In Exercises 55–58, use the given information to find the exact value of each of the following:
c. tan(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅

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Identify the given information: \(\sec \alpha = -3\) and \(\frac{\pi}{2} < \alpha < \pi\). This means \(\alpha\) is in the second quadrant where cosine is negative.

Recall the relationship between secant and cosine: \(\sec \alpha = \frac{1}{\cos \alpha}\). Use this to find \(\cos \alpha\) by taking the reciprocal of \(\sec \alpha\).

Since \(\cos \alpha\) is the reciprocal of \(\sec \alpha\), calculate \(\cos \alpha = \frac{1}{-3} = -\frac{1}{3}\).

Use the Pythagorean identity to find \(\sin \alpha\): \(\sin^2 \alpha + \cos^2 \alpha = 1\). Substitute \(\cos \alpha = -\frac{1}{3}\) and solve for \(\sin \alpha\).

Determine the sign of \(\sin \alpha\) based on the quadrant. Since \(\alpha\) is in the second quadrant, \(\sin \alpha\) is positive. Finally, find \(\tan \frac{\alpha}{2}\) using the half-angle formula: \(\tan \frac{\alpha}{2} = \frac{1 - \cos \alpha}{\sin \alpha}\) or \(\tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha}\). Choose the appropriate formula and substitute the values to express \(\tan \frac{\alpha}{2}\) exactly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Functions and Their Relationships

Trigonometric functions like sine, cosine, tangent, secant, etc., are ratios of sides in a right triangle or points on the unit circle. Understanding how these functions relate, such as tan(α) = sin(α)/cos(α) and sec(α) = 1/cos(α), is essential for finding unknown values from given information.

Using the Unit Circle and Angle Quadrants

The unit circle helps determine the sign and value of trig functions based on the angle's quadrant. Since α is between π/2 and π (second quadrant), cosine and secant are negative, while sine is positive. This knowledge guides the correct sign choice when calculating values.

Half-Angle Formulas

Half-angle formulas allow finding trig function values of α/2 using known values of α. For tangent, the formula tan(α/2) = ±√((1 - cos α)/(1 + cos α)) or tan(α/2) = sin α / (1 + cos α) is used, with the sign determined by the quadrant of α/2.

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