First slide

Trigonometric equations

Question

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

Difficult

A

2
B

3
C

4
D

0

Solution

$\begin{array}{l} W e h a v e \sin^{3} θ + \cos^{3} θ - 1 = - 3 \sin θ \cos θ \\ \Rightarrow \sin^{3} θ + \cos^{3} θ + {(- 1)}^{3} = 3 (\sin θ) (\cos θ) (- 1) \\ w h i c h i s i n t h e f o r m a^{3} + b^{3} + c^{3} = 3 a b c \\ \Rightarrow ∴ \sin θ = \cos θ = - 1 w h i c h i s i m p o s s i b l e \\ o r \sin θ + \cos θ + (- 1) = 0 \\ \Rightarrow \sin θ + \cos θ = 1 \\ \Rightarrow \frac{1}{\sqrt{2}} \sin θ + \frac{1}{\sqrt{2}} \cos θ = \frac{1}{\sqrt{2}} \\ \Rightarrow \cos (θ - \frac{π}{4}) = \cos \frac{π}{4} \\ \Rightarrow θ - \frac{π}{4} = 2 n π \pm \frac{π}{4} \\ \Rightarrow θ = 2 n π + \frac{π}{2} o r θ = 2 n π \\ \Rightarrow f o r n = 0, θ = \frac{π}{2}, 0 a n d f o r n = 1, θ = \frac{5 π}{2}, 2 π \end{array}$

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