Questions: Figure 9 Round your answer to four decimal places. θ ≈ Number radians

Solution Steps

In the given right triangle, the lengths of the sides are:

• Opposite side to angle θ: 9
• Adjacent side to angle θ: unknown
• Hypotenuse: 12

The Pythagorean theorem states that in a right triangle: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse.

Here, \( a = 9 \), \( c = 12 \), and we need to find \( b \): \[ 9^2 + b^2 = 12^2 \] \[ 81 + b^2 = 144 \] \[ b^2 = 144 - 81 \] \[ b^2 = 63 \] \[ b = \sqrt{63} \] \[ b = 3\sqrt{7} \]

The sine of angle θ is given by: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \sin(\theta) = \frac{9}{12} \] \[ \sin(\theta) = \frac{3}{4} \]

To find θ, take the inverse sine (arcsin) of \( \frac{3}{4} \): \[ \theta = \arcsin\left(\frac{3}{4}\right) \]

Final Answer