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Chapter 5: Problem 76

Explain why \(\sin 3^{\circ}+\sin 357^{\circ}=0\).

Short Answer

Expert verified
The given equation can be rewritten as \(\sin(3^{\circ}) + \sin(357^{\circ}) = \sin(3^{\circ}) + \sin(180^{\circ} - 3^{\circ})\). Using the sine property for negative angles, we have \(\sin(180^{\circ} - 3^{\circ}) = -\sin(3^{\circ})\). Substituting this back into the equation, we get \(\sin(3^{\circ}) + \sin(357^{\circ}) = \sin(3^{\circ}) - \sin(3^{\circ})\), which simplifies to \(0 = 0\), proving that \(\sin(3^{\circ}) + \sin(357^{\circ}) = 0\).

Step by step solution

01

Observe the relationship between the given angles

Notice that \(360^{\circ} - 3^{\circ} = 357^{\circ}\). This relationship will be important as it will allow us to take advantage of specific trigonometric properties.
02

Use the sine property for supplementary angles

Recall that \(\sin(180^{\circ} - x) = \sin(x)\) for any angle \(x\). Therefore, we can examine how this property applies to our angles: \(\sin(360^{\circ} - 357^{\circ}) = \sin(3^{\circ})\) Since \(360^{\circ}-180^{\circ}=180^{\circ}\), then \(\sin(180^{\circ} - 357^{\circ}) = \sin(3^{\circ})\)
03

Rewrite the original equation

Using the property demonstrated in Step 2, rewrite the original equation: \(\sin(3^{\circ}) + \sin(357^{\circ}) = \sin(3^{\circ}) + \sin(180^{\circ} - 3^{\circ})\)
04

Use the sine property for negative angles

Recall that \(\sin(-x) = -\sin(x)\) for any angle \(x\). Therefore, \(\sin(180^{\circ} - 3^{\circ}) = -\sin(3^{\circ})\)
05

Apply the results of Steps 3 and 4 to solve the equation

Combine the results of Steps 3 and 4 to rewrite the original equation: \(\sin(3^{\circ}) + \sin(357^{\circ}) = \sin(3^{\circ}) - \sin(3^{\circ})\) Now, we can see that the equation simplifies to: \(0 = 0\) So, we have proved that \(\sin(3^{\circ}) + \sin(357^{\circ}) = 0\).

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