Problem 90 - Precalculus

Question

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\csc \theta=-1.0001\) and the terminal side of \(\theta\) lies in quadrant III.

Step-by-Step Solution

Answer

The smallest positive measure of \( \theta \) is \( 270\degree \).

Step 1: Understanding the Problem

We need to find the smallest positive angle \( \theta \) for which the cosecant \( \csc \theta = -1.0001 \) and the terminal side of \( \theta \) lies in the third quadrant. Remember, cosecant is the reciprocal of sine, so \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Finding the Sine Value

Since \( \csc \theta = -1.0001 \), the sine value \( \sin \theta \) can be calculated as \( \sin \theta = \frac{1}{\csc \theta} = \frac{1}{-1.0001} \approx -0.9999 \).

Step 3: Determine Reference Angle

The reference angle is the angle with the same trigonometric ratios which is positive. Using an inverse function, \( \sin^{-1}(0.9999) \), we find the reference angle is approximately \( 90\degree \).

Step 4: Calculate the Smallest Positive Angle in Quadrant III

In quadrant III, sine is negative, which aligns with our value. The reference angle \( 90\degree \) is away from the \( 180\degree \) line of the third quadrant, resulting in an angle \( \theta = 180\degree + 90\degree = 270\degree \).

Step 5: Rounding to the Nearest Whole Number

The angle \( 270\degree \) is already at a whole number, so no further rounding is needed.

Key Concepts

Reference Angle

A reference angle is a positive acute angle that helps in determining the trigonometric values of angles located in different quadrants. It is always measured from the x-axis, making it a useful tool in trigonometry.

• The value of a reference angle is always between 0° and 90°.
• To find the reference angle, consider the absolute value of the angle's trigonometric function while ignoring any quadrant conditions.
• In our example, using \(\sin^{-1}(0.9999)\), we identify a reference angle which is close to 90°.

Reference angles are crucial for interpreting trigonometric functions, especially when angles reside outside the first quadrant. They help in transitioning back to familiar trigonometric values, enabling easier calculations for angles in quadrants II, III, and IV.

Sine Function

The sine function (\(\sin \theta\)) is a fundamental trigonometric function that outputs the ratio of the opposite side to the hypotenuse in a right triangle. Its range is bounded between -1 and 1, including all real numbers in the interval.
The sine value directly influences its reciprocal, the cosecant (\(\csc \theta = \frac{1}{\sin \theta}\)). When the sine value is negative, it indicates the angle lies in either the third or fourth quadrant.

• We found \(\sin \theta \approx -0.9999\), indicating a negative sine value.
• This further implies the angle \(\theta\) lies in a quadrant where sine is negative.
• In our task, it confirmed the placement of \(\theta\) in Quadrant III.

Understanding the sine function and its behavior across different quadrants is fundamental for accurately interpreting angles and their trigonometric functions.

Quadrant III

Placement in Quadrant III is distinctive in that both sine and cosine values are negative, while tangent is positive. This quadrant ranges from 180° to 270°.
For trigonometric purposes, knowing in which quadrant an angle lies is vital for determining the sign of its trigonometric values.

• In Quadrant III, because sine (\(\sin \theta\)) is negative, the cosecant (\(\csc \theta\)), as its reciprocal, is also negative.
• To locate the exact angle in this quadrant, we add the reference angle to 180°, aligning with our case to reach 270°.
• This stems from the property of the unit circle, reflecting the consistent negative nature of the sine and cosine functions within this quadrant.

Understanding quadrant placements facilitates smoother calculations and helps predict trigonometric value signs, an essential skill in mastering trigonometry.