Identify error in approximation usage

1. (a) Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that

$$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a).
Adele's working is shown below. Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\) Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\) (b) (i) Identify the mistake made by Adele in her working.
(ii) Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\) (2)

1. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$$ [3]

Adele uses \(\theta = 5°\) to test the approximation in part (a). Adele's working is shown below.

Using my calculator, \(1 + 4\cos(5°) + 3\cos^2(5°) = 7.962\), to 3 decimal places.

Using the approximation \(8 - 5\theta^2\) gives \(8 - 5(5)^2 = -117\)

Therefore, \(1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2\) is not true for \(\theta = 5°\)

1. 1. Identify the mistake made by Adele in her working.
   2. Show that \(8 - 5\theta^2\) can be used to give a good approximation to \(1 + 4\cos \theta + 3\cos^2 \theta\) for an angle of size \(5°\) [2]