Trigonometric Functions - I

Second Quadrant

Step 1: We need to determine where 5π/6 lies among the four quadrants. Step 2: Recall the quadrant boundaries: Q1: 0 to π/2, Q2: π/2 to π, Q3: π to 3π/2, Q4: 3π/2 to 2π. Step 3: Convert 5π/6 in comparison: π/2 = 3π/6 and π = 6π/6. Step 4: Since 3π/6 < 5π/6 < 6π/6, i.e., π/2 < 5π/6 < π, the angle lies in the Second Quadrant. ✓

Positive

Step 1: First find in which quadrant 5π/9 lies. Step 2: Compare with boundaries: π/2 = 4.5π/9 and π = 9π/9. Step 3: Since 4.5π/9 < 5π/9 < 9π/9, i.e., π/2 < 5π/9 < π, the angle is in the Second Quadrant. Step 4: In the Second Quadrant, sin is POSITIVE (remember: All-Sin-Tan-Cos rule — in Q2, only sin is positive). Therefore, sin(5π/9) is positive. ✓

1/2

Step 1: We need to recall the standard trigonometric values table. Step 2: The standard values for sin are: sin(0)=0, sin(π/6)=1/2, sin(π/4)=1/√2, sin(π/3)=√3/2, sin(π/2)=1. Step 3: A memory aid: sin values follow the pattern √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1. Step 4: Therefore, sin(π/6) = 1/2. Common mistake: Confusing sin(π/6) = 1/2 with cos(π/6) = √3/2. ✓

1/2

Step 1: Recall that cosine values go in the REVERSE order compared to sine values. Step 2: sin(π/3) = √3/2, and since cos values decrease as θ increases from 0 to π/2... Step 3: cos(0)=1, cos(π/6)=√3/2, cos(π/4)=1/√2, cos(π/3)=1/2, cos(π/2)=0. Step 4: Therefore, cos(π/3) = 1/2. Key Insight: Notice that cos(π/3) = sin(π/6) = 1/2. Cosine values are the reverse of sine values for standard angles. ✓