Problem 30 - Algebra 2 and Trigonometry

Question

In \(24-32,\) find the exact value of each expression. $$ \tan \left(\arccos \left(-\frac{\sqrt{2}}{2}\right)\right) $$

Step-by-Step Solution

Answer

The exact value is -1.

Step 1: Identify the Angle Corresponding to the Inverse Cosine

Recognize that \( \arccos\left(-\frac{\sqrt{2}}{2}\right) \) gives the angle whose cosine is \(-\frac{\sqrt{2}}{2}\). This angle is in the second quadrant, where cosine is negative. The corresponding reference angle in the unit circle is \( \frac{3\pi}{4} \).

Step 2: Use the Unit Circle to Find the Sine and Cosine

Recall that for the angle \( \frac{3\pi}{4} \), the coordinates on the unit circle are \( \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \). Here, the cosine is \(-\frac{\sqrt{2}}{2}\) and the sine is \(\frac{\sqrt{2}}{2}\).

Step 3: Calculate the Tangent of the Angle

The tangent of an angle is the ratio of the sine to the cosine. Thus, \( \tan\left(\frac{3\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for finding angles when the values of trigonometric ratios are known. They act as the "reverse" of normal trigonometric functions, allowing us to work backward from a ratio to the angle itself.
To illustrate, \(\arccos(x)\) gives us an angle whose cosine is \(x\). Likewise, \(\arcsin(x)\) finds the angle with a given sine, and \(\arctan(x)\) provides the angle for a specified tangent ratio. Their main purpose is to simplify problems where determining angles is necessary.
Keep in mind that inverse trigonometric functions have particular ranges.

• For \(\arccos(x)\), the output (range) lies between \[0, \pi\]\ (inclusive).
• For \(\arcsin(x)\), the range is \[-\frac{\pi}{2}, \frac{\pi}{2}\].
• For \(\arctan(x)\), the range extends from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Understanding these ranges helps you to correctly interpret results and ensures accurate calculations. In the exercise provided, recognizing that \(\arccos(-\frac{\sqrt{2}}{2})\) results in \(\frac{3\pi}{4}\), which places the angle in the correct quadrant, is key.
This correct recognition allows subsequent calculations to be precise.

Unit Circle

The unit circle is a fundamental concept in trigonometry linking angles to their respective trigonometric ratios. It is a circle with a radius of one, centered at the origin of a coordinate system.
Each angle on the unit circle corresponds to a point \( (\cos \theta, \sin \theta) \). This simplifies visualizing trigonometric identities and relationships. For example, the point on the circle at the angle \(\frac{3\pi}{4}\) has coordinates \(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\).
These coordinates immediately tell us that:

• Cosine of the angle \(\theta\) is \(-\frac{\sqrt{2}}{2}\)
• Sine of the angle \(\theta\) is \(\frac{\sqrt{2}}{2}\)

The unit circle also assists with understanding signs of trigonometric functions in various quadrants. For instance, in the second quadrant where \(\theta = \frac{3\pi}{4}\), cosine is negative and sine is positive. Recognizing this property ensures accurate assessment of trigonometric functions< especially within problems requiring precise evaluation of inverse and normal trigonometric functions.

Angle Reference

Angle reference is often used to determine the specific angle on the unit circle that corresponds to a given trigonometric value. This generally involves understanding which quadrants angles reside in based on the trigonometric function used.
In the context of inverse trigonometric functions, like \(\arccos\), knowing the reference angle helps to ascertain the correct quadrant and thus the correct sign of sine, cosine or tangent. For instance, in the given exercise, determining that the reference angle is \(\frac{3\pi}{4}\) helps map to the unit circle's coordinates.
Using angle reference:

• Decide in which quadrant your angle lies. For cosine and \(\arccos\), if your output is negative, the angle is usually in the second quadrant, \(\left(\pi/2, \pi\right)\).
• Compute or relate it to common angles. Here, the known values for \(45^\circ = \frac{\pi}{4}\) or its multiples, as shown, facilitate this.

In summarizing, understanding angle reference gives a clearer directional viewpoint on selecting rational points and rationing to the most simplified trigonometric functions' evaluation, significantly enhancing problem-solving efficiency and accuracy.