Question

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) 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 and the intervals for $$ \alpha $$ and $$ \beta $$:
- $$ \sin \alpha = -\frac{1}{3} $$ with $$ \pi \u003c \alpha \u003c \frac{3\pi}{2} $$
- $$ \cos \beta = -\frac{1}{2} $$ with $$ \pi \u003c \beta \u003c \frac{3\pi}{2} $$

These intervals indicate that both angles are in the third quadrant. Recall the formula for $$ \cos(\alpha - \beta) $$:

$$
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
$$ Since $$ \sin \alpha  is $$given, find $$ \cos \alpha $$ using the Pythagorean identity:

$$
\cos^2 \alpha = 1 - \sin^2 \alpha
$$

Calculate $$ \cos \alpha $$ considering the quadrant (third quadrant means $$ \cos \alpha \u003c 0 ). $$Similarly, since $$ \cos \beta  is $$given, find $$ \sin \beta $$ using the Pythagorean identity:

$$
\sin^2 \beta = 1 - \cos^2 \beta
$$

Determine the sign of $$ \sin \beta $$ based on the quadrant (third quadrant means $$ \sin \beta \u003c 0 ). $$Substitute the values of $$ \cos \alpha $$, $$ \cos \beta $$, $$ \sin \alpha $$, and $$ \sin \beta $$ into the formula for $$ \cos(\alpha - \beta) $$ and simplify to find the exact value.

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

Verified video answer for a similar problem:

Key Concepts

Cosine of a Difference Formula

Determining the Sign of Trigonometric Functions Based on Quadrants

Using Pythagorean Identity to Find Missing Values

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