A-Level Edexcel Maths Pure Solving Equations: 8. (a) Find all the values of $

8. (a) Find all the values of $ heta$, to 1 decimal place, in the interval $0^{ ext{o}} \leq \theta < 360^{ ext{o}}$ for which $$5 \sin (\theta + 30^{ ext{o}}) = 3.$$ (4) (b) Find all the values of $ heta$, to 1 decimal place, in the interval $0^{ ext{o}} \leq \theta < 360^{ ext{o}}$ for which $$\tan^{2} \theta = 4.$$ (5) - Edexcel - A-Level Maths Pure - Question 1 - 2006 - Paper 2

Question 1

$8.-(a)-Find-all-the-values-of-$-heta$,-to-1-decimal-place,-in-the-interval-$0^{-ext{o}}-\leq-\theta-<-360^{-ext{o}}$-for-which--$$5-\sin-(\theta-+-30^{-ext{o}})-=-3.$$-(4)--(b)-Find-all-the-values-of-$-heta$,-to-1-decimal-place,-in-the-interval-$0^{-ext{o}}-\leq-\theta-<-360^{-ext{o}}$-for-which--$$\tan^{2}-\theta-=-4.$$-(5)-Edexcel-A-Level Maths Pure-Question 1-2006-Paper 2.png$

8. (a) Find all the values of $ heta$, to 1 decimal place, in the interval $0^{ ext{o}} \leq \theta < 360^{ ext{o}}$ for which $$5 \sin (\theta + 30^{ ext{o}}) = 3.... show full transcript

Worked Solution & Example Answer:8. (a) Find all the values of $ heta$, to 1 decimal place, in the interval $0^{ ext{o}} \leq \theta < 360^{ ext{o}}$ for which $$5 \sin (\theta + 30^{ ext{o}}) = 3.$$ (4) (b) Find all the values of $ heta$, to 1 decimal place, in the interval $0^{ ext{o}} \leq \theta < 360^{ ext{o}}$ for which $$\tan^{2} \theta = 4.$$ (5) - Edexcel - A-Level Maths Pure - Question 1 - 2006 - Paper 2

Step 1

(a) Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which 5 sin(θ + 30°) = 3.

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Answer

To solve the equation, we first isolate the sine function:

$\sin (\theta + 30^{\text{o}}) = \frac{3}{5}.$

Next, we find the reference angle:

$\theta + 30^{\text{o}} = \arcsin\left(\frac{3}{5}\right) \approx 36.9^{\text{o}}.$

The angles in the interval for sine include:

First Quadrant:

ightarrow \theta \approx 6.9^{\text{o}}.$$

Second Quadrant:
$\theta + 30^{\text{o}} \approx 180^{\text{o}} - 36.9^{\text{o}} \implies \theta \approx 180^{\text{o}} - 36.9^{\text{o}} - 30^{\text{o}}\approx 143.1^{\text{o}}.$

Thus, the solutions for (a) are: $\theta = 6.9^{\text{o}} \text{ and } \theta = 143.1^{\text{o}}$ .

Step 2

(b) Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which tan² θ = 4.

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Answer

To find the values of $\theta$ , we first take the square root of both sides:

$\tan \theta = \pm 2.$

Next, we find the reference angles:

For $\tan \theta = 2$ :
$\theta = \arctan(2) \approx 63.4^{\text{o}}.$
Thus, the angles in the interval are:
- First Quadrant:
  $\theta = 63.4^{\text{o}}.$
- Third Quadrant:
  $\theta = 180^{\text{o}} + 63.4^{\text{o}} \approx 243.4^{\text{o}}.$
For $\tan \theta = -2$ :
Except for fourth quadrant:
- Second Quadrant:
  $\theta = 180^{\text{o}} - 63.4^{\text{o}} \approx 116.6^{\text{o}}.$
- Fourth Quadrant:
  $\theta = 360^{\text{o}} - 63.4^{\text{o}} \approx 296.6^{\text{o}}.$

Thus, the solutions for (b) are:
$\theta \approx 63.4^{\text{o}}, 243.4^{\text{o}}, 116.6^{\text{o}}, 296.6^{\text{o}}.$

(a) Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which 5 sin(θ + 30°) = 3.

(b) Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which tan² θ = 4.

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