A-Level Edexcel Maths Pure Modelling with Exponentials & Logarithms: For $-rac{ ext{π}}{2} ext{ ≤ }

For $-rac{ ext{π}}{2} ext{ ≤ } y ext{ ≤ } rac{ ext{π}}{2}$, sketch the graph of $y = g(y)$ where g(y) = ext{arcsin} x $-1 ext{ ≤ } x ext{ ≤ } 1$ Find the exact value of $x$ for which $3g(x + 1) + ext{π} = 0$ - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 3

Question 9

For-$--rac{-ext{π}}{2}--ext{-≤-}-y--ext{-≤-}--rac{-ext{π}}{2}$,-sketch-the-graph-of-$y-=-g(y)$-where----g(y)-=--ext{arcsin}-x---$-1--ext{-≤-}-x--ext{-≤-}-1$----Find-the-exact-value-of-$x$-for-which---$3g(x-+-1)-+--ext{π}-=-0$-Edexcel-A-Level Maths Pure-Question 9-2016-Paper 3.png

For $-rac{ ext{π}}{2} ext{ ≤ } y ext{ ≤ } rac{ ext{π}}{2}$, sketch the graph of $y = g(y)$ where g(y) = ext{arcsin} x $-1 ext{ ≤ } x ext{ ≤ } 1$ Find ... show full transcript

Worked Solution & Example Answer:For $-rac{ ext{π}}{2} ext{ ≤ } y ext{ ≤ } rac{ ext{π}}{2}$, sketch the graph of $y = g(y)$ where g(y) = ext{arcsin} x $-1 ext{ ≤ } x ext{ ≤ } 1$ Find the exact value of $x$ for which $3g(x + 1) + ext{π} = 0$ - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 3

Step 1

Sketch the graph of $y = g(y)$

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Answer

To sketch the graph of $y = g(y) = ext{arcsin} x$ , we first note that the function is defined for $x$ in the range $[-1, 1]$ . The arcsin function is increasing and maps these values as follows:

For $x = -1$ , $g(-1) = - rac{ ext{π}}{2}$ .
For $x = 0$ , $g(0) = 0$ .
For $x = 1$ , $g(1) = rac{ ext{π}}{2}$ .

Plotting these points, we can describe the curve from $(-1, - rac{ ext{π}}{2})$ to $(1, rac{ ext{π}}{2})$ . Make sure to depict it as a smooth, increasing curve. The final graph has the correct position and curvature, as required.

Step 2

Find the exact value of $x$ for which $3g(x + 1) + ext{π} = 0$

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Answer

Start with the equation:

$3g(x + 1) + ext{π} = 0$
1. Rearranging gives:
$3g(x + 1) = - ext{π}$
1. Therefore:
$g(x + 1) = - rac{ ext{π}}{3}$
1. Since $g(x) = ext{arcsin} x$ , we have:
$ext{arcsin}(x + 1) = - rac{ ext{π}}{3}$
1. Taking the sine of both sides results in:
$x + 1 = ext{sin}igg(- rac{ ext{π}}{3}igg)$
1. Using the property $ext{sin}(-θ) = - ext{sin}(θ)$ leads to:
$x + 1 = - ext{sin}igg( rac{ ext{π}}{3}igg)$
1. Knowing that $ext{sin}igg( rac{ ext{π}}{3}igg) = rac{ ext{\sqrt3}}{2}$ , we find:
$x + 1 = - rac{ ext{\sqrt3}}{2}$
1. Finally, solving for $x$ gives:
$x = - rac{ ext{\sqrt3}}{2} - 1$

For $- rac{ ext{π}}{2} ext{ ≤ } y ext{ ≤ } rac{ ext{π}}{2}$, sketch the graph of $y = g(y)$ where g(y) = ext{arcsin} x $-1 ext{ ≤ } x ext{ ≤ } 1$ Find the exact value of $x$ for which $3g(x + 1) + ext{π} = 0$ - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 3

Sketch the graph of $y = g(y)$

Find the exact value of $x$ for which $3g(x + 1) + ext{π} = 0$

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For $-rac{ ext{π}}{2} ext{ ≤ } y ext{ ≤ } rac{ ext{π}}{2}$, sketch the graph of $y = g(y)$ where g(y) = ext{arcsin} x $-1 ext{ ≤ } x ext{ ≤ } 1$ Find the exact value of $x$ for which $3g(x + 1) + ext{π} = 0$ - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 3