Problem 49 - Precalculus : Building Concepts and Connections

Question

Find the exact values of the given expressions in radian measure. $$\csc ^{-1} \sqrt{2}$$

Step-by-Step Solution

Answer

\(\pi / 4\)

Step 1: Identify Known Value

Here the \(\csc^{-1}(x)\) equals \(\sqrt{2}\) needs to be found. To do this, write \(\csc(y) = \sqrt{2}\) instead.

Step 2: Use Cosecant Definition

Cosecant is the reciprocal of the sine function. Therefore, you can express this problem as \(\sin(y) = 1 / \sqrt{2}\).

Step 3: Identify Corresponding Angle

From trigonometry, the angle which has sine equals 1/\(\sqrt{2}\) is \(\pi / 4\) or 45 degrees. But here it is required in radians, so the answer is \(\pi / 4\).

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as csc, is one of the six fundamental trigonometric functions. It is closely related to the sine function and is defined as the reciprocal of the sine function. This means if you have a sine value \( ext{sin}(x) \), then the cosecant function is \( ext{csc}(x) = \frac{1}{ ext{sin}(x)} \).
In trigonometry, the cosecant function is particularly useful when you need to work with the reciprocals of sine values, especially when solving triangles and in calculus calculations.

• The domain of the cosecant function excludes angles where the sine value is zero, such as \(0, \pi, 2\pi\), etc., due to division by zero being undefined.
• The range of the cosecant function is all real numbers except the interval \((-1, 1)\), as these are the outputs where sine is non-zero.
• Cosecant, like other trigonometric functions, can be expressed and calculated in both degrees and radians.

Radian Measure

Radian measure is a fundamental concept in trigonometry and is essential for working with angles in a pure mathematical context. Unlike degrees, where a circle is divided into 360 parts, radians make use of the circle's circumference. One radian is defined as the angle created when the arc length is equal to the radius of the circle.
This relationship comes from the formula \( \text{Angle in radians} = \frac{\text{Arc Length}}{\text{Radius}} \). A full circle in radians corresponds to \(2\pi\), which is approximately 6.2832 radians. Consequently, one full turn or 360 degrees equals \(2\pi\) radians, and thus 180 degrees equals \(\pi\) radians.

• Using radian measure can simplify mathematical expressions, especially in calculus.
• Many trigonometric identities, limits, and derivatives work more naturally and elegantly when angles are expressed in radians.
• For converting degrees to radians, multiply the degree value by \( \frac{\pi}{180} \).

Trigonometric Identities

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. These identities are used to simplify complicated trigonometric expressions and solve equations. Here are some of the most commonly used trigonometric identities:

• Reciprocal Identities: These include relations like \( \csc(x) = \frac{1}{\sin(x)} \), \( \sec(x) = \frac{1}{\cos(x)} \), and \( \cot(x) = \frac{1}{\tan(x)} \).
• Pythagorean Identities: The basic form is \( \sin^2(x) + \cos^2(x) = 1 \). It can be derived further to \( 1 + \tan^2(x) = \sec^2(x) \) and \( 1 + \cot^2(x) = \csc^2(x) \).
• Angle Sum and Difference Identities: Useful for calculating the sine, cosine, or tangent of an angle presented as the sum or difference of two other angles. For example, \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \).

Understanding and using these identities helps in solving trigonometric equations, simplifying expressions, and can be crucial in calculus fields such as integration and differentiation.