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tester [92]
3 years ago
11

I need help with number 7, and the second part of number 8. Thank you!

Mathematics
1 answer:
tangare [24]3 years ago
5 0

Answer:

7. \dfrac{\pi }{3},\ \pi ,\ \dfrac{5\pi }{3}

8. b. 0,\ \dfrac{2\pi }{3},\ \pi,\ \dfrac{4\pi }{3}

Step-by-step explanation:

7. Solve the equation

\cos x+\cos 2x=0

First, note that

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

Hence,

\cos x+2\cos ^2x-1=0

Use substitution

t=\cos x,

then

2t^2+t-1=0\\ \\D=1^2-4\cdot 2\cdot (-1)=1+8=9\\ \\\sqrt{D}=\sqrt{9}=3\\ \\t_{1,2}=\dfrac{-1\pm 3}{2\cdot 2}=-1,\ \dfrac{1}{2}

Now,

\cos x=-1\ \text{or}\ \cos x=\dfrac{1}{2}

Solve each equation separately for 0\le x

\cos x=-1\\ \\x=\pi+2\pi k, \in Z\\ \\\text{only } x=\pi \in [0,2\pi)

\cos x=\dfrac{1}{2}\\ \\x=\pm\arccos \dfrac{1}{2}+2\pi k,\ k\in Z\\ \\x=\dfrac{\pi }{3}\ \text{and}\ x=\dfrac{5\pi }{3}

8. b. Solve the equation

\sin x+2\sin x\cos x=0

Rewrite it:

\sin x(1+2\cos x)=0

By zero product property,

\sin x=0\ \text{or}\ 1+2\cos x=0

Solve each equation separately.

\sin x=0\\ \\x=\pi k,\ k\in Z\\ \\\text{for}\ x\in [0,2\pi),\ \ \ \ x_1=0\ \ x_2=\pi

1+2\cos x=0\\ \\\cos x=-\dfrac{1}{2}\\ \\x=\pm \arccos \left(- \dfrac{1}{2}\right)+2\pi k,\ k\in Z\\ \\x=\dfrac{2\pi }{3}\ \text{and}\ x=\dfrac{4\pi }{3}

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