Question

In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) sin α = 3/5, 0 \u003c α \u003c 𝝅/2, and sin β = 12/13, 𝝅/2 \u003c β \u003c 𝝅.

Accepted Answer

Identify the given information: $$ \sin \alpha = \frac{3}{5} $$ with $$ 0 \u003c \alpha \u003c \frac{\pi}{2} $$, and $$ \sin \beta = \frac{12}{13} $$ with $$ \frac{\pi}{2} \u003c \beta \u003c \pi . $$Determine the quadrants for $$ \alpha $$ and $$ \beta $$ based on the given interval conditions. Since $$ 0 \u003c \alpha \u003c \frac{\pi}{2} $$, $$ \alpha  is in $$the first quadrant where all trigonometric functions are positive. Since $$ \frac{\pi}{2} \u003c \beta \u003c \pi $$, $$ \beta  is in $$the second quadrant where sine is positive but cosine is negative. Find $$ \cos \alpha $$ using the Pythagorean identity: $$ \cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}ight)^2} . $$Since $$ \alpha  is in $$the first quadrant, $$ \cos \alpha  is $$positive. Find $$ \cos \beta $$ similarly: $$ \cos \beta = -\sqrt{1 - \sin^2 \beta} = -\sqrt{1 - \left(\frac{12}{13}ight)^2} . $$The negative sign is because $$ \beta  is in $$the second quadrant where cosine is negative. Use the tangent addition formula: $$ 	an(\alpha + \beta) = \frac{	an \alpha + 	an \beta}{1 - 	an \alpha 	an \beta} . $$Calculate $$ 	an \alpha = \frac{\sin \alpha}{\cos \alpha} $$ and $$ 	an \beta = \frac{\sin \beta}{\cos \beta} $$, then substitute these values into the formula to express $$ 	an(\alpha + \beta) $$ exactly.

In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

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Concetti chiave

Sum of Angles Formula for Tangent

Using Sine to Find Tangent

Quadrant and Sign Determination

In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.