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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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.

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Introduction to Trigonometric Functions

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.

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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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