Problems 3-8 require the Symbolic Math Toolbox. The others...

$$ \text { Problems 3-8 require the Symbolic Math Toolbox. The others do not. } $$ Compute to 15 digits the following quantities. (a) $\cosh (0.1)$. (b) $\ln (2)$. (Hint: the natural logarithm in MATLAB is called $l o g$, not $\ln$.) (c) $\arctan (1 / 2)$. (Hint: the inverse tangent function in MATLAB is called atan, not arctan.)

$$
\text { Problems 3-8 require the Symbolic Math Toolbox. The others do not. }
$$
Compute to 15 digits the following quantities.
(a) $\cosh (0.1)$.
(b) $\ln (2)$. (Hint: the natural logarithm in MATLAB is called $l o g$, not $\ln$.)
(c) $\arctan (1 / 2)$. (Hint: the inverse tangent function in MATLAB is called atan, not arctan.)

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as cosh, are analogs of the ordinary trigonometric functions but for a hyperbola instead of a circle. They are defined using exponential functions and arise in many areas including differential equations, complex analysis and physics. In high-precision computation, evaluating such functions accurately is crucial for ensuring correct results in sensitive calculations.

Logarithmic Functions

Logarithmic functions, specifically the natural logarithm, play a fundamental role in mathematics. They convert multiplicative relationships into additive relationships and are essential in solving exponential equations. Accurate high-precision computation of logarithms is important in numerical methods, scientific computing, and engineering applications.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, allow the determination of an angle given the ratio of sides of a right triangle. Their computation is central in many problems involving geometry, calculus and complex analysis. High-precision evaluation of these functions ensures that subsequent computations, like solving equations or modeling scenarios, are accurate.

High-Precision Arithmetic

High-precision arithmetic refers to the computational techniques and tools that calculate mathematical expressions to many significant digits. In fields that demand precise numerical results, standard floating-point arithmetic may be inadequate, thus employing arbitrary precision computing, as provided by tools like the Symbolic Math Toolbox, is necessary to avoid rounding errors and maintain accuracy.

Symbolic Computation

Symbolic computation involves manipulating mathematical expressions in symbolic form rather than in numerical form. This allows for exact representations and transformations, which is especially helpful when high precision is required. Tools offering symbolic capabilities enable computations that can achieve a specified number of digits of accuracy, beneficial in both theoretical and applied contexts.

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