Question

#1, continued
which statement is true?
$\sin 40^{\circ} = \frac{x}{24}$
$\sin 40^{\circ} = \frac{24}{x}$
therefore, which statement is true?
$x = 24 \cdot \sin 40^{\circ}$ $x = 24 / \sin 40^{\circ}$
what is the value of $x$? round to the nearest tenth.

Explanation:

First Question (Which statement is true?):

In right triangle \( ABC \) (right - angled at \( B \)), for angle \( C = 40^{\circ}\), the sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. The side opposite to \( 40^{\circ}\) is \( AB=x \), and the hypotenuse is \( AC = 24 \). By the definition of sine, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin40^{\circ}=\frac{x}{24}\).

Starting from the equation \(\sin40^{\circ}=\frac{x}{24}\), we can solve for \( x \) by multiplying both sides of the equation by \( 24 \). So, \( x = 24\times\sin40^{\circ}\).

Step 1: Recall the formula for \( x \)

We know that \( x = 24\times\sin40^{\circ}\).

Step 2: Calculate the value of \(\sin40^{\circ}\)

Using a calculator, \(\sin40^{\circ}\approx0.6428\).

Step 3: Multiply by 24

\( x=24\times0.6428 = 15.4272\)

Step 4: Round to the nearest tenth

Rounding \( 15.4272\) to the nearest tenth gives \( 15.4\).

Answer:

\(\boldsymbol{\sin 40^{\circ}=\frac{x}{24}}\)

Second Question (Therefore, which statement is true?):