Question

In Exercises 35–38, find the exact value of the following under the given conditions:e. cos( β/2)sin α = -1/3, 𝝅 \u003c α \u003c 3𝝅/2, and cos β = -1/3, 𝝅 \u003c β \u003c 3𝝅/2.

Accepted Answer

Identify the given information: $$ \sin \alpha = -\frac{1}{3} $$ with $$ \pi \u003c \alpha \u003c \frac{3\pi}{2} $$, and $$ \cos \beta = -\frac{1}{3} $$ with $$ \pi \u003c \beta \u003c \frac{3\pi}{2} . $$Both angles are in the third quadrant. Recall that in the third quadrant, both sine and cosine values are negative, which is consistent with the given values and intervals. Use the Pythagorean identity to find $$ \cos \alpha $$: $$ \cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} . $$The negative sign is chosen because cosine is negative in the third quadrant. Similarly, find $$ \sin \beta $$ using the Pythagorean identity: $$ \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} $$, since sine is also negative in the third quadrant. To find $$ \cos \frac{\beta}{2} $$, use the half-angle formula: $$ \cos \frac{\beta}{2} = \pm \sqrt{\frac{1 + \cos \beta}{2}}. $$ Determine the correct sign based on the quadrant where $$ \frac{\beta}{2} $$ lies, considering $$ \pi \u003c \beta \u003c \frac{3\pi}{2} .$$

In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos( β/2)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

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Key Concepts

Trigonometric Ratios and Their Signs in Different Quadrants

Exact Values of Trigonometric Functions

Using Trigonometric Identities to Find Unknown Values

In Exercises 35–38, find the exact value of the following under the given conditions:e. cos( β/2)sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.