Question

solve for x. round to the nearest tenth, if necessary. triangle with right angle at y, hypotenuse tz = 3.2, angle at z is 35°, side ty is x

Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right - triangle with hypotenuse \( TZ = 3.2 \) and angle \( \angle Z=35^{\circ} \). We want to find the length of the side opposite to angle \( Z \), which is \( x \) (side \( TY \)). The sine function relates the opposite side and the hypotenuse in a right - triangle, where \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). So, \( \sin(35^{\circ})=\frac{x}{3.2} \).

Step2: Solve for \( x \)

To solve for \( x \), we can multiply both sides of the equation by \( 3.2 \). So, \( x = 3.2\times\sin(35^{\circ}) \). We know that \( \sin(35^{\circ})\approx0.5736 \). Then \( x=3.2\times0.5736 \)
\( x = 3.2\times0.5736 = 1.83552 \)

Step3: Round to the nearest tenth

Rounding \( 1.83552 \) to the nearest tenth, we look at the hundredth place. The digit in the hundredth place is \( 3 \), which is less than \( 5 \). So we round down. \( x\approx1.8 \)

Answer:

\( 1.8 \)