Question

In Exercises 55–58, use the given information to find the exact value of each of the following: c. tan(α/2)tan α = 4/3, 180° \u003c α \u003c 270°

Accepted Answer

Identify the given information: $$ 	an \alpha = \frac{4}{3} $$ and the angle $$ \alpha $$ lies in the third quadrant where $$ 180^\circ \u003c \alpha \u003c 270^\circ . $$Recall that in the third quadrant, both sine and cosine values are negative, but tangent is positive, which is consistent with the given $$ 	an \alpha = \frac{4}{3} . $$Use the identity for tangent of half an angle: $$ 	an \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}}  or $$use the formula $$ 	an \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha}  or$$ $$ 	an \frac{\alpha}{2} = \frac{1 - \cos \alpha}{\sin \alpha} . $$Choose the form that is easiest based on known values. Find $$ \sin \alpha $$ and $$ \cos \alpha $$ using the given $$ 	an \alpha = \frac{4}{3} . $$Since $$ 	an \alpha = \frac{\sin \alpha}{\cos \alpha} $$, set $$ \sin \alpha = -4k $$ and $$ \cos \alpha = -3k  ($$both negative in the third quadrant), then use the Pythagorean identity $$ \sin^2 \alpha + \cos^2 \alpha = 1  to $$solve for $$ k . $$Substitute the values of $$ \sin \alpha $$ and $$ \cos \alpha $$ into the half-angle formula chosen in step 3 to find $$ 	an \frac{\alpha}{2} . $$Determine the correct sign of $$ 	an \frac{\alpha}{2} $$ based on the quadrant where $$ \frac{\alpha}{2} $$ lies.

In Exercises 55–58, use the given information to find the exact value of each of the following:
c. tan(α/2)
tan α = 4/3, 180° < α < 270°

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Key Concepts

Tangent Function and Its Properties

Reference Angles and Quadrant Location

Half-Angle Formulas

In Exercises 55–58, use the given information to find the exact value of each of the following: c. tan(α/2)tan α = 4/3, 180° < α < 270°