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12. (a) Prove \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 2 \cot 2\theta \] \( \theta \pm (90n)^\circ, n \in \mathbb{Z} \) (b) Hence solve, for \( 90^\circ < \theta < 180^\circ \), the equation \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 4 \] giving any solutions to one decimal place. - Edexcel - A-Level Maths Pure - Question 14 - 2019 - Paper 2
Question 14
![12.-(a)-Prove--\[-\frac{\cos-3\theta}{\sin-\theta}-+-\frac{\sin-3\theta}{\cos-\theta}-=-2-\cot-2\theta-\]--\(-\theta-\pm-(90n)^\circ,-n-\in-\mathbb{Z}-\)----(b)-Hence-solve,-for-\(-90^\circ-<-\theta-<-180^\circ-\),-the-equation--\[-\frac{\cos-3\theta}{\sin-\theta}-+-\frac{\sin-3\theta}{\cos-\theta}-=-4-\]--giving-any-solutions-to-one-decimal-place.-Edexcel-A-Level Maths Pure-Question 14-2019-Paper 2.png](https://cdn.simplestudy.io/assets/backend/uploads/question/MathsPure_2019_2_1_QP_12_2019.jpg)
12. (a) Prove \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 2 \cot 2\theta \] \( \theta \pm (90n)^\circ, n \in \mathbb{Z} \) (b) Henc... show full transcript
Worked Solution & Example Answer:12. (a) Prove \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 2 \cot 2\theta \] \( \theta \pm (90n)^\circ, n \in \mathbb{Z} \) (b) Hence solve, for \( 90^\circ < \theta < 180^\circ \), the equation \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 4 \] giving any solutions to one decimal place. - Edexcel - A-Level Maths Pure - Question 14 - 2019 - Paper 2
Step 1
Prove \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 2 \cot 2\theta \]
Answer
To prove the equation, we start with the left-hand side (LHS):
[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = \frac{\cos 3\theta \cos \theta + \sin 3\theta \sin \theta}{\sin \theta \cos \theta} ]
Using the angle addition formula, we simplify:
[ \text{LHS} = \frac{\cos(3\theta - \theta)}{\sin \theta \cos \theta} = \frac{\cos 2\theta}{\sin \theta \cos \theta} ]
We know that ( \cot 2\theta = \frac{\cos 2\theta}{\sin 2\theta} ), which leads us to:
[ \frac{\cos 2\theta}{\sin \theta \cos \theta} \cdot \frac{2}{2} = 2 \cot 2\theta ]
Thus, confirming the given identity.
Step 2
Hence solve, for \( 90^\circ < \theta < 180^\circ \), the equation \[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 4 \]
Answer
From the earlier derived formula:
[ \frac{\cos 3\theta}{\sin \theta} + \frac{\sin 3\theta}{\cos \theta} = 2 \cot 2\theta = 4 ]
We can simplify this to:
[ 2 \cot 2\theta = 4 \implies \cot 2\theta = 2 ]
Taking the cotangent inverse:
[ 2\theta = \arctan\left(\frac{1}{2}\right) ]
Thus:
[ \theta = \frac{1}{2} \arctan\left(\frac{1}{2}\right) + n(90)^\circ ]
Calculating the angle within ( 90^\circ < \theta < 180^\circ ):
The solution is: ( \theta \approx 103.3^\circ ).
Thus, the only solution to one decimal place is ( \theta = 103.3^\circ ).