key term - Csc(45°)

Definition

The term csc(45°) refers to the cosecant of a 45-degree angle, which is defined as the reciprocal of the sine function for that angle. In the context of the unit circle, this means that csc(45°) can be calculated by finding the sine value at 45 degrees and then taking its reciprocal. This concept connects to fundamental properties of right triangles, the unit circle's coordinates, and various trigonometric identities.

5 Must Know Facts For Your Next Test

1. csc(45°) is equal to 1 because sin(45°) is also equal to \(\frac{\sqrt{2}}{2}\), and its reciprocal is \(\frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\).
2. The angle 45 degrees corresponds to an angle in both the first quadrant and where sine and cosine values are equal.
3. In a 45-45-90 triangle, both legs are equal, which means that if the length of each leg is 1, the hypotenuse will be \(\sqrt{2}\).
4. Understanding csc(45°) helps reinforce the relationship between different trigonometric functions and their reciprocals.
5. This value can also be confirmed using the unit circle, where at 45 degrees (or \(\frac{\pi}{4}\) radians), both x and y coordinates are \(\frac{\sqrt{2}}{2}\).

Review Questions

• How does csc(45°) relate to the sine function and what is its significance in trigonometry?
  • csc(45°) is directly related to the sine function because it is defined as its reciprocal. Since sin(45°) equals \(\frac{\sqrt{2}}{2}\), it follows that csc(45°) equals \(\sqrt{2}\). This relationship highlights how different trigonometric functions interconnect and reinforces the understanding of their properties in solving problems involving right triangles.
• Explain how to find csc(45°) using both the unit circle and properties of a 45-45-90 triangle.
  • To find csc(45°), you can look at the unit circle where at this angle, the coordinates are both \(\frac{\sqrt{2}}{2}\). The sine value is equal to \(\frac{\sqrt{2}}{2}\), so taking its reciprocal gives you csc(45°) = \(\sqrt{2}\). Alternatively, in a 45-45-90 triangle with legs of length 1, the hypotenuse will be \(\sqrt{2}\), leading again to csc(45°) being \(\sqrt{2}\).
• Analyze how understanding csc(45°) enhances your grasp of other trigonometric functions and identities.
  • Understanding csc(45°) not only gives insight into this specific function but also builds a foundation for learning about other trigonometric identities and relationships. For example, knowing that csc is the reciprocal of sine helps in manipulating equations involving these functions. This knowledge can be applied to various scenarios in math and physics where angles and distances are calculated using trigonometric ratios, allowing for greater flexibility in problem-solving.

Related terms

Sine: The sine function, often abbreviated as sin, is a fundamental trigonometric function that gives the ratio of the length of the opposite side to the hypotenuse in a right triangle.

Unit Circle:

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane, used to define trigonometric functions in terms of angles.

Reciprocal: The reciprocal of a number is one divided by that number. For instance, the reciprocal of sin(θ) is csc(θ).