Questions: Given the triangle below, find the angle A and length of side x Note: picture is NOT drawn to scale, but you can assume an angle that appears acute is acute and angle that appears obtuse is obtuse. A= degrees x=

Solution Steps

• Angle given: 42°
• Side opposite to the given angle: 30
• Hypotenuse: 23

The sine function relates the angle to the ratio of the opposite side over the hypotenuse: \[ \sin(42^\circ) = \frac{30}{23} \]

\[ x = 23 \cdot \sin(42^\circ) \] Using a calculator: \[ \sin(42^\circ) \approx 0.6691 \] \[ x = 23 \cdot 0.6691 \approx 15.39 \]

The cosine function relates the angle to the ratio of the adjacent side over the hypotenuse: \[ \cos(42^\circ) = \frac{x}{23} \]

Using the inverse cosine function: \[ A = \cos^{-1}\left(\frac{x}{23}\right) \] \[ A = \cos^{-1}(0.6691) \approx 48^\circ \]

Final Answer