Double Angle and Half Angle Identities

Step 1: Find cos(θ) using Pythagorean identity: sin²(θ) + cos²(θ) = 1 (3/5)² + cos²(θ) = 1 9/25 + cos²(θ) = 1 cos²(θ) = 16/25 cos(θ) = ±4/5

Step 2: Determine sign of cos(θ): θ is in Quadrant II, so cos(θ) is negative cos(θ) = -4/5

Step 3: Use double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ)

Step 4: Substitute values: sin(2θ) = 2(3/5)(-4/5) = 2(-12/25) = -24/25

Answer: sin(2θ) = -24/25

Step 1: Choose a double angle formula for cosine: There are three options: • cos(2θ) = cos²(θ) - sin²(θ) • cos(2θ) = 2cos²(θ) - 1 • cos(2θ) = 1 - 2sin²(θ)

Step 2: Use the second formula (easiest with given info): cos(2θ) = 2cos²(θ) - 1

Step 3: Substitute cos(θ) = 5/13: cos(2θ) = 2(5/13)² - 1 = 2(25/169) - 1 = 50/169 - 169/169 = -119/169

Answer: cos(2θ) = -119/169

Step 1: Recognize the double angle pattern: This matches the double angle formula for sine

Step 2: Recall the double angle formula: sin(2A) = 2sin(A)cos(A)

Step 3: Compare to our expression: 2sin(x)cos(x) matches with A = x

Step 4: Apply the formula: 2sin(x)cos(x) = sin(2x)

Answer: sin(2x)

Step 1: Use the half angle formula for sine: sin(θ/2) = ±√[(1 - cos(θ))/2]

Step 2: Substitute cos(θ) = 7/25: sin(θ/2) = ±√[(1 - 7/25)/2] = ±√[(25/25 - 7/25)/2] = ±√[(18/25)/2] = ±√[18/50] = ±√[9/25] = ±3/5

Step 3: Determine the sign: If θ is in Quadrant I, then 0° < θ < 90° Therefore: 0° < θ/2 < 45° θ/2 is in Quadrant I, so sin(θ/2) is positive

Step 4: Choose the positive value: sin(θ/2) = 3/5

Answer: sin(θ/2) = 3/5

Step 1: Start with the double angle formula for sine and cosine: tan(2θ) = sin(2θ)/cos(2θ)

Step 2: Apply double angle formulas: tan(2θ) = [2sin(θ)cos(θ)]/[cos²(θ) - sin²(θ)]

Step 3: Divide numerator and denominator by cos²(θ): tan(2θ) = [2sin(θ)cos(θ)/cos²(θ)]/[(cos²(θ) - sin²(θ))/cos²(θ)]

Step 4: Simplify each part: Numerator: 2sin(θ)cos(θ)/cos²(θ) = 2sin(θ)/cos(θ) = 2tan(θ)

Denominator: (cos²(θ) - sin²(θ))/cos²(θ) = cos²(θ)/cos²(θ) - sin²(θ)/cos²(θ) = 1 - tan²(θ)

Step 5: Combine: tan(2θ) = 2tan(θ)/(1 - tan²(θ))

Identity proven! ✓

Answer: Proven ✓