Problem 59 - Algebra and Trigonometry

Question

Use a sketch to find the exact value of each expression. $$ \csc \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right] $$

Step-by-Step Solution

Answer

The exact value of \( \csc \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right] \) is \(2\).

Step 1: Determine the value of \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

From knowledge of the unit circle, the value that makes the cosine equal to \(-\frac{\sqrt{3}}{2}\) is \( \frac{5\pi}{6}\) or \( \frac{300^o}{1}\). Hence \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6}\).

Step 2: Determine the sine value of the angle obtained

From the right triangle definitions of sine and cosine, sine at angle \( \frac{5\pi}{6}\) or \( \frac{300^o}{1}\) can be determined by constructing a right triangle with an angle \( \frac{5\pi}{6}\). The sine of an angle in a right triangle is opposite/hypotenuse. The side opposite \( \frac{5\pi}{6}\) is \(1\) and the hypotenuse is \(2\). Hence \(\sin \left(\frac{5\pi}{6}\right) = \frac{1}{2}\).

Step 3: Determine the cosecant of the angle

The cosecant of an angle is the reciprocal of the sine value. Hence \(\csc \left(\frac{5\pi}{6}\right) = \frac{1}{\sin \left(\frac{5\pi}{6}\right)} = 2\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \(\csc\), is a key concept in trigonometry. It is the reciprocal of the sine function. When you have a sine value, you can find its cosecant by taking the reciprocal. For example, if \(\sin \theta = \frac{1}{2}\), then \(\csc \theta = \frac{1}{\frac{1}{2}} = 2\). This function is especially useful when dealing with reciprocal trigonometric identities and can simplify complex equations.
To remember:

• \(\csc \theta = \frac{1}{\sin \theta}\)
• The sine of an angle is needed to find the cosecant.
• Cosecant is undefined when sine is zero, because division by zero is impossible.

If you're given an angle, you always use the sine of that angle to find the cosecant. In our problem, for instance, the angle \(\frac{5\pi}{6}\) has a sine of \(\frac{1}{2}\), leading to a cosecant of 2.

Inverse Cosine

Inverse cosine, also known as \(\cos^{-1}\) or arccosine, is the function that helps find the angle whose cosine is a given number. It's the opposite of the cosine function. For example, if you know that \(\cos \theta = x\), the inverse cosine function can find the angle \(\theta\).
Key points to note:

• The range of \(\cos^{-1}(x)\) is from 0 to \(\pi\) (or 0 to 180 degrees), covering only the top half of the unit circle.
• \(\cos^{-1}(x)\) can only take values between -1 and 1, because these are the limits of the cosine function.

In the exercise shared, you determine that \(\cos^{-1}(-\frac{\sqrt{3}}{2})\) translates to the angle \(\frac{5\pi}{6}\), as it is known from trigonometric tables that \(\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}\).

Right Triangle

Right triangles are integral to trigonometry and are used to define the basic trigonometric ratios: sine, cosine, and tangent. A right triangle has one angle that is 90 degrees. In these triangles, the relationship between the sides can be described using the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
Trigonometric functions help you find unknown side lengths or angle measures in right triangles. When dealing with angles in the unit circle, you can use right triangle definitions to find sine and cosine. For the angle \(\frac{5\pi}{6}\), you visualize a right triangle where:

• The hypotenuse is 2 (a common radius in trigonometry).
• The side opposite the angle (sine aspect) is 1.

This setup allows you to find the sine as \(\sin(\frac{5\pi}{6})=\frac{1}{2}\), facilitating the calculation of other trigonometric functions like the cosecant.