Problem 135 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (if possible) the complement and supplement of each angle. (a) \(55^{\circ}\) (b) \(162^{\circ}\)

Step-by-Step Solution

Answer

The complement and supplement of \(55^{\circ}\) are \(35^{\circ}\) and \(125^{\circ}\) respectively. \(162^{\circ}\) has no complement and its supplement is \(18^{\circ}\).

Step 1: Calculate complement and supplement for \(55^{\circ}\)

Applying the formulas, the complement of \(55^{\circ}\) is \(90^{\circ} - 55^{\circ} = 35^{\circ}\) and the supplement of \(55^{\circ}\) is \(180^{\circ} - 55^{\circ} = 125^{\circ}\)

Step 2: Calculate complement and supplement for \(162^{\circ}\)

Since \(162^{\circ}\) is greater than \(90^{\circ}\), it has no complement. The supplement of \(162^{\circ}\) is \(180^{\circ} - 162^{\circ} = 18^{\circ}\).

Key Concepts

Complementary Angles

Complementary angles are two angles that add up to a total of 90 degrees. This means that if you know one angle, you can easily determine its complement by subtracting the given angle from 90. Imagine you have an angle measuring 30 degrees; its complement would be calculated as follows:

• Complement of 30 degrees: \( 90^{\circ} - 30^{\circ} = 60^{\circ} \)

Complementary angles are especially useful in geometry and trigonometry because they help us understand the relationships between angles—particularly in a right triangle, where the two non-right angles are complementary.
In the exercise provided, the angle of 55 degrees has a complement since it's less than 90 degrees:

• Complement of 55 degrees: \( 90^{\circ} - 55^{\circ} = 35^{\circ} \)

However, if an angle itself is greater than 90 degrees, like 162 degrees, it cannot have a complement because such a scenario would exceed the 90-degree limit.

Supplementary Angles

Supplementary angles are two angles that together make a straight line, adding up to a total of 180 degrees. This is a key concept in understanding and solving many problems in trigonometry and geometry.
To find the supplement of an angle, you subtract the known angle from 180 degrees. For example, if you have an angle measuring 110 degrees, you would find its supplement like this:

• Supplement of 110 degrees: \( 180^{\circ} - 110^{\circ} = 70^{\circ} \)

Supplementary angles are often involved in the study of linear pairs and can be found in various geometrical shapes, such as triangles and quadrilaterals.
In the original exercise, the angle of 55 degrees is less than 180 degrees, so it can have a supplement:

• Supplement of 55 degrees: \( 180^{\circ} - 55^{\circ} = 125^{\circ} \)

Even for angles greater than 90 but less than 180, like 162 degrees, the concept still applies:

• Supplement of 162 degrees: \( 180^{\circ} - 162^{\circ} = 18^{\circ} \)

Angle Calculation

Understanding how to calculate angles is a fundamental skill in trigonometry. To effectively solve angle-related problems, you need to be familiar with both complementary and supplementary angles and apply them to various scenarios.
The calculation is straightforward:

• For complementary angles, subtract the angle from 90 degrees.
• For supplementary angles, subtract the angle from 180 degrees.

These calculations can be particularly useful when dealing with real-world applications like designing objects and understanding structures that require precise measurements, such as bridges and buildings.
Let's look at the problem in the exercise:

• 55 degrees has a complement: \( 90^{\circ} - 55^{\circ} = 35^{\circ} \) and a supplement: \( 180^{\circ} - 55^{\circ} = 125^{\circ} \).
• 162 degrees cannot have a complement but has a supplement: \( 180^{\circ} - 162^{\circ} = 18^{\circ} \).

Mastering these concepts allows for easy problem-solving and enhances one’s understanding of geometrical principles.