Or press and hold the picture and choose “Save image”.
for Whoobee's shop
Double-Angle Formulas — Easy Worksheet 1
Grade 12 · MathSheets Name: ____________________ Date: __________
Give exact answers: fractions in simplest form, or radicals like √3/2.
How to solve
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Example
cos θ = 817 and θ is acute. Find the exact value of sin 2θ.
Answer: 240289
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (817)2 = 1 − 64289 = 225289, so cos θ = ±1517. θ is acute, so cos θ = 1517. Double-angle formula: cos 2θ = cos2 θ − sin2 θ = (1517)2 − (817)2 = 225289 − 64289 = 161289. So cos 2θ = 161289.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (817)2 = 1 − 64289 = 225289, so cos θ = ±1517. θ is acute, so cos θ = 1517. Double-angle formula: cos 2θ = cos2 θ − sin2 θ = (1517)2 − (817)2 = 225289 − 64289 = 161289. So cos 2θ = 161289.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 725 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (45)2 = 1 − 1625 = 925, so cos θ = ±35. θ is acute, so cos θ = 35. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (45) × (35) = 2425. So sin 2θ = 2425.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (45)2 = 1 − 1625 = 925, so cos θ = ±35. θ is acute, so cos θ = 35. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (45) × (35) = 2425. So sin 2θ = 2425.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 2425 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 This matches the double-angle formula cos 2θ = cos2 θ − sin2 θ with θ = 67.5°, so it equals cos(2 × 67.5°) = cos(135°). 135° is in quadrant II, so the reference angle is 180° − 135° = 45°. cos(45°) = √22, and cosine is negative in quadrant II, so cos(135°) = -√22.
✓ Correct!
Not quite. The answer is
💡 This matches the double-angle formula cos 2θ = cos2 θ − sin2 θ with θ = 67.5°, so it equals cos(2 × 67.5°) = cos(135°). 135° is in quadrant II, so the reference angle is 180° − 135° = 45°. cos(45°) = √22, and cosine is negative in quadrant II, so cos(135°) = -√22.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
cos θ = 35 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: sin2 θ = 1 − (2425)2 = 1 − 576625 = 49625, so sin θ = ±725. θ is acute, so sin θ = 725. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (725) × (2425) = 336625. So sin 2θ = 336625.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: sin2 θ = 1 − (2425)2 = 1 − 576625 = 49625, so sin θ = ±725. θ is acute, so sin θ = 725. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (725) × (2425) = 336625. So sin 2θ = 336625.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 1213 and θ is acute. Find the exact value of sin 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: sin2 θ = 1 − (35)2 = 1 − 925 = 1625, so sin θ = ±45. θ is acute, so sin θ = 45. Double-angle formula: cos 2θ = cos2 θ − sin2 θ = (35)2 − (45)2 = 925 − 1625 = -725. So cos 2θ = -725.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: sin2 θ = 1 − (35)2 = 1 − 925 = 1625, so sin θ = ±45. θ is acute, so sin θ = 45. Double-angle formula: cos 2θ = cos2 θ − sin2 θ = (35)2 − (45)2 = 925 − 1625 = -725. So cos 2θ = -725.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 35 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 This matches the double-angle formula cos 2θ = cos2 θ − sin2 θ with θ = 22.5°, so it equals cos(2 × 22.5°) = cos(45°). From the 45°-45°-90° triangle, cos(45°) = √22.
✓ Correct!
Not quite. The answer is
💡 This matches the double-angle formula cos 2θ = cos2 θ − sin2 θ with θ = 22.5°, so it equals cos(2 × 22.5°) = cos(45°). From the 45°-45°-90° triangle, cos(45°) = √22.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
cos θ = 45 and θ is acute. Find the exact value of sin 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: sin2 θ = 1 − (35)2 = 1 − 925 = 1625, so sin θ = ±45. θ is acute, so sin θ = 45. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (45) × (35) = 2425. So sin 2θ = 2425.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: sin2 θ = 1 − (35)2 = 1 − 925 = 1625, so sin θ = ±45. θ is acute, so sin θ = 45. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (45) × (35) = 2425. So sin 2θ = 2425.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
cos θ = 2425 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 This matches the double-angle formula sin 2θ = 2 sin θ cos θ with θ = 15°, so it equals sin(2 × 15°) = sin(30°). From the 30°-60°-90° triangle, sin(30°) = 12.
✓ Correct!
Not quite. The answer is
💡 This matches the double-angle formula sin 2θ = 2 sin θ cos θ with θ = 15°, so it equals sin(2 × 15°) = sin(30°). From the 30°-60°-90° triangle, sin(30°) = 12.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
cos θ = 817 and θ is acute. Find the exact value of sin 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Log in to save your progress.
Double-Angle Formulas — Easy Worksheet 1
Give exact answers: fractions in simplest form, or radicals like √3/2.