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Chapter 4: Problem 55

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$750^{\circ}$$

Short Answer

Expert verified
The sine of \(750^\circ\) is \(1/2\), the cosine of \(750^\circ\) is \(\sqrt{3}/2\), and the tangent of \(750^\circ\) is \(1/\sqrt{3}\).

Step by step solution

01

Reduce the given angle to fall within \(360^\circ\)

The given angle is \(750^\circ\). Start subtracting \(360^\circ\) until the angle is less than \(360^\circ\). \(750^\circ\) is greater than \(720^\circ = 2 \times 360^\circ\), but less than \(1080^\circ = 3 \times 360^\circ\). Therefore, subtract \(2 \times 360^\circ\) from the given angle, getting the result \(750^\circ - 2 \times 360^\circ = 390^\circ - 360^\circ = 30^\circ\).
02

Determine the sine, cosine, and tangent of the reduced angle \(30^\circ\)

Recall the standard values. The sine of \(30^\circ\) is \(1/2\), the cosine of \(30^\circ\) is \(\sqrt{3}/2\), and the tangent of \(30^\circ\) is \(\sqrt{3}/3\) or \(1/\sqrt{3}\).
03

Determine the sine, cosine, and tangent of the given angle \(750^\circ\)

Because the values of sine, cosine and tangent repeat every \(360^\circ\), then the sine, cosine and tangent of \(750^\circ\) are the same as for \(30^\circ\). Thus, \(\sin750^\circ = 1/2\), \(\cos750^\circ = \sqrt{3}/2\), and \(\tan750^\circ = 1/\sqrt{3}\).

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