Question

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) tan α = 3/4, 𝝅 \u003c α \u003c 3𝝅/2, and cos β = 1/4, 3𝝅/2 \u003c β \u003c 2𝝅

Accepted Answer

Identify the given information: $$ 	an \alpha = \frac{3}{4} $$ with $$ \pi \u003c \alpha \u003c \frac{3\pi}{2} $$, and $$ \cos \beta = \frac{1}{4} $$ with $$ \frac{3\pi}{2} \u003c \beta \u003c 2\pi . $$Determine the signs of sine and cosine for angles $$ \alpha $$ and $$ \beta $$ based on their quadrant locations. Since $$ \pi \u003c \alpha \u003c \frac{3\pi}{2} $$, $$ \alpha  is in $$the third quadrant where sine and cosine are both negative. Since $$ \frac{3\pi}{2} \u003c \beta \u003c 2\pi $$, $$ \beta  is in $$the fourth quadrant where cosine is positive and sine is negative. Use the identity $$ 	an \alpha = \frac{\sin \alpha}{\cos \alpha}  to $$find $$ \sin \alpha $$ and $$ \cos \alpha . $$Given $$ 	an \alpha = \frac{3}{4} $$, set $$ \sin \alpha = 3k $$ and $$ \cos \alpha = 4k $$ for some $$ k . $$Use the Pythagorean identity $$ \sin^2 \alpha + \cos^2 \alpha = 1  to $$solve for $$ k $$, then apply the correct signs based on the quadrant. Similarly, find $$ \sin \beta $$ using the Pythagorean identity $$ \sin^2 \beta + \cos^2 \beta = 1 $$ with $$ \cos \beta = \frac{1}{4} . $$Determine the sign of $$ \sin \beta $$ based on the quadrant of $$ \beta . $$Apply the cosine addition formula: $$ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta . $$Substitute the values of $$ \cos \alpha $$, $$ \cos \beta $$, $$ \sin \alpha $$, and $$ \sin \beta $$ found in previous steps to express $$ \cos(\alpha + \beta) $$ exactly.

In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅

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Key Concepts

Sum of Angles Formula for Cosine

Determining Trigonometric Ratios from Given Conditions

Quadrant Sign Rules for Trigonometric Functions

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