Graph of cosh

Definition

The graph of cosh, or the hyperbolic cosine function, is a smooth curve that represents the relationship between an angle and its hyperbolic cosine value. This function is defined as $$ ext{cosh}(x) = \frac{e^x + e^{-x}}{2}$$, where $$e$$ is Euler's number. The graph has a characteristic shape resembling a parabola that opens upwards, highlighting its unique properties in hyperbolic geometry and connecting to concepts such as hyperbolic trigonometric identities and transformations.

5 Must Know Facts For Your Next Test

1. The graph of cosh is symmetric with respect to the y-axis, indicating that it is an even function.
2. The minimum value of cosh(x) occurs at x = 0, where cosh(0) equals 1.
3. As x approaches positive or negative infinity, the graph of cosh(x) increases without bound, demonstrating exponential growth.
4. The graph of cosh intersects the y-axis at (0, 1) and does not have any x-intercepts.
5. The relationship between cosh and sinh can be expressed through the identity $$\text{cosh}^2(x) - \text{sinh}^2(x) = 1$$.

Review Questions

• How does the shape of the graph of cosh relate to its properties as a function?
  • The graph of cosh resembles a parabola that opens upwards, illustrating that it is an even function with symmetry about the y-axis. Its lowest point at (0, 1) shows that the hyperbolic cosine never dips below 1. The growth of the graph towards infinity on both sides indicates that as x becomes large or small, the values of cosh(x) will increase exponentially.
• Discuss how the graph of cosh interacts with other hyperbolic functions, specifically sinh.
  • The graph of cosh and sinh are closely related, as they both represent fundamental hyperbolic functions. The graphs intersect at the origin and share similar growth patterns; however, while cosh(x) is always positive and symmetric, sinh(x) has points where it is negative and crosses the x-axis. The connection can be seen through their identities and how they both describe different aspects of hyperbolic geometry.
• Evaluate the significance of the graph of cosh in understanding hyperbolic identities and transformations.
  • The graph of cosh is crucial for visualizing hyperbolic identities and transformations because it illustrates how these functions behave geometrically. By understanding its properties and relationships with other functions like sinh, one can better grasp complex concepts in hyperbolic geometry. This understanding facilitates applications in various fields such as physics and engineering where these principles are applied to model real-world phenomena.

Related terms

sinh: The hyperbolic sine function, defined as $$\text{sinh}(x) = \frac{e^x - e^{-x}}{2}$$, which complements the cosh function and helps describe the relationships in hyperbolic geometry.

hyperbola: A type of conic section formed by the intersection of a plane and a double cone, which is closely related to the graphs of hyperbolic functions.

hyperbolic identities: Equations that establish relationships between hyperbolic functions, similar to trigonometric identities for regular sine and cosine functions.