Trigonometry

60 m

Step 1: Height of lighthouse = 60 m, angle of depression = 45°. Step 2: The angle of depression equals the angle of elevation from the ship (alternate angles). So angle of elevation from ship = 45°. Step 3: tan 45° = height/horizontal distance = 60/d. Step 4: 1 = 60/d, so d = 60 m. A common mistake is forgetting that angle of depression = angle of elevation (alternate interior angles).

sec θ - cos θ = 1/cos θ - cos θ = (1 - cos²θ)/cos θ

Step 1: Replace sec θ with 1/cos θ to get 1/cosθ - cosθ. Step 2: Take LCM: (1 - cos²θ)/cosθ. Step 3: Use identity 1 - cos²θ = sin²θ: sin²θ/cosθ. Step 4: Write as (sinθ/cosθ) × sinθ = tanθ × sinθ = tanθ sinθ. The key is substituting sec θ = 1/cos θ first and then using the Pythagorean identity.

40√3 m

Step 1: Height of kite = 60 m (opposite side), angle with ground = 60°, string = hypotenuse. Step 2: sin 60° = height/length of string = 60/L. Step 3: √3/2 = 60/L, so L = 120/√3 = 120√3/3 = 40√3 m. A common error is using cos instead of sin, or not rationalizing the denominator.

sec θ cosec θ

Step 1: cot θ + tan θ = cosθ/sinθ + sinθ/cosθ. Step 2: Take LCM (sinθ cosθ): (cos²θ + sin²θ)/(sinθ cosθ). Step 3: Use sin²θ + cos²θ = 1: = 1/(sinθ cosθ). Step 4: Write as (1/sinθ)(1/cosθ) = cosecθ × secθ. Many students stop at step 2 and don't simplify using the Pythagorean identity.