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Step-by-step solution for: High School Math Worksheets | Math Worksheets PDF
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Step-by-step solution for: High School Math Worksheets | Math Worksheets PDF
To solve the given trigonometric equations for \(0^\circ \leq x \leq 360^\circ\), we will use the properties of trigonometric functions and their inverses. Let's solve each equation step by step.
- The sine function reaches its maximum value of 1 at \(x = 90^\circ\).
- Solution: \(x = 90^\circ\).
- The tangent function equals 1 at \(x = 45^\circ\) and \(x = 225^\circ\) (since \(\tan(180^\circ + \theta) = \tan \theta\)).
- Solution: \(x = 45^\circ, 225^\circ\).
- The cosine function equals 1 at \(x = 0^\circ\) and \(x = 360^\circ\) (which is the same as \(0^\circ\)).
- Solution: \(x = 0^\circ\).
- The sine function equals 0.5 at \(x = 30^\circ\) and \(x = 150^\circ\) (since \(\sin(180^\circ - \theta) = \sin \theta\)).
- Solution: \(x = 30^\circ, 150^\circ\).
- Use the inverse cosine function: \(x = \cos^{-1}(0.6)\).
- \(x \approx 53.1^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution is \(x = 360^\circ - 53.1^\circ = 306.9^\circ\).
- Solution: \(x \approx 53.1^\circ, 306.9^\circ\).
- Use the inverse tangent function: \(x = \tan^{-1}(0.2)\).
- \(x \approx 11.3^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ + 11.3^\circ = 191.3^\circ\).
- Solution: \(x \approx 11.3^\circ, 191.3^\circ\).
- Simplify: \(\cos x = \frac{3}{7}\).
- Use the inverse cosine function: \(x = \cos^{-1}\left(\frac{3}{7}\right)\).
- \(x \approx 64.6^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution is \(x = 360^\circ - 64.6^\circ = 295.4^\circ\).
- Solution: \(x \approx 64.6^\circ, 295.4^\circ\).
- Simplify: \(\tan x = \frac{1}{8}\).
- Use the inverse tangent function: \(x = \tan^{-1}\left(\frac{1}{8}\right)\).
- \(x \approx 7.1^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ + 7.1^\circ = 187.1^\circ\).
- Solution: \(x \approx 7.1^\circ, 187.1^\circ\).
- Simplify: \(3 \sin x = 2 \Rightarrow \sin x = \frac{2}{3}\).
- Use the inverse sine function: \(x = \sin^{-1}\left(\frac{2}{3}\right)\).
- \(x \approx 41.8^\circ\).
- Since \(\sin(180^\circ - \theta) = \sin \theta\), the other solution is \(x = 180^\circ - 41.8^\circ = 138.2^\circ\).
- Solution: \(x \approx 41.8^\circ, 138.2^\circ\).
- The tangent function equals -1 at \(x = 135^\circ\) and \(x = 315^\circ\) (since \(\tan(180^\circ + \theta) = \tan \theta\)).
- Solution: \(x = 135^\circ, 315^\circ\).
- Use the inverse sine function: \(x = \sin^{-1}(-0.9)\).
- \(x \approx -64.2^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 295.8^\circ\).
- Since \(\sin(360^\circ - \theta) = -\sin \theta\), the other solution is \(x = 180^\circ + 64.2^\circ = 244.2^\circ\).
- Solution: \(x \approx 244.2^\circ, 295.8^\circ\).
- Use the inverse tangent function: \(x = \tan^{-1}(-0.1)\).
- \(x \approx -5.7^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 354.3^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ - 5.7^\circ = 174.3^\circ\).
- Solution: \(x \approx 174.3^\circ, 354.3^\circ\).
- Simplify: \(8 \sin x = -3 \Rightarrow \sin x = -\frac{3}{8}\).
- Use the inverse sine function: \(x = \sin^{-1}\left(-\frac{3}{8}\right)\).
- \(x \approx -22.0^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 338.0^\circ\).
- Since \(\sin(360^\circ - \theta) = -\sin \theta\), the other solution is \(x = 180^\circ + 22.0^\circ = 202.0^\circ\).
- Solution: \(x \approx 202.0^\circ, 338.0^\circ\).
- Use the inverse sine function: \(2x = \sin^{-1}(0.2886)\).
- \(2x \approx 16.7^\circ\). So, \(x \approx 8.4^\circ\).
