Problem 55 - Algebra 2 and Trigonometry

Question

In \(51-58,\) find the smallest positive value of \(\theta\) to the nearest minute. $$ \sin \theta=0.6534 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is approximately 40°43'.

Step 1: Use Inverse Sine Function

Find the angle \( \theta \) by using the inverse sine function. The expression becomes \( \theta = \sin^{-1}(0.6534) \). Use a calculator to find the approximate value of \( \theta \).

Step 2: Calculate \( \theta \) in Degrees

Evaluate the expression \( \theta = \sin^{-1}(0.6534) \) using a scientific calculator to obtain \( \theta \) in degrees.

Step 3: Convert Decimal Degrees to Degrees and Minutes

The calculator might show \( \theta \) as a decimal, such as \( 40.717 \) degrees. Convert this by taking the whole number as degrees and multiplying the decimal part by 60 to convert to minutes: \( 0.717 \times 60 = 43.02 \) minutes.

Step 4: Round Minutes to the Nearest Whole Number

Since \( 43.02 \) minutes is closest to 43 minutes, we round the minutes to 43.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions used to relate angles to side lengths in right-angled triangles.
It is particularly useful due to its periodic and oscillating nature, which finds applications in waves and circular motion.

• The sine of an angle, denoted as \( \sin \theta \), represents the ratio of the length of the opposite side to the hypotenuse of a right triangle.
• When dealing with the unit circle, which is a circle with a radius of 1, the sine of an angle relates directly to the \(y\)-coordinate of a point on the circle.

The inverse sine function, written as \( \sin^{-1} \) or \( \arcsin \), is used to determine the angle \( \theta \) whose sine is a given number.
For example, finding \( \theta = \sin^{-1}(0.6534) \) means identifying the angle whose sine value equals 0.6534.
This angle is usually found using a scientific calculator because it requires precise computation of many decimal places.

Angle Conversion

In trigonometry, angles can be measured in different units, typically using either degrees or radians.
Degrees are more commonly used in everyday situations, while radians are often used in higher-level mathematics.

• To convert radians to degrees, use the formula:
  \[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]
• Conversely, to convert degrees to radians, apply:
  \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

This exercise specifically focuses on working with angles in degrees.
When using the inverse sine function to find an angle, the result is typically provided in degrees by a calculator.
Understanding how to convert between different angle measurements is crucial when dealing with trigonometric functions.

Degree to Minute Conversion

Degrees can be further divided into smaller units called minutes and seconds for more precise angle measurements.
One degree is equivalent to 60 minutes, and each minute can be divided into 60 seconds.

• After finding the angle in degrees, you might get a decimal that requires conversion to minutes for accuracy.
• For example, the decimal degrees of \(40.717\) need to be converted into a more precise format by multiplying the fractional part (0.717) by 60, yielding \(43.02\) minutes.
• Finally, when a more accurate measurement is needed, rounding to the nearest whole minute is standard practice, as seen in rounding 43.02 to 43 minutes.

This process helps in conveying angle measurements that are precise and easy to understand, especially in practical applications such as navigation and astronomy.