1, there is one solution in each half period.
• Secant, Cosecant, and Cotangent: These functions are reciprocals of cosine, sine, and tangent respectively. Their solutions are derived from their base functions.

Understanding these periodic properties is crucial for determining the number of solutions and their locations within any given domain.

Specific Domains

The domain of an equation specifies where you're looking for the solutions on the trigonometric circle. The standard period for trigonometric functions is from 0 to \( 2\pi \), but you may be asked to find solutions in non-standard ranges like \( -2\pi \leq \theta < 2\pi \) or even over multiple periods:

• For sine and cosine over \( 0 \leq \theta < 2\pi \), you expect to find two solutions.
• If the domain extends over \( -2\pi \leq \theta < 2\pi \), then you may need to double the number of solutions you originally found within one period.
• Similarly, an extended domain like \( -360^{\circ} \leq \theta \leq 180^{\circ} \) may cover more than one full period, resulting in additional solutions.

Always ensure to convert degrees to radians if needed and check all possible multiples within the given domain.

Analyzing Equations

To analyze and solve trigonometric equations effectively, you need to follow a systematic approach:

• Identify the fundamental trigonometric identity or equation presented.
• Find the angles that satisfy the equation within one complete period of the function.
• Extend these solutions across the entire specified domain by considering the periodicity of the trigonometric function.

For instance, to solve \( \sin \theta = \frac{\sqrt{3}}{2} \) in the domain \( 0 \leq \theta < 2\pi \), you first find the angles \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \). Since the domain covers just one period for sine, only these two solutions are valid. Always write out each step and make sure no potential solutions are overlooked.