- Since \(\sin(180^\circ - \theta) = \sin \theta\), the other solution for \(2x\) is \(2x = 180^\circ - 16.7^\circ = 163.3^\circ\). So, \(x \approx 81.7^\circ\).
- Since the period of \(\sin 2x\) is \(180^\circ\), we also have solutions in the next cycle: \(2x = 360^\circ + 16.7^\circ = 376.7^\circ\) and \(2x = 360^\circ + 163.3^\circ = 523.3^\circ\). So, \(x \approx 188.4^\circ\) and \(x \approx 261.7^\circ\).
- Solution: \(x \approx 8.4^\circ, 81.7^\circ, 188.4^\circ, 261.7^\circ\).
- Use the inverse cosine function: \(3x = \cos^{-1}(-0.3321)\).
- \(3x \approx 109.5^\circ\). So, \(x \approx 36.5^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution for \(3x\) is \(3x = 360^\circ - 109.5^\circ = 250.5^\circ\). So, \(x \approx 83.5^\circ\).
- Since the period of \(\cos 3x\) is \(120^\circ\), we also have solutions in the next cycles: \(3x = 360^\circ + 109.5^\circ = 469.5^\circ\) and \(3x = 360^\circ + 250.5^\circ = 610.5^\circ\). So, \(x \approx 156.5^\circ\) and \(x \approx 203.5^\circ\).
- Solution: \(x \approx 36.5^\circ, 83.5^\circ, 156.5^\circ, 203.5^\circ\).
- Use the inverse sine function: \(x + 20^\circ = \sin^{-1}(0.6551)\).
- \(x + 20^\circ \approx 41.0^\circ\). So, \(x \approx 21.0^\circ\).
- Since \(\sin(180^\circ - \theta) = \sin \theta\), the other solution for \(x + 20^\circ\) is \(x + 20^\circ = 180^\circ - 41.0^\circ = 139.0^\circ\). So, \(x \approx 119.0^\circ\).
- Solution: \(x \approx 21.0^\circ, 119.0^\circ\).
- Use the inverse tangent function: \(x - 15^\circ = \tan^{-1}(-0.9128)\).
- \(x - 15^\circ \approx -42.3^\circ\). So, \(x \approx -27.3^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 332.7^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution for \(x - 15^\circ\) is \(x - 15^\circ = 180^\circ - 42.3^\circ = 137.7^\circ\). So, \(x \approx 152.7^\circ\).
- Solution: \(x \approx 152.7^\circ, 332.7^\circ\).
- Use the inverse cosine function: \(2x + 33^\circ = \cos^{-1}(0.306)\).
- \(2x + 33^\circ \approx 72.0^\circ\). So, \(2x \approx 39.0^\circ\) and \(x \approx 19.5^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution for \(2x + 33^\circ\) is \(2x + 33^\circ = 360^\circ - 72.0^\circ = 288.0^\circ\). So, \(2x \approx 255.0^\circ\) and \(x \approx 127.5^\circ\).
- Since the period of \(\cos(2x + 33^\circ)\) is \(180^\circ\), we also have solutions in the next cycle: \(2x + 33^\circ = 360^\circ + 72.0^\circ = 432.0^\circ\) and \(2x + 33^\circ = 360^\circ + 288.0^\circ = 648.0^\circ\). So, \(2x \approx 399.0^\circ\) and \(x \approx 199.5^\circ\), and \(2x \approx 615.0^\circ\) and \(x \approx 307.5^\circ\).
- Solution: \(x \approx 19.5^\circ, 127.5^\circ, 199.5^\circ, 307.5^\circ\).
- Take the square root: \(\tan x = \pm \frac{1}{2}\).
- For \(\tan x = \frac{1}{2}\): Use the inverse tangent function: \(x = \tan^{-1}\left(\frac{1}{2}\right)\).
- \(x \approx 26.6^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ + 26.6^\circ = 206.6^\circ\).
- For \(\tan x = -\frac{1}{2}\): Use the inverse tangent function: \(x = \tan^{-1}\left(-\frac{1}{2}\right)\).
- \(x \approx -26.6^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 333.4^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ - 26.6^\circ = 153.4^\circ\).
- Solution: \(x \approx 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ\).
- Take the square root: \(\cos x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}\).
