Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Find the angle that is complementary to \(\theta\). (a) \(\theta=63^{\circ} 4^{\prime} 15^{N}\) (b) \(\theta=82.73^{\circ}\)

Step-by-Step Solution

Answer

(a) 26.9291° (b) 7.27°.

Step 1: Understand Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. Therefore, if you are given an angle \( \theta \), the angle that is complementary to it can be calculated as \( 90^\circ - \theta \).

Step 2: Convert Angle to Degrees (if necessary)

In some cases, the given angle is not in standard degree form with decimals or minutes and seconds. For part (a), the angle is \( 63^{\circ} 4' 15'' \). Convert this to degrees by noting that \( 1' = \frac{1}{60} \) degree and \( 1'' = \frac{1}{3600} \) degree.Calculate the total degrees: \[ 63^{\circ} + \frac{4}{60}^{\circ} + \frac{15}{3600}^{\circ} = 63^{\circ} + 0.0667^{\circ} + 0.0042^{\circ} \approx 63.0709^{\circ} \]

Step 3: Calculate the Complementary Angle for Part (a)

Use the formula for complementary angles, \( 90^\circ - \theta \), where \( \theta = 63.0709^{\circ} \).\[ 90^{\circ} - 63.0709^{\circ} = 26.9291^{\circ} \]

Step 4: Calculate the Complementary Angle for Part (b)

For part (b), \( \theta = 82.73^{\circ} \). Use the complementary angle formula:\[ 90^{\circ} - 82.73^{\circ} = 7.27^{\circ} \]

Key Concepts

Angle Conversion

To fully grasp complementary angles, you may need to convert them. Angles can be expressed in degrees, minutes, and seconds. This system uses the same logic as hours, minutes, and seconds in telling time. Degrees are the main unit. Minutes are one-sixtieth (\(60 \)) of a degree, and seconds are one-sixtieth (\(60 \)) of a minute. Here's how to convert them:

• Minutes to Degrees: Divide the number of minutes by 60.
• Seconds to Degrees: Divide the number of seconds by 3,600.

In our exercise, you convert the angle \(63^{\circ} 4' 15''\) to degrees:

• Convert minutes: \(\frac{4}{60} = 0.0667^{\circ}\)
• Convert seconds: \(\frac{15}{3600} = 0.0042^{\circ}\)

Add these conversions to the whole degree to get the full angle in degrees, \(63.0709^{\circ}\). This simplifies further calculations like finding complementary angles.

Degree Calculation

After understanding conversion, calculating the complementary angle becomes straightforward. Complementary angles are simply two angles that add up to precisely 90 degrees. To find the complementary angle of a given angle, subtract it from 90 degrees. Here's the process:

• Start with \(90^{\circ}\).
• Subtract your given angle.
• The result is the complementary angle.

For example:

• If \(\theta = 63.0709^{\circ}\), the complementary angle is:\(90^{\circ} - 63.0709^{\circ} = 26.9291^{\circ}\)
• For \(\theta = 82.73^{\circ}\), the calculation becomes:\(90^{\circ} - 82.73^{\circ} = 7.27^{\circ}\)

In both cases, subtracting the known angle from \(90^{\circ}\) gives the needed complementary angle correctly. This simple mathematical property ensures quick results.

Angle Measures

Angles are fundamental in mathematics and come in various forms such as degrees, minutes, and seconds. Understanding these measures helps in tasks such as geometry and trigonometry.

• Degrees (\(^{\circ}\)): The most common unit for angles.
• Minutes (\('\)): Used for finer measurements; \(1' = \frac{1}{60}^{\circ}\).
• Seconds (\(''\)): Even finer measurements; \(1'' = \frac{1}{3600}^{\circ}\).

When working with angles, it’s crucial to know how to identify and convert these units, especially if you are calculating complementary angles. This is why converting everything to degrees simplifies these mathematical operations, ensuring consistent and clear results.In practical applications, understanding angle measures helps in everything from construction to navigation, reinforcing why clear measures are vital in diverse fields.