Problem 45 - Algebra and Trigonometry

Question

Use a half-angle formula to find the exact value of each expression. $$ \tan \frac{7 \pi}{8} $$

Step-by-Step Solution

Answer

\(\tan(\frac{7 \pi }{8})\) is undefined

Step 1: Determine the Current Angle

First, determine which standardized angle, in terms of pi, matches up with \(7 \pi / 8\). It's \( \pi \), or 180 degrees, because \(7 \pi / 8\) is greater than half of pi but smaller than pi.

Step 2: Apply Half-Angle Formula

Next, apply the half-angle formula which is \(\tan(\frac{x}{2}) = \pm \sqrt{ \frac{1 - \cos(x)}{1 + \cos(x)}}\). So, let's replace x with \( \pi \) in the formula: \(\tan(\frac{7 \pi }{8}) = \sqrt{ \frac{1 - \cos(\pi)}{1 + \cos(\pi)}}\).

Step 3: Determine Cosine Value

The value of \(\cos(\pi)\) is -1. Substitute -1 into the formula: \(\tan(\frac{7 \pi }{8}) = \sqrt{ \frac{1 - (-1)}{1 + (-1)}}\).

Step 4: Calculate the Value

After substitution, we simplify within the square root to obtain \(\tan(\frac{7 \pi }{8}) = \sqrt{\frac{2}{0}}\), which means \(\tan(\frac{7 \pi }{8})\) is undefined.

Step 5: Check Quadrant

The final factor to consider is the quadrant of our original angle. A simple way to do this is to realise that \(7 \pi / 8\) is more than \(\pi / 2\) but less than \(\pi \). This places it in the second quadrant, where tangent is positive. Thus, the final answer will be positive.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that describe relationships between the angles and sides of a triangle. They are essential tools in trigonometry for simplifying expressions and solving equations.

In the context of the half-angle formula for tangent, we deal with the identity:

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

This identity allows us to calculate the tangent of half an angle given the cosine of the full angle.

The "+" or "-" sign in the formula depends on the angle's quadrant, which affects the sign of the tangent.By understanding trigonometric identities, you can solve a wide variety of problems like the one posed in our exercise with greater ease. They form the backbone of trigonometric problem-solving strategies and are widely applied in fields such as physics, engineering, and navigation.

Tangent Function

The tangent function, often abbreviated as \(\tan\), is one of the primary trigonometric functions. It can be defined as the ratio of the sine and cosine of an angle:

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

Understanding the tangent function involves knowing its properties, like where it is positive or negative across different quadrants.

The half-angle formula helps us find values of tangent at angles that are not easily accessible through the basic unit circle or standard angle measures.In the original problem, we apply the tangent half-angle identity to determine the exact value for \(\tan\left(\frac{7 \pi}{8}\right)\). This angle doesn't have a standard sine or cosine value, making the half-angle formula an effective method for calculation.

This deepens your understanding of how tangent varies not just along the standard angles but also at their halves and quarters.

Quadrant Analysis

Understanding quadrant analysis is vital when dealing with trigonometric functions because they change signs based on their position in the unit circle. Each quadrant determines whether the values for sine, cosine, and tangent are positive or negative.

The unit circle is divided into four quadrants:

• Quadrant I: both sine and cosine are positive, so \( \tan \) is positive.
• Quadrant II: sine is positive, cosine is negative, so \( \tan \) is negative.
• Quadrant III: both sine and cosine are negative, so \( \tan \) is positive.
• Quadrant IV: sine is negative, cosine is positive, so \( \tan \) is negative.

In our exercise, \(\frac{7\pi}{8}\) lies in the second quadrant (since it's between \(\frac{\pi}{2}\) and \(\pi\)), which normally means the tangent is negative. However, due to a calculation quirk with half-angle formulas, checking the quadrant ensures that the final result aligns with the defined rules of trigonometric signs.