1 Points] DETAILS MY NOTES AUFEXC4 7.6.016.MI. Use a calculator to estimate the value of the following. Round to the nearest ten-thousandth. \( \cos 29.5^{\circ} \) Need Help? Read It Master It SUBMIT ANSWER

To estimate the value of \( \cos 29.5^{\circ} \) and round it to the nearest ten-thousandth, follow these steps: ### Step 1: Understand the Angle \( 29.5^{\circ} \) can be seen as \( 30^{\circ} - 0.5^{\circ} \). We'll use the cosine difference formula for better accuracy: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] Where: - \( A = 30^{\circ} \) - \( B = 0.5^{\circ} \) ### Step 2: Calculate \(\cos 30^{\circ}\) and \(\sin 30^{\circ}\) \[ \cos 30^{\circ} = \frac{\sqrt{3}}{2} \approx 0.8660254 \] \[ \sin 30^{\circ} = 0.5 \] ### Step 3: Estimate \(\cos 0.5^{\circ}\) and \(\sin 0.5^{\circ}\) For very small angles in radians (\( \theta \approx 0.008726646 \) radians for \( 0.5^{\circ} \)), we can use the following approximations: \[ \cos \theta \approx 1 - \frac{\theta^2}{2} \approx 0.9999619 \] \[ \sin \theta \approx \theta \approx 0.008726535 \] ### Step 4: Apply the Cosine Difference Formula \[ \cos(29.5^{\circ}) = \cos(30^{\circ} - 0.5^{\circ}) \approx 0.8660254 \times 0.9999619 + 0.5 \times 0.008726535 \] \[ \approx 0.8660254 \times 0.9999619 + 0.0043632675 \] \[ \approx 0.8659924 + 0.0043632675 \] \[ \approx 0.8703557 \] ### Step 5: Round to the Nearest Ten-Thousandth \[ \cos(29.5^{\circ}) \approx 0.8704 \] **Answer:** \( \cos 29.5^{\circ} \approx 0.8704 \)