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.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

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.