Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$225^{\circ}$$

Step-by-Step Solution

Answer

The sine of \(225^\circ\) is \(-\sqrt{2}/2\), the cosine of \(225^\circ\) is \(-\sqrt{2}/2\), and the tangent of \(225^\circ\) is 1.

Step 1: Identify the quadrant

The given angle is \(225^\circ\). This angle lies in the third quadrant of the unit circle where an angle between \(180^\circ\) and \(270^\circ\) falls.

Step 2: Find the related angle

To calculate the sine, cosine and tangent of the angle in the third quadrant we find the related acute angle. The related acute angle, \(A\), can be found by subtracting \(180^\circ\) from the angle: \(225^\circ - 180^\circ = 45^\circ\)

Step 3: Determine the sine

The sine of an angle in the third quadrant is negative and the sine of \(45^\circ\) is \(\sqrt{2}/2\). Therefore, the sine of \(225^\circ\) is \(-\sqrt{2}/2\).

Step 4: Determine the cosine

The cosine of an angle in the third quadrant is negative and the cosine of \(45^\circ\) is \(\sqrt{2}/2\). Therefore, the cosine of \(225^\circ\) is \(-\sqrt{2}/2\).

Step 5: Determine the tangent

The tangent of an angle in the third quadrant is positive. Tangent is equal to sine divided by cosine. Therefore, the tangent of \(225^\circ\) is \(-\sqrt{2}/2\) divided by \(-\sqrt{2}/2\), which equals 1.