- For \(\cos x = \frac{\sqrt{2}}{2}\): The cosine function equals \(\frac{\sqrt{2}}{2}\) at \(x = 45^\circ\) and \(x = 315^\circ\).
- For \(\cos x = -\frac{\sqrt{2}}{2}\): The cosine function equals \(-\frac{\sqrt{2}}{2}\) at \(x = 135^\circ\) and \(x = 225^\circ\).
- Solution: \(x = 45^\circ, 135^\circ, 225^\circ, 315^\circ\).
\[
\boxed{
\begin{array}{ll}
1) & 90^\circ \\
2) & 45^\circ, 225^\circ \\
3) & 0^\circ \\
4) & 30^\circ, 150^\circ \\
5) & 53.1^\circ, 306.9^\circ \\
6) & 11.3^\circ, 191.3^\circ \\
7) & 64.6^\circ, 295.4^\circ \\
8) & 7.1^\circ, 187.1^\circ \\
9) & 41.8^\circ, 138.2^\circ \\
10) & 135^\circ, 315^\circ \\
11) & 244.2^\circ, 295.8^\circ \\
12) & 174.3^\circ, 354.3^\circ \\
13) & 202.0^\circ, 338.0^\circ \\
14) & 8.4^\circ, 81.7^\circ, 188.4^\circ, 261.7^\circ \\
15) & 36.5^\circ, 83.5^\circ, 156.5^\circ, 203.5^\circ \\
16) & 21.0^\circ, 119.0^\circ \\
17) & 152.7^\circ, 332.7^\circ \\
18) & 19.5^\circ, 127.5^\circ, 199.5^\circ, 307.5^\circ \\
19) & 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ \\
20) & 45^\circ, 135^\circ, 225^\circ, 315^\circ \\
\end{array}
}
\]
1) \(\sin x = 1\)
- The sine function reaches its maximum value of 1 at \(x = 90^\circ\).
- Solution: \(x = 90^\circ\).
2) \(\tan x = 1\)
- The tangent function equals 1 at \(x = 45^\circ\) and \(x = 225^\circ\) (since \(\tan(180^\circ + \theta) = \tan \theta\)).
- Solution: \(x = 45^\circ, 225^\circ\).
3) \(\cos x = 1\)
- The cosine function equals 1 at \(x = 0^\circ\) and \(x = 360^\circ\) (which is the same as \(0^\circ\)).
- Solution: \(x = 0^\circ\).
4) \(\sin x = 0.5\)
- The sine function equals 0.5 at \(x = 30^\circ\) and \(x = 150^\circ\) (since \(\sin(180^\circ - \theta) = \sin \theta\)).
- Solution: \(x = 30^\circ, 150^\circ\).
5) \(\cos x = 0.6\)
- Use the inverse cosine function: \(x = \cos^{-1}(0.6)\).
- \(x \approx 53.1^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution is \(x = 360^\circ - 53.1^\circ = 306.9^\circ\).
- Solution: \(x \approx 53.1^\circ, 306.9^\circ\).
6) \(\tan x = 0.2\)
- Use the inverse tangent function: \(x = \tan^{-1}(0.2)\).
- \(x \approx 11.3^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ + 11.3^\circ = 191.3^\circ\).
- Solution: \(x \approx 11.3^\circ, 191.3^\circ\).
7) \(7 \cos x = 3\)
- Simplify: \(\cos x = \frac{3}{7}\).
- Use the inverse cosine function: \(x = \cos^{-1}\left(\frac{3}{7}\right)\).
- \(x \approx 64.6^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution is \(x = 360^\circ - 64.6^\circ = 295.4^\circ\).
- Solution: \(x \approx 64.6^\circ, 295.4^\circ\).
8) \(2 \tan x = \frac{1}{4}\)
- Simplify: \(\tan x = \frac{1}{8}\).
- Use the inverse tangent function: \(x = \tan^{-1}\left(\frac{1}{8}\right)\).
- \(x \approx 7.1^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ + 7.1^\circ = 187.1^\circ\).
- Solution: \(x \approx 7.1^\circ, 187.1^\circ\).
9) \(8 + 3 \sin x = 10\)
- Simplify: \(3 \sin x = 2 \Rightarrow \sin x = \frac{2}{3}\).
- Use the inverse sine function: \(x = \sin^{-1}\left(\frac{2}{3}\right)\).
- \(x \approx 41.8^\circ\).
