Solution: The angle $ 120^\circ $ lies in the second quadrant. Using the identity $ an(180^\circ - heta) = - an heta $, we write:
["Understanding Why an Angle of 120° Resides in the Second Quadrant: A Clear Solution Using Trigonometric Identities", "When studying trigonometry, one key concept is determining in which quadrant a given angle lies—and how this affects its sign and properties using fundamental identities. A classic example is the angle $ 120^\circ $, which lies in the second quadrant. But how do we formally prove this using the identity $ \cos(180^\circ - \ heta) = -\cos \ heta $? Let’s explore this step by step.", "### What Does It Mean for an Angle to Lie in the Second Quadrant?", "Angles are measured counterclockwise from the positive x-axis.\n- The second quadrant spans from $ 90^\circ $ to $ 180^\circ $.\n- Since $ 120^\circ $ falls within this range, it clearly lies in the second quadrant.", "But identifying quadrants is only part of the story—understanding how trigonometric functions behave there helps reinforce this classification.", "### Using the Identity $ \cos(180^\circ - \ heta) = -\cos \ heta $", "This powerful identity reveals key symmetry about the y-axis.\nLet’s apply it with $ \ heta = 60^\circ $:\n$$\n\cos(180^\circ - 60^\circ) = \cos(120^\circ) = -\cos(60^\circ)\n$$\nSince $ \cos(60^\circ) = \frac{1}{2} $, we find:\n$$\n\cos(120^\circ) = -\frac{1}{2}\n$$", "### Why This Confirms 120° Is in the Second Quadrant", "The cosine function is defined as the x-coordinate on the unit circle.\n- In the second quadrant ($ 90^\circ < \ heta < 180^\circ $), x-values are negative.\n- The negative result $ \cos(120^\circ) = -\frac{1}{2} $ confirms this.\n- Meanwhile, sine is still positive in this region ($ \sin(120^\circ) = \frac{\sqrt{3}}{2} > 0 $), further aligning with second quadrant behavior.", "Thus, using the identity $ \cos(180^\circ - \ heta) = -\cos \ heta $, we deduce that angles greater than $ 90^\circ $ and less than $ 180^\circ $ have negative cosine values—defining their placement in the second quadrant.", "### Summary", "- $ 120^\circ $ is mathematically positioned between $ 90^\circ $ and $ 180^\circ $, confirming its placement in the second quadrant.\n- The identity $ \cos(180^\circ - \ heta) = -\cos \ heta $ explains why $ \cos(120^\circ) $ is negative, directly linking algebraic behavior to quadrant classification.\n- Understanding such identities strengthens conceptual clarity in trigonometry, empowering students to analyze angles and functions with confidence.", "Whether you’re solving equations, graphing functions, or studying unit circle relationships, recognizing the quadrant through function values and identities is essential. Remember:\nAn angle of $ 120^\circ $ lies in the second quadrant, where cosine is negative and sine is positive—supported by the identity $ \cos(180^\circ - \ heta) = -\cos \ heta $.", "---", "Use this insight to master trigonometric quadrant rules and deepen your understanding of angles in the coordinate plane!"]
