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Chapter 6: Problem 23

Find the complement and the supplement. $$45.2^{\circ}$$

Short Answer

Expert verified
Complement: \(44.8^{\text{掳}}\), Supplement: \(134.8^{\text{掳}}\)

Step by step solution

01

Understand the Definitions

The complement of an angle is what, when added to the angle, will equal \(90^{\text{掳}}\). The supplement of an angle is what, when added to the angle, will equal \(180^{\text{掳}}\).
02

Find the Complement

Subtract \(45.2^{\text{掳}}\) from \(90^{\text{掳}}\) to find the complement. So, we have: \(90^{\text{掳}} - 45.2^{\text{掳}} = 44.8^{\text{掳}}\)
03

Find the Supplement

Subtract \(45.2^{\text{掳}}\) from \(180^{\text{掳}}\) to find the supplement. So, we have: \(180^{\text{掳}} - 45.2^{\text{掳}} = 134.8^{\text{掳}}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complement of an Angle
The complement of an angle is an important concept in geometry. When we talk about the complement, we mean another angle that, when added to the original angle, equals \(90^{\text{掳}}\). For example, if we have an angle of \(45.2^{\text{掳}}\), we can find its complement by subtracting it from \(90^{\text{掳}}\). This gives us:
  • \(90^{\text{掳}} - 45.2^{\text{掳}} = 44.8^{\text{掳}}\)

This means that the complement of \(45.2^{\text{掳}}\) is \(44.8^{\text{掳}}\). By finding the complement, we see how two angles can come together to form a right angle.
Complementary angles always add up to \(90^{\text{掳}}\), making them useful to know for various problems.
Supplement of an Angle
Another significant concept is the supplement of an angle. Unlike complements, supplements refer to another angle that, when combined with the original angle, equals \(180^{\text{掳}}\). For the angle \(45.2^{\text{掳}}\), the supplement is found by subtracting it from \(180^{\text{掳}}\):
  • \(180^{\text{掳}} - 45.2^{\text{掳}} = 134.8^{\text{掳}}\)

So, the supplement of \(45.2^{\text{掳}}\) is \(134.8^{\text{掳}}\).
Knowing about supplementary angles helps in understanding how angles relate, especially in polygons and other geometric shapes where the total degrees around a point needs to be \(360^{\text{掳}}\) or half that for a straight line.
Basic Trigonometry
Basic trigonometry may seem daunting at first, but it鈥檚 all about the relationships between the angles and sides of triangles. Here are some key points to remember:
  • Trigonometric functions such as sine, cosine, and tangent connect angles to the ratios of the sides of a right triangle.

  • Sine (sin) of an angle is the ratio of the length of the opposite side to the hypotenuse.

  • Cosine (cos) of an angle is the ratio of the length of the adjacent side to the hypotenuse.

  • Tangent (tan) of an angle is the ratio of the length of the opposite side to the adjacent side.

To solve problems involving trigonometry, it's critical to be comfortable with these functions and understand how to use them in different scenarios.
For example, to find how long a ramp should be to reach a certain height at a specific angle, we might use the sine function. Basic trigonometry forms the foundation for more advanced math and physics, making it an essential tool in your math toolbox.

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