Problem 45 - Precalculus Essentials

Question

In Exercises \(39-46,\) use a half-angle formula to find the exact value of each expression. $$\tan \frac{7 \pi}{8}$$

Step-by-Step Solution

Answer

\(\tan \frac{7\pi}{8} = \sqrt{2} - 1\)

Step 1: Find the cos value

Determine the cos value of \(7\pi /4\), which is the double angle for \(7\pi /8\). We know that \(\cos(7\pi /4) = \sqrt{2}/2\).

Step 2: Identify the quadrant

Recognize that the half-angle, \(7\pi /8\), is in the second quadrant where tangent values are positive.

Step 3: Apply half-angle formula

Substitute \(\cos 7\pi /4\) into the half-angle identity for tangent. Therefore \[ \tan \left(\frac{7\pi }{8}\right) = \sqrt{\frac{1- \cos (7\pi /4)}{1+ \cos (7\pi /4)}} = \sqrt{\frac{1-\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}}\].

Step 4: Simplify the expression

Simplify the above expression to get your answer. \[\tan \left(\frac{7\pi }{8}\right) = \sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}} = \sqrt{2}-1\] .

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities help us to simplify complex trigonometric expressions and solve equations. The half-angle formulas are a special type of trigonometric identities used to find the sine, cosine, or tangent of an angle that is half of a known angle.
For example, if you know the cosine of a particular angle, the half-angle formula can be utilized to find the tangent. In our exercise, the half-angle formula for tangent is applied. It states:

• \( \tan \left( \frac{\theta}{2} \right) = \pm \sqrt{ \frac{1 - \cos \theta}{1 + \cos \theta} } \)

The plus or minus sign depends on the quadrant in which \( \frac{\theta}{2} \) lies. This identity helps us turn a problem into a step-by-step process and find exact values rather than decimal approximations.

Exploring the Tangent Function

The tangent function, denoted as \( \tan \), is one of the main trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. Geometrically, it represents the slope of the angle with respect to the horizontal.
When dealing with radians and the unit circle, tangent can also be represented by the ratio of sine to cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This property of tangent is especially useful in analytical approaches, such as when using half-angle identities. Remembering that tangent has a periodicity of \( \pi \), the function repeats its values every \( \pi \) radians. This understanding helps in predicting its behavior across different angles. Additionally, knowing that tangent changes signs depending on the quadrant it is located in, aids in determining whether we're dealing with positive or negative values.

Navigating Trigonometric Quadrants

The unit circle is divided into four quadrants, each hosting specific behaviors for the trigonometric functions sine, cosine, and tangent. Identifying the correct quadrant is crucial for determining the signs of these functions:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, but cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, with sine and tangent negative.

In our specific exercise, the angle \( \frac{7\pi}{8} \) places us in the second quadrant. Here, the sine function remains positive, but tangent changes sign due to its dependence on both sine and cosine. Therefore, understanding the quadrant location helps determine the sign of the solution. In general, being comfortable with the idea of trigonometric quadrants ensures accurate problem-solving for angles located in different quadrants.