Problem 56 - Algebra and Trigonometry

Question

Use a sketch to find the exact value of each expression. $$ \sec \left[\sin ^{-1}\left(-\frac{1}{2}\right)\right] $$

Step-by-Step Solution

Answer

The exact value of the expression \( \sec \left[\sin ^{-1}\left(-\frac{1}{2}\right)\right] \) is \( \frac{2\sqrt{3}}{3} \).

Step 1: Understand The Inverse Sine Function

The inverse sine function, also known as arcsin, is the inverse of the sine function. The arcsin of -1/2 is \(-\frac{\pi }{6}\) because \(sin(-\frac{\pi }{6}) = -\frac{1}{2}\). Therefore, we are looking the value of secant when \(\theta = -\frac{\pi }{6}\).

Step 2: Calculate the Value of Cosine

The secant of an angle in a right triangle is the hypotenuse divided by the adjacent side. Luckily, we know that cosine represents this ratio. Also, the cosine of \(-\frac{\pi }{6}\) can be found from its corresponding (30-60-90) right triangle where the cosine of the angle is equal to \(\frac{\sqrt{3}}{2}\).

Step 3: The Secant from Cosine

Since the secant function is the reciprocal function of the cosine function, we can find it by inverting the cosine value. Formally, \(\sec(x) = \frac{1}{\cos(x)}\). Thus, the secant of \(-\frac{\pi }{6}\) is the reciprocal of the cosine of \(-\frac{\pi }{6}\), giving \(\frac{2}{\sqrt{3}}\).

Step 4: Rationalize the Denominator

To express the result in standard mathematical form, multiply the numerator and denominator by \(\sqrt{3}\) to get rid of the square root in the denominator. This gives us \(\frac{2\sqrt{3}}{3}\).

Key Concepts

Secant Function

One of the six fundamental trigonometric functions is the secant, designated as 'sec'. Unlike the more commonly known sine and cosine functions, which relate to the sides of a right-angled triangle, the secant function is slightly different. It is defined as the reciprocal of the cosine function, meaning that for any angle \theta, the secant of \theta is the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle. In formula terms, it is expressed as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

When solving problems involving the secant function, remember that since it is the reciprocal of cosine, any time you have the cosine of an angle, you can easily find the secant by flipping the cosine ratio. However, there's an important aspect to consider: the domain and range of the secant function differ from those of cosine, as secant is not defined wherever cosine is zero to prevent division by zero.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities are invaluable tools for simplifying and calculating complex trigonometric expressions and solving trigonometric equations.

Some common trigonometric identities include the reciprocal identities like \( \sec(\theta) = \frac{1}{\cos(\theta)} \) as discussed earlier, the Pythagorean identities such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \), and angle sum identities like \( \sin(\theta + \phi) = \sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi) \). For students and mathematicians alike, memorizing these identities can prove extremely useful, particularly when solving trigonometric equations, proving other identities, or simplifying expressions before proceeding with calculus operations.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals from the bottom of a fraction. The main reason for doing this is to achieve a form that is typically considered more simplified or more aesthetically pleasing. It also can prevent round-off errors in numerical operations and make it easier to carry out further algebraic manipulations.

To rationalize a denominator that contains a square root, you multiply the numerator and denominator by the same square root that is in the denominator. For example, to rationalize \( \frac{2}{\sqrt{3}} \), multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \) to obtain \( \frac{2\sqrt{3}}{3} \). This technique is employed when the solution to an expression results in an 'unrationalized' form, such as in our exercise where the secant value initially had an irrational denominator.