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artcher [175]
3 years ago
15

2sin²(x)-5cos(x)+1=0

Mathematics
1 answer:
Margarita [4]3 years ago
7 0
Remember the basic trig identity

<u />\sin^2 x + \cos^2 x = 1

We can move the cosine term to the right side to get

\sin^2 x = 1 - \cos^2 x

We can then replace \sin^2 x in the original equation and basically factor the resulting equation like a quadratic:

2(1 - \cos^2 x) - 5 \cos x + 1 = 0
(2 - 2 \cos^2 x) - 5 \cos x + 1 = 0
{-2} \cos^2 x - 5 \cos x  + 3 = 0
2 \cos^2 x + 5 \cos x - 3 = 0
(2 \cos x - 1)(\cos x + 3) = 0&#10;&#10;

Then, we can see that \cos x either equals \frac{1}{2} or -3. But since the range of cosine is only from [-1, 1], cosine can't equal -3 at all! So, you just have to solve for x when \cos x =  \frac{1}{2}, which is when

x = \{ \frac{\pi}{3}, \frac{5\pi}{3} \} + 2 \pi k | k \in \mathbb{Z}
(The last part of the solution says that k can be any integer.)

If you only want solutions in the range [0, 2\pi), then your answer is just

x = \frac{\pi}{3}, \frac{5\pi}{3}









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