7. (i) Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$ giving your answers to 1 decimal place. You must show each step of your working. (ii) Solve, for $-\pi < x < \pi$, the equation $$2 \tan x - 3 \sin x = 0$$ giving your answers to 2 decimal places where appropriate. [Solutions based entirely on graphical or numerical methods are not acceptable.]

A-Level Edexcel Maths Pure Circles: 7. (i) Solve, for $0 \leq \theta

7. (i) Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$ giving your answers to 1 decimal place - Edexcel - A-Level Maths Pure - Question 9 - 2014 - Paper 1

Question 9

$7.-(i)-Solve,-for-$0-\leq-\theta-<-360^\circ$,-the-equation--$$9-\sin(\theta-+-60^\circ)-=-4$$-giving-your-answers-to-1-decimal-place-Edexcel-A-Level Maths Pure-Question 9-2014-Paper 1.png$

7. (i) Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$ giving your answers to 1 decimal place. You must show each step of you... show full transcript

Worked Solution & Example Answer:7. (i) Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$ giving your answers to 1 decimal place - Edexcel - A-Level Maths Pure - Question 9 - 2014 - Paper 1

Step 1

Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$

96%

114 rated

Answer

To solve the equation, first isolate the sine term:

$\sin(\theta + 60^\circ) = \frac{4}{9}$

Using a calculator, we can find the angle:

$\theta + 60^\circ = \arcsin\left(\frac{4}{9}\right)$

Calculating:

$\arcsin\left(\frac{4}{9}\right) \approx 26.3877^\circ$

We can also find the second angle in the range $0 \leq \theta < 360^\circ$ :

$\theta + 60^\circ = 180^\circ - 26.3877^\circ = 153.6123^\circ$

Now, let's isolate $\theta$ :

For the first angle:

$\theta = 26.3877^\circ - 60^\circ = -33.6123^\circ$ (not in range)

For the second angle:

$\theta = 153.6123^\circ - 60^\circ = 93.6123^\circ$

Find the other equivalent angle:

$\theta + 60^\circ = 360^\circ - 26.3877^\circ$

So:

$\theta + 60^\circ = 326.6123^\circ$

Thus,

$\theta = 326.6123^\circ - 60^\circ = 266.6123^\circ$

Finally, rounding to 1 decimal place, we have:

$\theta \approx 93.6^\circ, 326.6^\circ$

Step 2

Solve, for $-\pi < x < \pi$, the equation $$2 \tan x - 3 \sin x = 0$$

99%

104 rated

Answer

Rearranging the equation:

$2 \tan x = 3 \sin x$

Using the identity $\tan x = \frac{\sin x}{\cos x}$ , we have:

$2 \frac{\sin x}{\cos x} = 3 \sin x$

This simplifies to:

$2 \sin x = 3 \sin x \cos x$

Rearranging gives:

$3 \sin x \cos x - 2 \sin x = 0$

Factoring out $\sin x$ leads to:

$\sin x(3 \cos x - 2) = 0$

Setting each factor equal to zero provides:

$\sin x = 0$ :
- Solutions are: $x = 0, -\pi, \pi$ (but $-\pi$ and $\pi$ are outside the range)
$3 \cos x - 2 = 0$ :
- So: $\cos x = \frac{2}{3}$
- Taking the inverse cosine:
- We find: $x = \arccos\left(\frac{2}{3}\right) \approx 0.8410$
- The second solution in the given range: $x = -\arccos\left(\frac{2}{3}\right) \approx -0.8410$

Thus, the solutions to 2 decimal places are:

$x \approx 0.84, -0.84$

7. (i) Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$ giving your answers to 1 decimal place - Edexcel - A-Level Maths Pure - Question 9 - 2014 - Paper 1

Worked Solution & Example Answer:7. (i) Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$ giving your answers to 1 decimal place - Edexcel - A-Level Maths Pure - Question 9 - 2014 - Paper 1

Solve, for $0 \leq \theta < 360^\circ$, the equation $$9 \sin(\theta + 60^\circ) = 4$$

Solve, for $-\pi < x < \pi$, the equation $$2 \tan x - 3 \sin x = 0$$

Join the A-Level students using SimpleStudy...