Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$300^{\circ}$$

Step-by-Step Solution

Answer

The sine, cosine, and tangent values for \(300^{\circ}\) are \(-\sqrt{3}/2\), \(1/2\), and \(-\sqrt{3}\) respectively.

Step 1: Convert Degree Measurement to Radian Measurement

For converting degrees into radians, we can use the formula \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\). Hence, for an angle of \(300^{\circ}\), the radian measurement will be \(300 \times \frac{\pi}{180}\). However, in the context of finding trigonometric values, it is more convenient to use the degree measure and place the angle on the trigonometric circle.

Step 2: Find the Corresponding Angle in the First Quadrant

Since \(300^{\circ}\) lies in the fourth quadrant and sine, cosine and tangent show different signs in each quadrant, we need to find the corresponding angle in the first quadrant. Subtract the angle from \(360^{\circ}\) to find the corresponding angle. In this case, it will be \(360^{\circ} - 300^{\circ} = 60^{\circ}\). We can now use this angle to find sine, cosine, and tangent as all are positive in the first quadrant.

Step 3: Evaluate the Sine, Cosine, and Tangent Values

For an angle of \(60^{\circ}\), the sine, cosine, and tangent values are well known. We have \(\sin(60^{\circ}) = \sqrt{3}/2\), \(\cos(60^{\circ}) = 1/2\), and \(\tan(60^{\circ}) = \sqrt{3}\). However, since our original angle is in the fourth quadrant where sine is negative and cosine is positive, we need to reverse the sign of the sine value. Therefore, for \(300^{\circ}\) we have \(\sin(300^{\circ}) = -\sqrt{3}/2\), \(\cos(300^{\circ}) = 1/2\), and \(\tan(300^{\circ}) = -\sqrt{3}\).