A-Level Edexcel Maths Pure Integration: Find all the values of θ, to 1 d

Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which 5 sin(θ + 30°) = 3 - Edexcel - A-Level Maths Pure - Question 9 - 2006 - Paper 2

Question 9

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

Worked Solution & Example Answer:Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which 5 sin(θ + 30°) = 3 - Edexcel - A-Level Maths Pure - Question 9 - 2006 - Paper 2

Step 1

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 for θ, start by isolating sin(θ + 30°):

Divide both sides by 5:
$rac{3}{5} = ext{sin}(θ + 30°)$
Use the inverse sine function:
$θ + 30° = ext{sin}^{-1}igg( rac{3}{5}igg)$
This calculates to approximately:
$θ + 30° = 36.9°$
Consider the second solution in the sine function:
$θ + 30° = 180° - 36.9° = 143.1°$
Now, solving for θ in both cases:
$θ = 36.9° - 30° = 6.9°$ $θ = 143.1° - 30° = 113.1°$

Thus, the values of θ are 6.9° and 113.1°.

Step 2

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 θ, first take the square root of both sides:

$an θ = ±2$
Solving for θ gives:
- For $an θ = 2$ : $θ = ext{tan}^{-1}(2)$

ightarrow θ ≈ 63.4° $$

For the second quadrant solution: $θ = 180° - 63.4°$

ightarrow θ ≈ 116.6° $$

For $an θ = -2$ $an θ = - 2$ :
- Third quadrant:
$θ = 180° + 63.4°$

ightarrow θ ≈ 243.4° $- Fourth quadrant:$ θ = 360° - 63.4° ightarrow θ ≈ 296.6° $$

Thus, the values of θ are 63.4°, 116.6°, 243.4°, and 296.6°.

Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which 5 sin(θ + 30°) = 3 - Edexcel - A-Level Maths Pure - Question 9 - 2006 - Paper 2

Worked Solution & Example Answer:Find all the values of θ, to 1 decimal place, in the interval 0° ≤ θ < 360° for which 5 sin(θ + 30°) = 3 - Edexcel - A-Level Maths Pure - Question 9 - 2006 - Paper 2

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

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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