Problem 77 - Precalculus

Question

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\sin \theta=0.9397\) and the terminal side of \(\theta\) lies in quadrant II.

Step-by-Step Solution

Answer

The smallest positive measure of \( \theta \) is 110°.

Step 1: Find the Reference Angle

Use the inverse sine function to find the reference angle in degrees. The equation is given by \( \theta_{ref} = \arcsin(0.9397) \). Computing this, we get \( \theta_{ref} \approx 69.998 \). Rounding this to the nearest degree, we get \( \theta_{ref} = 70° \).

Step 2: Determine the Quadrant II Angle

Since sin(θ) is positive and the angle lies in quadrant II, calculate the angle \( \theta \) using the reference angle. In Quadrant II, \( \theta = 180° - \theta_{ref} \). Substitute the reference angle: \( \theta = 180° - 70° \).

Step 3: Compute the Smallest Positive Measure of \( \theta \)

Compute the final angle by performing the subtraction from the previous step: \( \theta = 180° - 70° = 110° \). Thus, the smallest positive measure of \( \theta \) is 110°.

Key Concepts

Inverse Sine Function

The inverse sine function, often denoted as \( \arcsin \), is a way of determining the angle whose sine value is known. This means if you know the sine value of an angle, you can find the angle itself by using this function.
For example, if \( \sin(\theta) = 0.9397 \), then you can use \( \arcsin(0.9397) \) to find the reference angle.
This function is essential in trigonometry to move from a known ratio back to an angle.

• It gives angles usually in the range of \(-90°\) to \(90°\).
• These angles give the primary solution as angle calculations adjust for different quadrants later.

Understanding \( \arcsin \) is key to solving problems where you're working backwards from a sine value back to the angle measure.

Reference Angle

A reference angle is the smallest angle between the terminal side of an angle and the x-axis.
It is always a positive angle and helps in calculating angles in different quadrants by using their signs.
Here's how it works for sine:

• Find the inverse sine, which gives you the reference angle \( \theta_{ref} \).
• In our case, \( \theta_{ref} = 70° \) because \( \arcsin(0.9397) \approx 69.998° \) and rounds to \( 70° \).

The reference angle is crucial because it's a helper in defining the specific angle in any quadrant easily, which might otherwise need complex calculations.

Quadrant II

In trigonometry, the coordinate plane is divided into four quadrants. Quadrant II is the second one, where the angles range from \(90°\) to \(180°\). In this quadrant:

• Sine values are positive, making it relevant for our exercise.
• To find the angle in this quadrant using a reference angle \( (\theta_{ref}) \), subtract it from \(180°\), giving \( \theta=180°-\theta_{ref} \).

For instance, if \(\theta_{ref} = 70°\), then \(\theta = 180° - 70° = 110°\). Thus, \(110°\) is the smallest positive angle measure satisfying both the sine value and the constraint of being in Quadrant II.
This understanding is pivotal when solving trigonometric problems where negative or larger angles need simplification to positive measures within expected ranges.