- Since \(\sin(180^\circ - \theta) = \sin \theta\), the other solution is \(x = 180^\circ - 41.8^\circ = 138.2^\circ\).
- Solution: \(x \approx 41.8^\circ, 138.2^\circ\).
10) \(\tan x = -1\)
- The tangent function equals -1 at \(x = 135^\circ\) and \(x = 315^\circ\) (since \(\tan(180^\circ + \theta) = \tan \theta\)).
- Solution: \(x = 135^\circ, 315^\circ\).
11) \(\sin x = -0.9\)
- Use the inverse sine function: \(x = \sin^{-1}(-0.9)\).
- \(x \approx -64.2^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 295.8^\circ\).
- Since \(\sin(360^\circ - \theta) = -\sin \theta\), the other solution is \(x = 180^\circ + 64.2^\circ = 244.2^\circ\).
- Solution: \(x \approx 244.2^\circ, 295.8^\circ\).
12) \(\tan x = -0.1\)
- Use the inverse tangent function: \(x = \tan^{-1}(-0.1)\).
- \(x \approx -5.7^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 354.3^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ - 5.7^\circ = 174.3^\circ\).
- Solution: \(x \approx 174.3^\circ, 354.3^\circ\).
13) \(7 + 8 \sin x = 4\)
- Simplify: \(8 \sin x = -3 \Rightarrow \sin x = -\frac{3}{8}\).
- Use the inverse sine function: \(x = \sin^{-1}\left(-\frac{3}{8}\right)\).
- \(x \approx -22.0^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 338.0^\circ\).
- Since \(\sin(360^\circ - \theta) = -\sin \theta\), the other solution is \(x = 180^\circ + 22.0^\circ = 202.0^\circ\).
- Solution: \(x \approx 202.0^\circ, 338.0^\circ\).
14) \(\sin 2x = 0.2886\)
- Use the inverse sine function: \(2x = \sin^{-1}(0.2886)\).
- \(2x \approx 16.7^\circ\). So, \(x \approx 8.4^\circ\).
- Since \(\sin(180^\circ - \theta) = \sin \theta\), the other solution for \(2x\) is \(2x = 180^\circ - 16.7^\circ = 163.3^\circ\). So, \(x \approx 81.7^\circ\).
- Since the period of \(\sin 2x\) is \(180^\circ\), we also have solutions in the next cycle: \(2x = 360^\circ + 16.7^\circ = 376.7^\circ\) and \(2x = 360^\circ + 163.3^\circ = 523.3^\circ\). So, \(x \approx 188.4^\circ\) and \(x \approx 261.7^\circ\).
- Solution: \(x \approx 8.4^\circ, 81.7^\circ, 188.4^\circ, 261.7^\circ\).
15) \(\cos 3x = -0.3321\)
- Use the inverse cosine function: \(3x = \cos^{-1}(-0.3321)\).
- \(3x \approx 109.5^\circ\). So, \(x \approx 36.5^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution for \(3x\) is \(3x = 360^\circ - 109.5^\circ = 250.5^\circ\). So, \(x \approx 83.5^\circ\).
- Since the period of \(\cos 3x\) is \(120^\circ\), we also have solutions in the next cycles: \(3x = 360^\circ + 109.5^\circ = 469.5^\circ\) and \(3x = 360^\circ + 250.5^\circ = 610.5^\circ\). So, \(x \approx 156.5^\circ\) and \(x \approx 203.5^\circ\).
- Solution: \(x \approx 36.5^\circ, 83.5^\circ, 156.5^\circ, 203.5^\circ\).
16) \(\sin(x + 20^\circ) = 0.6551\)
- Use the inverse sine function: \(x + 20^\circ = \sin^{-1}(0.6551)\).
- \(x + 20^\circ \approx 41.0^\circ\). So, \(x \approx 21.0^\circ\).
- Since \(\sin(180^\circ - \theta) = \sin \theta\), the other solution for \(x + 20^\circ\) is \(x + 20^\circ = 180^\circ - 41.0^\circ = 139.0^\circ\). So, \(x \approx 119.0^\circ\).
- Solution: \(x \approx 21.0^\circ, 119.0^\circ\).
17) \(\tan(x - 15^\circ) = -0.9128\)
- Use the inverse tangent function: \(x - 15^\circ = \tan^{-1}(-0.9128)\).
