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Understanding Tangent Ratios in Geometry

Learn about the tangent ratio, its uses in finding angles and sides in right triangles, solve examples, and grasp concepts of sine and cosine functions. Lab work and homework assignments included.

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Understanding Tangent Ratios in Geometry

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • HW Questions (Asgs #75-#77)? • Tangent Ratios • Lab Work • Homework / Questions • Clean-up

  3. The Tangent Ratio • The tangentof an acute angle of a right triangle is the ratio of the length of the opposite leg to the length of the adjacent leg. • Abbreviated: • tan A = opposite leg⁄adjacent leg • tan A = a/b • “the tangent of A is a/b” a b

  4. Tangents • Uses of the tangent ratio: • Given a and b, find tan A or tan B. • Given A, B,tan A, or tan Band either leg, find the other leg. • Given a and b, find A or B. a b

  5. Examples • a = 12 and b = 8; find tan A and tan B. • tan A = 12⁄8 = 3⁄2 • tan B = b⁄a = 8⁄12 = 2⁄3 • tan A = 20 and b = 3; find a. • tan A = a⁄b 20 = a⁄3  a = 60 • tan B = ½ and b = 4; find a. • tan B = b⁄a ½ = 4⁄a  a = 8 a b

  6. Examples – using the calculator • A = 21° and a = 20. Find b. • tan 21° = 20⁄b • tan 21°  0.384 • b  20⁄0.384  52.08 • A = 33° and b = 9. Find a. • tan 33° = a⁄9 • tan 33°  0.649 • a 9  0.649  5.84 a b

  7. Tangent Review Answer: tan 18.92° = 12⁄ON tan 18.92°  0.3428 0.3428 12⁄ON ON  12⁄0.3428  35.01 • MNO is a right triangle with N = 18.92° and leg MO = 12. • Sketch the triangle and label the known parts. • Set up an equation using tan to find leg ON. M N O

  8. Velocity & Motion Q: How far should the ball move, horizontally & vertically? 4px ? 61.2° ? Problem: tan requires a leg.

  9. The Sine Function • In a right triangle, the sine of an acute angle A is the ratio of the opposite leg to the hypotenuse. • sin A = BC/AB • In the game problem, what ratio does sin 61.2° represent? B A C 61.2°

  10. Cosine • The cosine of A is defined as the ratio of the adjacent leg to the hypotenuse. • cos A = AC/AB • In the game problem, what ratiodoes cos 61.2° represent? B A C 61.2°

  11. Summary of Trig Functions • In a right triangle: • The sine of an acute angle is the opposite leg over the hypotenuse. • The cosine of an acute angle is the adjacent leg over the hypotenuse. • The tangent of an acute angle is the opposite leg over the adjacent leg. • Mnemonic: “SOH-CAH-TOA.”

  12. Lab / Homework • 50+ minutes over the weekend: • Assignments #75-80. • Proof practice • Asgs #75-76 have problems about similar right triangles. • Lab: • Work on proof write-up and portfolio.

  13. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

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