Determine the value of lengths y and z. Determine the value of the following. Show all calculations sin a sin2 a + cos2 a sin a / cos a

We know the opposite side (XY = 20) and the angle $\angle Z = 30^\circ$. We use the sine function.

$$\sin(\angle Z) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$ $$\sin(30^\circ) = \frac{XY}{XZ}$$ $$\frac{1}{2} = \frac{20}{y}$$ $$y = 2 \times 20$$ $$y = \boxed{\text{40}}$$

We know the opposite side (XY = 20) and the angle $\angle Z = 30^\circ$. We use the tangent function.

$$\tan(\angle Z) = \frac{\text{Opposite}}{\text{Adjacent}}$$ $$\tan(30^\circ) = \frac{XY}{YZ}$$ $$\frac{1}{\sqrt{3}} = \frac{20}{z}$$ $$z = 20\sqrt{3}$$ $$z \approx \boxed{\text{34.64}}$$

Using the Pythagorean theorem:

$$AB^2 = AC^2 + BC^2$$ $$AB^2 = 6^2 + 8^2$$ $$AB^2 = 36 + 64$$ $$AB^2 = 100$$ $$AB = \sqrt{100}$$ $$AB = 10$$

• 3.3.1 $\sin \alpha$ For angle $\alpha$ (at B), the opposite side is AC and the hypotenuse is AB.

  $$\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AC}{AB}$$ $$\sin \alpha = \frac{6}{10}$$ $$\sin \alpha = \boxed{\frac{3}{5}}$$

• 3.3.2 $\sin^2 \alpha + \cos^2 \alpha$ First, calculate $\cos \alpha$. For angle $\alpha$, the adjacent side is BC and the hypotenuse is AB.

  $$\cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AB}$$ $$\cos \alpha = \frac{8}{10} = \frac{4}{5}$$

  Now substitute into the expression:

  $$\sin^2 \alpha + \cos^2 \alpha = \left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2$$ $$= \frac{9}{25} + \frac{16}{25}$$ $$= \frac{9+16}{25}$$ $$= \frac{25}{25}$$ $$= \boxed{1}$$

• 3.3.3 $\frac{\sin \alpha}{\cos \alpha}$ Using the values calculated for $\sin \alpha$ and $\cos \alpha$:

  $$\frac{\sin \alpha}{\cos \alpha} = \frac{\frac{3}{5}}{\frac{4}{5}}$$ $$= \frac{3}{5} \times \frac{5}{4}$$ $$= \boxed{\frac{3}{4}}$$