Trig Equations worksheet 5.1 featuring ten trigonometric equations to solve within the interval 0 ≤ x < 2π.

Let’s solve each trig equation step by step for $0 \le x < 2\pi$. We’ll go one at a time.

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1. $\sin x = \frac{1}{2}$

We know sine is positive in Quadrants I and II.

Reference angle: $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$

So solutions:
- QI: $x = \frac{\pi}{6}$
- QII: $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$

✔ Final Answer for #1: $\boxed{\frac{\pi}{6}, \frac{5\pi}{6}}$

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2. $\tan x = 1$

Tangent is positive in QI and QIII.

Reference angle: $\tan^{-1}(1) = \frac{\pi}{4}$

Solutions:
- QI: $x = \frac{\pi}{4}$
- QIII: $x = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$

✔ Final Answer for #2: $\boxed{\frac{\pi}{4}, \frac{5\pi}{4}}$

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3. $2\sin x + \sqrt{3} = 0$

Subtract $\sqrt{3}$:
$2\sin x = -\sqrt{3}$

Divide by 2:
$\sin x = -\frac{\sqrt{3}}{2}$

Sine is negative in QIII and QIV.

Reference angle: $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$

Solutions:
- QIII: $x = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$
- QIV: $x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$

✔ Final Answer for #3: $\boxed{\frac{4\pi}{3}, \frac{5\pi}{3}}$

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4. $2\cos x + \sqrt{3} = 0$

Subtract $\sqrt{3}$:
$2\cos x = -\sqrt{3}$

Divide by 2:
$\cos x = -\frac{\sqrt{3}}{2}$

Cosine is negative in QII and QIII.

Reference angle: $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}$

Solutions:
- QII: $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$
- QIII: $x = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$

✔ Final Answer for #4: $\boxed{\frac{5\pi}{6}, \frac{7\pi}{6}}$

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5. $2\cos x - \sqrt{3} = 0$

Add $\sqrt{3}$:
$2\cos x = \sqrt{3}$

Divide by 2:
$\cos x = \frac{\sqrt{3}}{2}$

Cosine is positive in QI and QIV.

Reference angle: $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}$

Solutions:
- QI: $x = \frac{\pi}{6}$
- QIV: $x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$

✔ Final Answer for #5: $\boxed{\frac{\pi}{6}, \frac{11\pi}{6}}$

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6. $\tan x + 1 = 0$

Subtract 1:
$\tan x = -1$

Tangent is negative in QII and QIV.

Reference angle: $\tan^{-1}(1) = \frac{\pi}{4}$

Solutions:
- QII: $x = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$
- QIV: $x = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$

✔ Final Answer for #6: $\boxed{\frac{3\pi}{4}, \frac{7\pi}{4}}$

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7. $\sqrt{3}\cot x + 1 = 0$

Subtract 1:
$\sqrt{3}\cot x = -1$

Divide by $\sqrt{3}$:
$\cot x = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$ (rationalized)

Cotangent is reciprocal of tangent, so same sign behavior.

$\cot x = -\frac{\sqrt{3}}{3}$ → $\tan x = -\sqrt{3}$

Tangent is negative in QII and QIV.

Reference angle: $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

Solutions:
- QII: $x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$
- QIV: $x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$

✔ Final Answer for #7: $\boxed{\frac{2\pi}{3}, \frac{5\pi}{3}}$

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8. $\csc^2 x = 1$

Take square root:
$\csc x = \pm 1$

But $\csc x = \frac{1}{\sin x}$, so:

If $\csc x = 1$, then $\sin x = 1$ → $x = \frac{\pi}{2}$

If $\csc x = -1$, then $\sin x = -1$ → $x = \frac{3\pi}{2}$

Check domain: both are within $[0, 2\pi)$

✔ Final Answer for #8: $\boxed{\frac{\pi}{2}, \frac{3\pi}{2}}$

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9. $\sec^2 x - 1 = 0$

Add 1:
$\sec^2 x = 1$

Take square root:
$\sec x = \pm 1$

$\sec x = \frac{1}{\cos x}$, so:

If $\sec x = 1$ → $\cos x = 1$ → $x = 0$

If $\sec x = -1$ → $\cos x = -1$ → $x = \pi$

Both in range.

✔ Final Answer for #9: $\boxed{0, \pi}$

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10. $4\sec^2 x = 8$

Divide by 4:
$\sec^2 x = 2$

Take square root:
$\sec x = \pm \sqrt{2}$

So $\cos x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}$

Now find all angles where cosine is $\frac{\sqrt{2}}{2}$ or $-\frac{\sqrt{2}}{2}$

Reference angle: $\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$

Cases:

- $\cos x = \frac{\sqrt{2}}{2}$ → QI and QIV: $x = \frac{\pi}{4}, \frac{7\pi}{4}$
- $\cos x = -\frac{\sqrt{2}}{2}$ → QII and QIII: $x = \frac{3\pi}{4}, \frac{5\pi}{4}$

All four are valid.

✔ Final Answer for #10: $\boxed{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}}$

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Final Answer:
1. $\boxed{\frac{\pi}{6}, \frac{5\pi}{6}}$
2. $\boxed{\frac{\pi}{4}, \frac{5\pi}{4}}$
3. $\boxed{\frac{4\pi}{3}, \frac{5\pi}{3}}$
4. $\boxed{\frac{5\pi}{6}, \frac{7\pi}{6}}$
5. $\boxed{\frac{\pi}{6}, \frac{11\pi}{6}}$
6. $\boxed{\frac{3\pi}{4}, \frac{7\pi}{4}}$
7. $\boxed{\frac{2\pi}{3}, \frac{5\pi}{3}}$
8. $\boxed{\frac{\pi}{2}, \frac{3\pi}{2}}$
9. $\boxed{0, \pi}$
10. $\boxed{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}}$