<Text-field style="Heading 1" layout="Heading 1">Problem 2-2</Text-field>

Trig Equation

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For the equation 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, find all solutions in the interval 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Graphical Solution</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Control-drag the equation. Press the Enter key.

Context Menu: Move to Left

Context Menu: Left-hand Side

Context Menu: Plots\342\211\273Plot Builder Set 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 Options: Range to/from\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSI2RidGLy1GLDYtUSIsRidGL0YyL0Y2USV0cnVlRidGN0Y5RjtGPUY/L0ZCUSYwLjBlbUYnL0ZFUSwwLjMzMzMzMzNlbUYnLUZHNiRRIjBGJ0YvRi8=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Numerical Solution</Text-field>

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Control-drag the equation or reference it by its equation label.

Context Menu: Solve\342\211\273Solve Numerically from a Point Provide appropriate starting value(s) for ensuing Newton iteration(s).

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Solution by Task Template</Text-field>

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Tools\342\211\273Tasks\342\211\273Browse: Algebra\342\211\273Solve Analytically in a Specified Interval

Replace the equation in the task template with the equation label of the given equation.

Press the Enter key.

Solve Analytically in a Specified Interval

Enter an expres sion:

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

LCgqJiIjNyIiIiktSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kmX194X19HRiwiIiNGJUYlKiYiIiZGJUYnRiUhIiIiIiRGMg==

Find the roots in a specified interval:

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Express the roots in floating-point form:

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

NyYkIishej8xWykhIzUkIit2MGAkSCMhIiokIitrJkg5WyRGKCQiKylSW0wlZkYo

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGMg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">General Solution</Text-field>

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Control-drag the equation and press the Enter key.

Context Menu: Solve\342\211\273Solve (general solution)

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 while LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjX1pGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GLzYlUSJrRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw== is an integer

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<Text-field style="Heading 1" layout="Heading 1">Solution from First Principles</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Create a quadratic in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== and factor

Control-drag the equation; press the Enter key.

Select LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEieEYnRj5GQEY1RjVGNQ==: Smart Pop-Up\342\211\273 change LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEieEYnRj5GQEY1RjVGNQ== to 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

Select 3: Smart Pop-Up\342\211\273Subtract from each side

Select left-hand side: Smart Pop-Up\342\211\273Factor

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Set each factor to zero and solve

Control-drag the first factor; equate to zero.

Select LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg==: Isolate via Smart Pop-Up

Select LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: Isolate via Smart Pop-Up

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Control-drag the second factor; equate to zero.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Apply standard quadrant analysis to obtain the additional two solutions.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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