1.05e - OCR Spec

3 marks Standard +0.3

Use small angle approximations to find the small negative root of the equation $$\sin x + \cos x = 0.5.$$ [3]

6 marks Standard +0.3

Given that $\theta$ is small, show that $2\cos\theta + \sin\theta - 1 \approx 1 + \theta - \theta^2$. [2]
Hence, when $\theta$ is small, show that $$\frac{1}{2\cos\theta + \sin\theta - 1} \approx 1 + a\theta + b\theta^2,$$ where $a$, $b$ are constants to be found. [4]

5 marks Moderate -0.8

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. 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]

5 marks Standard +0.3

For a small angle $\theta$, where $\theta$ is in radians, show that $2\cos\theta + (1 - \tan\theta)^2 \approx 3 - 2\theta$. [3]
Hence determine an approximate solution to $2\cos\theta + (1 - \tan\theta)^2 = 28\sin\theta$. [2]

4 marks Standard +0.3

For a small angle $\theta$, where $\theta$ is in radians, show that $1 + \cos \theta - 3\cos^2 \theta \approx -1 + \frac{3}{2}\theta^2$. [4]

1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x