Problem 85 - Precalculus

Question

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\sin \theta=-0.3420\) and the terminal side of \(\theta\) lies in quadrant IV.

Step-by-Step Solution

Answer

340 degrees.

Step 1: Understanding the Sinus Function in Quadrants

The sine function is negative in Quadrants III and IV. Since the problem indicates that the terminal side of \(\theta\) lies in Quadrant IV, we're focused on finding the angle there.

Step 2: Finding the Reference Angle

To find \(\theta\), first calculate the reference angle. Use the inverse sine (arcsine) to determine the reference angle: \(\theta_{ref} = \arcsin(0.3420)\). Calculate and round to the nearest degree. \(\theta_{ref} \approx 20\) degrees.

Step 3: Determine the Angle in the Fourth Quadrant

In Quadrant IV, the relationship for converting reference angles is \(\theta = 360^\circ - \theta_{ref}\). Using our reference angle, \(\theta = 360^\circ - 20^\circ = 340^\circ\).

Step 4: Concluding the Smallest Positive Measure

After calculating, the smallest positive measure of \(\theta\) that fits the given conditions is 340 degrees.

Key Concepts

sine function

The sine function is one of the fundamental trigonometric functions, alongside cosine and tangent. It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. However, sine is not limited to only right triangles. It is a periodic function that is defined for all real numbers using the unit circle.
The sine function is represented as \( \sin \theta \), where \( \theta \) is the angle. The range of \( \sin \theta \) is between -1 and 1, inclusive. This means that the sine of any angle will always fall within this range.
Trigonometric functions, including sine, are periodic in nature. Specifically, the sine function repeats every 360 degrees or \( 2\pi \) radians. This periodicity implies that the same sine value can appear for multiple angles. Thus, when we solve problems involving sine functions, it's crucial to consider how these angles correspond within specific quadrants on the coordinate plane.

quadrants

When discussing angles, especially in trigonometry, we often refer to quadrants. The coordinate plane is divided into four quadrants:

• Quadrant I: Both \( x \) and \( y \) coordinates are positive.
• Quadrant II: \( x \) is negative, \( y \) is positive.
• Quadrant III: Both \( x \) and \( y \) are negative.
• Quadrant IV: \( x \) is positive, \( y \) is negative.

The sine function's sign is determined by the \( y \)-coordinate in each quadrant. In Quadrants I and II, where \( y \) is positive, \( \sin\theta \) is positive. Conversely, in Quadrants III and IV, where \( y \) is negative, \( \sin\theta \) is negative. In the original problem, given \( \sin \theta = -0.3420 \) and \( \theta \) in Quadrant IV, this makes perfect sense because sine values are negative here. Understanding quadrants is essential for determining not just the sign of trigonometric functions, but also how to calculate angles effectively.

reference angle

A reference angle is a helpful concept in trigonometry, especially when determining specific values of angles in various quadrants. It is defined as the smallest angle formed by the terminal side of the angle and the horizontal axis, and it is always positive and less than 90 degrees or \( \pi/2 \) radians.
To find the reference angle for a given angle \( \theta \):

• In Quadrant I, the angle itself is the reference angle.
• In Quadrant II, the reference angle is \( 180^\circ - \theta \).
• In Quadrant III, the reference angle is \( \theta - 180^\circ \).
• In Quadrant IV, the reference angle is \( 360^\circ - \theta \).

In the exercise provided, we find that the reference angle for \( \theta \) is \( \arcsin(0.3420) \approx 20^\circ \). By using this reference angle, we can then calculate the smallest positive measure of the angle in Quadrant IV to be \( 360^\circ - 20^\circ = 340^\circ \).
Reference angles make it easier to work with trigonometric values and convert angles within different quadrants.