Trigonometry

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Step 1: Find hypotenuse using Pythagoras theorem: AC² = AB² + BC² = 5² + 12² = 25 + 144 = 169, so AC = 13 cm. Step 2: Identify sides relative to angle C: - Opposite side to C = AB = 5 cm - Adjacent s…

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Using the reciprocal trigonometric ratios from FC1: - sin(θ) = 5/13, cos(θ) = 12/13, tan(θ) = 5/12 Step 1: Find sec(θ) (reciprocal of cos): sec(θ) = 1/cos(θ) = 1/(12/13) = 13/12 Step 2: Find cosec(θ…

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Proof using right-angled triangle: Consider right-angled triangle OMP with right angle at M. Let OP = hypotenuse, OM = adjacent side, MP = opposite side. By Pythagoras theorem: OP² = OM² + MP² Divi…

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Step 1: Use the fundamental identity: sin²θ + cos²θ = 1 Step 2: Rearrange to find cos²θ: cos²θ = 1 - sin²θ Step 3: Substitute sin(θ) = 3/5: cos²θ = 1 - (3/5)² cos²θ = 1 - 9/25 cos²θ = 25/25 - 9/25 c…

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Use 1 + tan²θ = sec²θ when: - You know tan(θ) and need to find sec(θ) - You need to simplify expressions containing both tan and sec - You're proving other trigonometric identities - You need to manip…

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Step 1: Start with the left side and express tan in terms of sin and cos: tan²θ - sin²θ = sin²θ/cos²θ - sin²θ Step 2: Find common denominator: = sin²θ/cos²θ - sin²θ × cos²θ/cos²θ = (sin²θ - sin²θ × c…

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Complementary Angles Relationships: If two angles are complementary (sum = 90°), their trigonometric ratios are related: 1. sin(90° - θ) = cos(θ) 2. cos(90° - θ) = sin(θ) 3. tan(90° - θ) = cot(θ) 4.

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Step 1: Recall standard angle values from the trigonometric table: sin(30°) = 1/2 cos(30°) = √3/2 sin(60°) = √3/2 cos(60°) = 1/2 Step 2: Substitute values into the expression: sin(30°) × cos(60°) + c…