A-Level Edexcel Maths Pure Exponential & Logarithms: 7. (a) Prove that $$\frac{\sin

7. (a) Prove that $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} + \frac{\sin \theta}{\sin \theta} = 2 \csc 2\theta, \quad \theta \neq 90^\circ.$$ (b) On the axes on page 20, sketch the graph of $$y = 2 \csc 2\theta$$ for $$0^\circ < \theta < 360^\circ.$$ (c) Solve, for $$0^\circ < \theta < 360^\circ$$, the equation $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} \cdot \frac{\sin \theta}{\sin \theta} = 3,$$ giving your answers to 1 decimal place. - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 5

Question 1

$7.-(a)-Prove-that---$$\frac{\sin-\theta-\cdot-\cos-\theta}{\cos^2-\theta}-+-\frac{\sin-\theta}{\sin-\theta}-=-2-\csc-2\theta,-\quad-\theta-\neq-90^\circ.$$----(b)-On-the-axes-on-page-20,-sketch-the-graph-of---$$y-=-2-\csc-2\theta$$-for-$$0^\circ-<-\theta-<-360^\circ.$$----(c)-Solve,-for-$$0^\circ-<-\theta-<-360^\circ$$,-the-equation---$$\frac{\sin-\theta-\cdot-\cos-\theta}{\cos^2-\theta}-\cdot-\frac{\sin-\theta}{\sin-\theta}-=-3,$$---giving-your-answers-to-1-decimal-place.-Edexcel-A-Level Maths Pure-Question 1-2007-Paper 5.png$

7. (a) Prove that $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} + \frac{\sin \theta}{\sin \theta} = 2 \csc 2\theta, \quad \theta \neq 90^\circ.$$ (b) On... show full transcript

Worked Solution & Example Answer:7. (a) Prove that $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} + \frac{\sin \theta}{\sin \theta} = 2 \csc 2\theta, \quad \theta \neq 90^\circ.$$ (b) On the axes on page 20, sketch the graph of $$y = 2 \csc 2\theta$$ for $$0^\circ < \theta < 360^\circ.$$ (c) Solve, for $$0^\circ < \theta < 360^\circ$$, the equation $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} \cdot \frac{\sin \theta}{\sin \theta} = 3,$$ giving your answers to 1 decimal place. - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 5

Step 1

Prove that $ \frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} + \frac{\sin \theta}{\sin \theta} = 2 \csc 2\theta $

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Answer

To prove the identity, we'll start by simplifying the left-hand side:

Combine the fractions onto a common denominator: $\frac{\sin \theta \cdot \cos \theta + \cos^2 \theta}{\cos^2 \theta}$
Notice that we can express the numerator as: $\sin \theta \cdot \cos \theta + \cos^2 \theta = \sin \theta \cdot \cos \theta + \frac{1}{2} \cdot 2 \cos^2 \theta$
Use the double angle sine identity: $\sin 2\theta = 2 \sin \theta \cos \theta \\ \Rightarrow \frac{\sin 2\theta}{2} = \sin \theta \cos \theta$

Thus, our expression becomes: $\frac{1}{\cos^2 \theta} (\sin \theta \cdot \cos \theta + \frac{1}{2}(2)\cos^2 \theta) = 2 \csc 2\theta$ .

This proves the identity correctly.

Step 2

On the axes on page 20, sketch the graph of $ y = 2 \csc 2\theta $ for $ 0^\circ < \theta < 360^\circ $

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Answer

To sketch the graph of ( y = 2 \csc 2\theta ):

Identify vertical asymptotes: The function ( \csc , \theta ) is undefined where ( \sin 2\theta = 0 ). This occurs at: $2\theta = n\pi, n \in \mathbb{Z} \Rightarrow \theta = \frac{n\pi}{2}$ Hence, the vertical asymptotes are at ( \theta = 0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ ).
Determine values of the function: The maximum and minimum points of the function occur when ( \sin 2\theta = 1 ) and ( \sin 2\theta = -1 ), giving maximum and minimum values of 2 and -2 respectively.
Sketch the graph: Plot the points, asymptotes, and minimum/maximum values to create a repeated wave pattern with the specified range.

Step 3

Solve, for $ 0^\circ < \theta < 360^\circ $, the equation $ \frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} \cdot \frac{\sin \theta}{\sin \theta} = 3 $

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Answer

To solve the equation:

The given equation simplifies to: $\sin^2 \theta = 3 \cos^2 \theta$
Use the identity ( \sin^2 \theta + \cos^2 \theta = 1 ): Substitute ( \sin^2 \theta = 3(1 - \sin^2 \theta) ) leading to: $4\sin^2 \theta = 3$ $\sin^2 \theta = \frac{3}{4} \Rightarrow \sin \theta = \pm\frac{\sqrt{3}}{2}$
Find angles: Using the principal values for ( \sin \theta = \frac{\sqrt{3}}{2} ) gives:
- ( \theta = 60^\circ )
- ( \theta = 120^\circ ) and for ( \sin \theta = -\frac{\sqrt{3}}{2} ):
- ( \theta = 240^\circ )
- ( \theta = 300^\circ )

Thus, the solutions in the range are approximately:

( \theta = 60.0^\circ, 120.0^\circ, 240.0^\circ, 300.0^\circ )

Prove that \( \frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} + \frac{\sin \theta}{\sin \theta} = 2 \csc 2\theta \)

On the axes on page 20, sketch the graph of \( y = 2 \csc 2\theta \) for \( 0^\circ < \theta < 360^\circ \)

Solve, for \( 0^\circ < \theta < 360^\circ \), the equation \( \frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} \cdot \frac{\sin \theta}{\sin \theta} = 3 \)

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