Csc(60°)

Definition

The term csc(60°) refers to the cosecant of a 60-degree angle, which is the reciprocal of the sine function for that angle. In the context of the unit circle, it represents the ratio of the length of the hypotenuse to the length of the opposite side of a right triangle formed by that angle. This connection to the unit circle helps in visualizing and understanding how trigonometric functions relate to angles and their respective ratios.

5 Must Know Facts For Your Next Test

1. The sine of 60 degrees is $$\frac{\sqrt{3}}{2}$$, so csc(60°), being its reciprocal, equals $$\frac{2}{\sqrt{3}}$$.
2. In its rationalized form, csc(60°) can also be expressed as $$\frac{2\sqrt{3}}{3}$$ after multiplying the numerator and denominator by $$\sqrt{3}$$.
3. Cosecant is undefined for angles where sine equals zero; however, csc(60°) is defined because sine is non-zero at this angle.
4. Understanding csc(60°) helps in solving problems involving right triangles where angle measures and side lengths are involved.
5. Cosecant can also be used in calculus, particularly in integration and differentiation involving trigonometric functions.

Review Questions

• How do you derive csc(60°) using its relationship to sine?
  • To derive csc(60°), start by recalling that cosecant is the reciprocal of sine. First, find sin(60°), which equals $$\frac{\sqrt{3}}{2}$$. Since cosecant is defined as csc(θ) = $$\frac{1}{sin(θ)}$$, substituting gives csc(60°) = $$\frac{1}{\frac{\sqrt{3}}{2}}$$. This simplifies to csc(60°) = $$\frac{2}{\sqrt{3}}$$.
• Explain how csc(60°) can be visualized using the unit circle.
  • In the unit circle, an angle of 60 degrees corresponds to a point at coordinates $$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$. The y-coordinate represents sin(60°), which is $$\frac{\sqrt{3}}{2}$$. Cosecant being the reciprocal means we take the hypotenuse (which is 1 in the unit circle) divided by this y-coordinate. Thus, csc(60°) can be visualized as measuring how many times the opposite side (the height) fits into the hypotenuse in a right triangle formed at this angle.
• Evaluate how understanding csc(60°) enhances your ability to solve trigonometric equations or real-world problems.
  • Understanding csc(60°) not only aids in calculating values for trigonometric equations but also strengthens your grasp on concepts such as triangle properties and relationships between angles and sides. When faced with real-world problems involving heights and distances, knowing that csc(60°) relates to a triangle's dimensions allows you to apply this knowledge practically. It also provides a foundation for more complex applications in calculus, physics, and engineering where trigonometric functions are prevalent.

Related terms

Sine Function: A trigonometric function that gives the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

Unit Circle:

A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions in terms of angles and coordinates.

Reciprocal Functions: Functions that give the reciprocal value of another function; for example, cosecant is the reciprocal of sine.