Trigonometric Ratios Of Some Special Angles

1

Step 1: cot θ = 1/tan θ or cot θ = cos θ/sin θ. Step 2: We know tan 45° = 1, so cot 45° = 1/tan 45° = 1/1 = 1. Step 3: Alternatively, cot 45° = cos 45°/sin 45° = (1/√2)/(1/√2) = 1. Step 4: In a 45-45-90 triangle, adjacent = opposite, so cot 45° = adjacent/opposite = 1. The answer is 1.

1

Step 1: At 90°, the angle is a right angle. Step 2: In this case, the opposite side equals the hypotenuse (the angle opens completely). Step 3: sin 90° = opposite/hypotenuse = hypotenuse/hypotenuse = 1. Step 4: This is a standard value that represents the maximum value of sine function. The answer is 1.

tan 90°

Step 1: tan θ = sin θ/cos θ. Step 2: At 90°, sin 90° = 1 and cos 90° = 0. Step 3: tan 90° = sin 90°/cos 90° = 1/0, which is undefined (division by zero). Step 4: All other ratios are defined: sin 90° = 1, cos 90° = 0, csc 90° = 1. The answer is tan 90°.

1

Step 1: At 0°, there is no angle - the ray lies along the positive x-axis. Step 2: In this position, the adjacent side equals the hypotenuse. Step 3: cos 0° = adjacent/hypotenuse = hypotenuse/hypotenuse = 1. Step 4: This represents the maximum value of the cosine function. The answer is 1.