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Compute the exact value of tan(255 °)

Compute the exact value of tan(255 °)

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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**Compute the Exact Value of \(\tan(255^\circ)\)** To solve this problem, we need to understand how to find the tangent of an angle that isn't one of the common angles found on the unit circle. We can use trigonometric identities and angle transformations to compute this value precisely. 1. **Identify the Reference Angle**: - The reference angle for \(\theta = 255^\circ\) is \(255^\circ - 180^\circ = 75^\circ\). 2. **Use Trigonometric Identities**: - \(\tan(255^\circ) = \tan(180^\circ + 75^\circ)\). - By using the identity \(\tan(180^\circ + \theta) = \tan(\theta)\), we find that: - \(\tan(255^\circ) = \tan(75^\circ)\). 3. **Acquiring the Exact Value of \(\tan(75^\circ)\)**: - \(\tan(75^\circ) = \tan(45^\circ + 30^\circ)\). - Using the tangent addition formula \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\): - Let \(a = 45^\circ\) and \(b = 30^\circ\). - \(\tan 45^\circ = 1\) and \(\tan 30^\circ = \frac{\sqrt{3}}{3}\). - \(\tan(75^\circ) = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \times \frac{\sqrt{3}}{3}}\). - Simplifying gives \(\tan(75^\circ) = 2 + \sqrt{3}\). 4. **Conclusion**: - Therefore, \(\tan(255^\circ) = 2 + \sqrt{3}\). This approach combines multiple trigonometric properties and identities to achieve the exact value of the tangent function for a non-standard angle.
Transcribed Image Text:**Compute the Exact Value of \(\tan(255^\circ)\)** To solve this problem, we need to understand how to find the tangent of an angle that isn't one of the common angles found on the unit circle. We can use trigonometric identities and angle transformations to compute this value precisely. 1. **Identify the Reference Angle**: - The reference angle for \(\theta = 255^\circ\) is \(255^\circ - 180^\circ = 75^\circ\). 2. **Use Trigonometric Identities**: - \(\tan(255^\circ) = \tan(180^\circ + 75^\circ)\). - By using the identity \(\tan(180^\circ + \theta) = \tan(\theta)\), we find that: - \(\tan(255^\circ) = \tan(75^\circ)\). 3. **Acquiring the Exact Value of \(\tan(75^\circ)\)**: - \(\tan(75^\circ) = \tan(45^\circ + 30^\circ)\). - Using the tangent addition formula \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\): - Let \(a = 45^\circ\) and \(b = 30^\circ\). - \(\tan 45^\circ = 1\) and \(\tan 30^\circ = \frac{\sqrt{3}}{3}\). - \(\tan(75^\circ) = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \times \frac{\sqrt{3}}{3}}\). - Simplifying gives \(\tan(75^\circ) = 2 + \sqrt{3}\). 4. **Conclusion**: - Therefore, \(\tan(255^\circ) = 2 + \sqrt{3}\). This approach combines multiple trigonometric properties and identities to achieve the exact value of the tangent function for a non-standard angle.
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