In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Verified step by step guidance

1. Identify the double angle formula for sine: $\sin 2\alpha =2\sin \alpha \cos \alpha$.
2. Given $\sin \alpha =\frac{3}{5}$, use the Pythagorean identity ${\sin }^2\alpha +{\cos }^2\alpha =1$ to find $\cos \alpha$.
3. Calculate $\cos \alpha$ by rearranging the Pythagorean identity: $\cos \alpha =\sqrt{1-{\sin }^2\alpha }=\sqrt{1-{\left(\frac{3}{5}\right)}^2}$.
4. Substitute $\sin \alpha$ and $\cos \alpha$ into the double angle formula: $\sin 2\alpha =2\times \frac{3}{5}\times \cos \alpha$.
5. Simplify the expression to find the exact value of $\sin 2\alpha$.

Key Concepts

Double Angle Formula for Sine

The double angle formula for sine states that sin(2α) = 2sin(α)cos(α). This formula allows us to express the sine of double an angle in terms of the sine and cosine of the original angle, which is essential for solving problems involving angles that are multiples of a given angle.

Understanding Sine Values

The sine function relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Knowing the value of sin(α) is crucial for finding sin(2α), as it directly influences the calculation of both sin(α) and cos(α) in the double angle formula.

Range of Angles

The specified ranges for α and β indicate the quadrants in which these angles lie, affecting the signs of their sine and cosine values. For example, if 0 < α < π/2, both sin(α) and cos(α) are positive, which is important for accurately calculating sin(2α) using the double angle formula.