Question

In Exercises 35–38, find the exact value of the following under the given conditions:  e. cos(β/2)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 $$ \frac{\pi}{2} \u003c \beta \u003c \pi . $$Since $$ \alpha  is in $$the first quadrant (between 0 and $$ \frac{\pi}{2} $$), both $$ \sin \alpha $$ and $$ \cos \alpha $$ are positive. Use the Pythagorean identity to find $$ \cos \alpha $$:

$$\cos \alpha = \sqrt{1 - \sin^2$ \alpha} = \sqrt{1 - \left(\frac{3}{5}\$$right)^2$$}$$ Since $$ \beta  is in $$the second quadrant (between $$ \frac{\pi}{2} $$ and $$ \pi $$), $$ \sin \beta  is $$positive but $$ \cos \beta  is $$negative. Use the Pythagorean identity to find $$ \cos \beta $$:

$$\cos \beta = -\sqrt{1 - \sin^2$ \beta} = -\sqrt{1 - \left(\frac{12}{13}\$$right)^2$$}$$ Use the cosine of difference formula to find $$ \cos(\beta - \alpha) $$:

$$\cos(\beta - \alpha) = \cos \beta \cos \alpha + \sin \beta \sin \alpha$$ Substitute the values of $$ \cos \alpha $$, $$ \cos \beta $$, $$ \sin \alpha $$, and $$ \sin \beta $$ into the formula and simplify to find the exact value of $$ \cos(\beta - \alpha) .$$

In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos(β/2)
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

Trigonometric Ratios and Their Definitions

Quadrant and Angle Restrictions

Using Pythagorean Identity 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 = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.