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Chapter 4: Problem 58

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$510^{\circ}$$

Short Answer

Expert verified
The sine, cosine and tangent of \(510^{\circ}\) are \(\sin(510^{\circ})= \frac{1}{2}\), \(\cos(510^{\circ})= -\frac{\sqrt{3}}{2}\) and \(\tan(510^{\circ})= -\frac{\sqrt{3}}{3}\) respectively.

Step by step solution

01

Convert to a value less than 360

Subtract multiples of 360 from 510 until a value less than 360 is obtained. In this case: \(510 - 1(360) = 150^{\circ}\)
02

Calculate sine of the angle

Remember sin values for standard angles and note that \(sin(150^{\circ}) = sin(180^{\circ} - 150^{\circ}) = sin(30^{\circ}) = \frac{1}{2}\)
03

Calculate cosine of the angle

Similarly, \(cos(150^{\circ}) = cos(180^{\circ} - 150^{\circ}) = -cos(30^{\circ}) = -\frac{\sqrt{3}}{2}\)
04

Calculate tangent of the angle

Lastly, \(tan(150^{\circ}) = \frac{sin(150^{\circ})}{cos(150^{\circ})} = \frac{1/2}{-\sqrt{3}/2} = -\frac{\sqrt{3}}{3}\)

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