Problem 42 - Precalculus

Question

Evaluate each expression, if possible. $$\sin \left(-\frac{5 \pi}{2}\right)+\csc \left(\frac{3 \pi}{2}\right)$$

Step-by-Step Solution

Answer

The expression evaluates to -2.

Step 1: Understanding Negative Angles

Evaluate the sine of negative angles: The sine function is an odd function, meaning \[\sin(-x) = -\sin(x). \] Applying this to the given angle: \[\sin\left(-\frac{5\pi}{2}\right) = -\sin\left(\frac{5\pi}{2}\right). \]

Step 2: Finding the Sine of an Angle Beyond \(2\pi\)

Since \(\frac{5\pi}{2}\) is beyond \(2\pi\), reduce it by \(2\pi\) to find an equivalent angle within \([0, 2\pi)\). Calculating: \[ \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2}. \] Hence, \(\sin\left(\frac{5\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1. \)

Step 3: Calculating Sine of the Negative Angle

Substitute back to get sine of the negative angle: \[\sin\left(-\frac{5\pi}{2}\right) = -\sin\left(\frac{\pi}{2}\right) = -1.\]

Step 4: Evaluate Cosecant \(\csc\left(\frac{3\pi}{2}\right)\)

The cosecant is the reciprocal of the sine, i.e., \(\csc(x) = \frac{1}{\sin(x)}. \) Determine the sine first: \[ \sin\left(\frac{3\pi}{2}\right) = -1. \] Hence, the cosecant: \[\csc\left(\frac{3\pi}{2}\right) = \frac{1}{-1} = -1.\]

Step 5: Combine Results

Now add the results: \[\sin\left(-\frac{5\pi}{2}\right) + \csc\left(\frac{3\pi}{2}\right) = -1 + (-1) = -2.\]

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that expresses the relationship between an angle in a right triangle and the ratio of the length of the opposite side to the hypotenuse. Its importance extends far beyond triangles, as it's recurringly useful in wave mechanics, alternating current theory, and much more. Understanding the characteristics of the sine function can vastly improve your grasp of various mathematical concepts.

Here are some crucial properties of the sine function:

• Periodicity: It is periodic, repeating its values every \(2\pi\). This means \(\sin(x + 2\pi) = \sin(x)\), for any angle \(x\).
• Odd Function: The sine function is an odd function, so \(\sin(-x) = -\sin(x)\).
• Range: The function's output lies between -1 and 1.
• Key Angles: At 0, \(\pi\), and \(2\pi\), \(\sin(x) = 0\). At \(\frac{\pi}{2}\), \(\sin(x) = 1\), and at \(\frac{3\pi}{2}\),
  \(\sin(x) = -1\).

In trigonometric identities and various calculations, recognizing these properties simplifies resolving complex angle expressions. For example, knowing \(\sin(-x) = -\sin(x)\) allows for quick computation of negative angles.

Cosecant Function

The cosecant function is the reciprocal of the sine function. It's usually symbolized as \(\csc(x)\), which equals \(\frac{1}{\sin(x)}\). Understanding the cosecant function requires a foundation in the sine function, because it directly builds upon those values.

Here’s what you need to know about the cosecant function:

• Reciprocal Nature: Being the reciprocal, \(\csc(x)\) is undefined wherever \(\sin(x) = 0\). Thus, at values like \(x = 0, \pi, 2\pi\), which make the denominator zero and the function undefined.
• Range of Values: The cosecant values are greater than 1 or less than -1, reflecting its reciprocal nature (as long as the sine value is nonzero).
• Key Angles: Understanding the sine's behavior leads to deductions about cosecant; for example, in \(\csc\left(\frac{3\pi}{2}\right) = \frac{1}{-1} = -1\).

Use the cosecant function in problems involving trigonometric identities or when simplifying expressions where \(\frac{1}{\sin(x)}\) is present. Grasping its properties streamlines those tasks, aiding in a wide array of applications from pre-calculus to engineering mathematics.

Angle Reduction

Angle reduction is an essential technique when dealing with angles that exceed a full rotational cycle or need simplification. It ensures that trigonometric functions provide equivalent values within a manageable range, usually between \(0\) and \(2\pi\) or \(0\) and \(360^\circ\) degrees.

Here's how angle reduction typically works:

• Reduction by Full Circles: Because trigonometric functions are periodic, reducing angles by \(2\pi\) or \(360^\circ\) brings them within the range for simpler calculation. For instance, \(\frac{5\pi}{2}\) can be reduced by subtracting \(2\pi\) once to become \(\frac{\pi}{2}\).
• Utilizing Symmetry: Functions like sine and cosine showcase symmetry, allowing for reflection and rotation observations, supporting angle reduction even if initial values are negative or beyond standard ranges.
• Practical Calculation: Frequently reducing angles aids in solving equations or evaluating expressions accurately.Consider \(\sin\left(-\frac{5\pi}{2}\right)\) where \(\sin\left(\frac{5\pi}{2}\right)\) translates into \(\sin\left(\frac{\pi}{2}\right)\) because of the periodicity after reduction.

Mastering angle reduction fuses seamlessly with the principles of periodicity and symmetry of trigonometric functions, making computations less daunting and more intuitive in complex problems.