First slide

Trigonometric equations

Question

The number of values of $θ$ in $[0, 3 π]$ satisfying $\sin^{3} θ - \cos^{3} θ + 1 + 3 \sin θ \cos θ = 0$

Moderate

A

2
B

3
C

4
D

5

Solution

$W e h a v e \sin^{3} θ + {(- \cos θ)}^{3} + 1 = 3 \sin θ (- \cos θ) .1$
$\Rightarrow \sin θ - \cos θ + 1 = 0$ ( $∵$ $a^{3} + b^{3} + c^{3} = 3 a b c \Rightarrow a + b + c = 0 o r a = b = c)$
$\Rightarrow \cos θ - \sin θ = 1$ $\Rightarrow \frac{1}{\sqrt{2}} \cos θ - \frac{1}{\sqrt{2}} \sin θ = \frac{1}{\sqrt{2}}$ ,
$\Rightarrow \cos (θ + \frac{π}{4}) = \cos \frac{π}{4}$
$\Rightarrow θ + \frac{π}{4} = 2 n π \pm \frac{π}{4}$
$\Rightarrow θ = 2 n π$ or $2 n π - \frac{π}{2}$
$\Rightarrow θ = 0, 2 π, \frac{3 π}{2}$

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