In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)

Verified step by step guidance

1. Identify the angle given: the expression is \( \tan\left(\frac{7\pi}{8}\right) \). Notice that \( \frac{7\pi}{8} \) is an angle between \( \frac{\pi}{2} \) and \( \pi \), so it is in the second quadrant.
2. Express the angle \( \frac{7\pi}{8} \) as a half-angle. Since \( \frac{7\pi}{8} = \frac{1}{2} \times \frac{7\pi}{4} \), we can write \( \theta = \frac{7\pi}{4} \) and use the half-angle formula for tangent.
3. Recall the half-angle formula for tangent: \[ \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \] The sign depends on the quadrant of \( \frac{\theta}{2} \). Since \( \frac{7\pi}{8} \) is in the second quadrant, tangent is negative there.
4. Calculate \( \cos(\theta) \) where \( \theta = \frac{7\pi}{4} \). Use known values or the unit circle to find \( \cos\left(\frac{7\pi}{4}\right) \).
5. Substitute \( \cos(\theta) \) into the half-angle formula and apply the correct sign to find \( \tan\left(\frac{7\pi}{8}\right) \). This will give the exact value of the expression.

Key Concepts

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the functions of the original angle. For tangent, the formula is tan(θ/2) = ±√((1 - cos θ) / (1 + cos θ)) or tan(θ/2) = sin θ / (1 + cos θ). These formulas help find exact values when the angle is a fraction of a known angle.

Reference Angles and Angle Reduction

To apply half-angle formulas effectively, it's important to recognize the given angle in terms of a known angle or multiple of π. Reducing 7π/8 to a related angle helps identify the correct cosine or sine values needed for the formula, ensuring the exact value is found.

Sign Determination in Trigonometric Functions

When using half-angle formulas, the sign of the result depends on the quadrant where the half-angle lies. Since 7π/8 is in the second quadrant, its half (7π/16) lies in the first quadrant, where tangent is positive. Correct sign choice is crucial for the exact value.