Introduction To Trigonometry

1/√3

Step 1: In a 30°-60°-90° triangle, the sides are in ratio 1:√3:2. Step 2: For the 30° angle, opposite side = 1 and adjacent side = √3. Step 3: tan 30° = opposite/adjacent = 1/√3. Step 4: We can rationalize: 1/√3 = √3/3, but 1/√3 is the standard form. Step 5: Therefore, tan 30° = 1/√3.

2

Step 1: We know that sec θ = 1/cos θ. Step 2: First find cos 60° = 1/2 (standard value). Step 3: Therefore, sec 60° = 1/cos 60° = 1/(1/2). Step 4: sec 60° = 2. Step 5: Verification: In a 30°-60°-90° triangle with hypotenuse 2 and adjacent side to 60° angle = 1, sec 60° = 2/1 = 2.

4/3

Step 1: Given cot θ = 3/4. Step 2: We know that cot θ and tan θ are reciprocals, so cot θ × tan θ = 1. Step 3: Therefore, tan θ = 1/cot θ. Step 4: tan θ = 1/(3/4) = 4/3. Step 5: Verification: tan θ × cot θ = (4/3) × (3/4) = 1 ✓.

1

Step 1: Consider a right triangle where one angle approaches 90°. Step 2: As the angle becomes 90°, the opposite side becomes equal to the hypotenuse. Step 3: sin 90° = opposite/hypotenuse = hypotenuse/hypotenuse = 1. Step 4: This is also evident from the unit circle where at 90°, the y-coordinate is 1. Step 5: Therefore, sin 90° = 1.