- \(x - 15^\circ \approx -42.3^\circ\). So, \(x \approx -27.3^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 332.7^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution for \(x - 15^\circ\) is \(x - 15^\circ = 180^\circ - 42.3^\circ = 137.7^\circ\). So, \(x \approx 152.7^\circ\).
- Solution: \(x \approx 152.7^\circ, 332.7^\circ\).
18) \(\cos(2x + 33^\circ) = 0.306\)
- Use the inverse cosine function: \(2x + 33^\circ = \cos^{-1}(0.306)\).
- \(2x + 33^\circ \approx 72.0^\circ\). So, \(2x \approx 39.0^\circ\) and \(x \approx 19.5^\circ\).
- Since \(\cos(360^\circ - \theta) = \cos \theta\), the other solution for \(2x + 33^\circ\) is \(2x + 33^\circ = 360^\circ - 72.0^\circ = 288.0^\circ\). So, \(2x \approx 255.0^\circ\) and \(x \approx 127.5^\circ\).
- Since the period of \(\cos(2x + 33^\circ)\) is \(180^\circ\), we also have solutions in the next cycle: \(2x + 33^\circ = 360^\circ + 72.0^\circ = 432.0^\circ\) and \(2x + 33^\circ = 360^\circ + 288.0^\circ = 648.0^\circ\). So, \(2x \approx 399.0^\circ\) and \(x \approx 199.5^\circ\), and \(2x \approx 615.0^\circ\) and \(x \approx 307.5^\circ\).
- Solution: \(x \approx 19.5^\circ, 127.5^\circ, 199.5^\circ, 307.5^\circ\).
19) \(\tan^2 x = \frac{1}{4}\)
- Take the square root: \(\tan x = \pm \frac{1}{2}\).
- For \(\tan x = \frac{1}{2}\): Use the inverse tangent function: \(x = \tan^{-1}\left(\frac{1}{2}\right)\).
- \(x \approx 26.6^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ + 26.6^\circ = 206.6^\circ\).
- For \(\tan x = -\frac{1}{2}\): Use the inverse tangent function: \(x = \tan^{-1}\left(-\frac{1}{2}\right)\).
- \(x \approx -26.6^\circ\). Since we need solutions in \(0^\circ \leq x \leq 360^\circ\), we add \(360^\circ\) to get \(x \approx 333.4^\circ\).
- Since \(\tan(180^\circ + \theta) = \tan \theta\), the other solution is \(x = 180^\circ - 26.6^\circ = 153.4^\circ\).
- Solution: \(x \approx 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ\).
20) \(\cos^2 x = \frac{1}{2}\)
- Take the square root: \(\cos x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}\).
- For \(\cos x = \frac{\sqrt{2}}{2}\): The cosine function equals \(\frac{\sqrt{2}}{2}\) at \(x = 45^\circ\) and \(x = 315^\circ\).
- For \(\cos x = -\frac{\sqrt{2}}{2}\): The cosine function equals \(-\frac{\sqrt{2}}{2}\) at \(x = 135^\circ\) and \(x = 225^\circ\).
- Solution: \(x = 45^\circ, 135^\circ, 225^\circ, 315^\circ\).
Final Answer:
\[
\boxed{
\begin{array}{ll}
1) & 90^\circ \\
2) & 45^\circ, 225^\circ \\
3) & 0^\circ \\
4) & 30^\circ, 150^\circ \\
5) & 53.1^\circ, 306.9^\circ \\
6) & 11.3^\circ, 191.3^\circ \\
7) & 64.6^\circ, 295.4^\circ \\
8) & 7.1^\circ, 187.1^\circ \\
9) & 41.8^\circ, 138.2^\circ \\
10) & 135^\circ, 315^\circ \\
11) & 244.2^\circ, 295.8^\circ \\
12) & 174.3^\circ, 354.3^\circ \\
13) & 202.0^\circ, 338.0^\circ \\
14) & 8.4^\circ, 81.7^\circ, 188.4^\circ, 261.7^\circ \\
15) & 36.5^\circ, 83.5^\circ, 156.5^\circ, 203.5^\circ \\
16) & 21.0^\circ, 119.0^\circ \\
17) & 152.7^\circ, 332.7^\circ \\
18) & 19.5^\circ, 127.5^\circ, 199.5^\circ, 307.5^\circ \\
19) & 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ \\
20) & 45^\circ, 135^\circ, 225^\circ, 315^\circ \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of junior high math worksheet.
