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.

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Key Concepts

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.