Write $i^p$ in the form $a + ib$ where $a$ and $b$ are real - HSC - SSCE Mathematics Extension 2 - Question 2 - 2009 - Paper 1
Question 2
Write $i^p$ in the form $a + ib$ where $a$ and $b$ are real.
Write $-2 + 3i$ in the form $a + ib$ where $a$ and $b$ are real.
The points $P$ and $Q$ on the Argand ... show full transcript
Worked Solution & Example Answer:Write $i^p$ in the form $a + ib$ where $a$ and $b$ are real - HSC - SSCE Mathematics Extension 2 - Question 2 - 2009 - Paper 1
Step 1
Write $i^p$ in the form $a + ib$ where $a$ and $b$ are real
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Answer
To express ip in the form a+ib, we can use the fact that i = e^{i rac{ au}{2}}. Therefore, we can write:
i^p &= ig(e^{i rac{ au}{2}} ig)^p \
&= e^{i rac{p au}{2}} \ \
&= ext{cos}igg( rac{p au}{2} igg) + i ext{sin}igg( rac{p au}{2} igg) \ \
&= a + ib,
ext{where } a = ext{cos}igg( rac{p au}{2} igg) ext{ and } b = ext{sin}igg( rac{p au}{2} igg).
\
ext{Thus, } i^p ext{ is expressed in the desired form.}
\
\
ext{This answer depends on the value of } p.
Step 2
Write $-2 + 3i$ in the form $a + ib$ where $a$ and $b$ are real
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Answer
The complex number −2+3i is already in the form a+ib. We can directly identify:
a=−2,extandb=3.
Step 3
the point $R$ representing $iz$
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Given the point P represents the complex number z, the point R representing iz can be found using:
R=iimesz=i(x+iy)=−y+ix.
Step 4
the point $S$ representing $w$
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The point S will be directly translated from the complex number w, represented simply as:
S=w.
Step 5
the point $T$ representing $z + w$
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To find the point T, we add the complex numbers z and w:
T=z+w.
Step 6
Sketch the region in the complex plane where the inequalities $|z - 1|
leq 2$ and $- rac{ au}{4}
leq arg(z - 1)
leq rac{ au}{4}$ hold simultaneously.
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To sketch this region, first identify the circle with center (1,0) and radius 2. Then, consider the angular restrictions given by the argument. The shaded region will be the intersection of the circle and the sector defined by the arguments.
Step 7
Find all the 5th roots of $-1$ in modulus-argument form.
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Answer
To find the 5th roots of −1, we express −1 in polar form:
−1=eiau.
The 5th roots are given by:
ext{root}_k = e^{i rac{ au + 2k ext{ } au}{5} } ext{ for } k = 0, 1, 2, 3, 4.
Step 8
Sketch the 5th roots of $-1$ on an Argand diagram.
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On the Argand diagram, plot the points corresponding to the angles:
