Trigonometric Equation Help (1 Viewer)

FesuY · Mar 12, 2022

Everytime a question that has the domain (-π, π) always stuffs me up. Could anyone help me out with these types of questions?

Screen Shot 2022-03-12 at 10.24.43 pm.png

Lith_30 · Mar 12, 2022

You could shift the domain to something that is easier for questions for example $-\pi\leq{x}\leq\pi\implies{0}\leq{x+\pi}\leq{2}\pi$

Say for question a originally you would have to solve $2\cos(2x)=1$ for $-\pi\leq{x}\leq\pi$ but by shifting the domain by pi forward you get $2\cos(2(x+\pi))=1$ for $0\leq{x+\pi}\leq{2}\pi$ which can be further modified to match what is in the cos function $0\leq{2}(x+\pi)\leq{4}\pi$ .

After solving the equation a little, you should arrive at
$\\2(x+\pi)=\frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}, \frac{11\pi}{3}\\\\x=-\frac{5\pi}{6},-\frac{\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6}$
Which is in the required domain.

FesuY · Mar 12, 2022

Lith_30 said:
You could shift the domain to something that is easier for questions for example $-\pi\leq{x}\leq\pi\implies{0}\leq{x+\pi}\leq{2}\pi$

Say for question a originally you would have to solve $2\cos(2x)=1$ for $-\pi\leq{x}\leq\pi$ but by shifting the domain by pi forward you get $2\cos(2(x+\pi))=1$ for $0\leq{x+\pi}\leq{2}\pi$ which can be further modified to match what is in the cos function $0\leq{2}(x+\pi)\leq{4}\pi$ .

After solving the equation a little, you should arrive at
$\\2(x+\pi)=\frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}, \frac{11\pi}{3}\\\\x=-\frac{5\pi}{6},-\frac{\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6}$
Which is in the required domain.

Thats what I thought, however for this question the answer is 90, -150 and -30, and my answer was 90, 210, -150 and -30

Lith_30 · Mar 12, 2022

FesuY said:
Thats what I thought, however for this question the answer is 90, -150 and -30, and my answer was 90, 210, -150 and -30

Assuming the same domain applies as the previous equation $-180^{\circ}\leq{x}\leq180^{\circ}\implies-630^{\circ}\leq3(x-30^{\circ})\leq450^{\circ}$

Thus solving for $\cos(3(x-30^{\circ}))=-1$ will get us

$\\3(x-30^{\circ})=\ -540^{\circ},\ -180^{\circ},\ 180^{\circ}\\\\x=-150^{\circ},\ -30^{\circ},\ 90^{\circ}$

So the 210 is not included.

You could also look at the original domain and see of the answers fit within it, since 210 does not fit within the original domain you you can disregard it.

Trigonometric Equation Help (1 Viewer)

FesuY

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Lith_30

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FesuY

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