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Trigonometry Primer

Learn the basics of trigonometry and how it helps analyze vectors, motion, and triangle relationships. Discover how to calculate side lengths, angles, and apply the Pythagorean theorem.

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Trigonometry Primer

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  1. Trigonometry Primer

  2. What is Trigonometry • Trig deals with triangles. • Usu. right triangles. • Why is trig useful in physics? • Because many kinds of motion can be broken down into components at right angles to each other. • Allows us to analyze vectors in two or three dimensions.

  3. N W E S Motion in Two Dimensions • The arrow indicates the velocity of a bird in flight. • We can picture the bird’s velocity as the sum of two imaginary velocities. • Trig allows us to make the necessary calculations.

  4. Motion in Two Dimensions • A boat is moving across a river at 10 m/s. • The river’s current is pushing the boat downstream at 5 m/s. • Trig allows us to calculate how fast the boat is moving and in what direction.

  5. The Sides of a Triangle hypotenuse opposite adjacent

  6. opposite adjacent opposite hypotenuse hypotenuse adjacent Basic Trig Functions • Sine (or sin) = • Cosine (or cos) = • Tangent (or tan) =

  7. a 50.0 m a 50.0 m 30º Basic Trig Functions • Find the length of side a, given the angle and hypotenuse. • sin30º = • 50.0 m(sin30º) = a • a = 25.0 m

  8. b 50.0 m 50.0 m 30º b Basic Trig Functions • Find the length of side b, given the angle and hypotenuse. • cos30º = • 50.0 m(cos30º) = b • b = 43.3 m

  9. a a 10.0 m 70º 10.0 m Basic Trig Functions • Find side a, given the angle and adjacent side. • tan70º = • 10.0 m(tan70º) = a • a = 27.5 m

  10. Inverse Trig Functions • Can be used to find the angle when the appropriate sides are known. • Written as arcsin, arccos, and arctan • Sometimes as sin-1, cos-1, and tan-1. • Generally avoid the latter as it can be confusing. • If sin(x) = y, then arcsin(y) = x

  11. 25.0 m 25.0 m 35.0 m A 35.0 m Inverse Trig Functions • Find the measure of angle A, given the opposite and adjacent sides. • tanA = • tanA = 0.714 • arctan(0.714) = A • A = 35.5º

  12. The Pythagorean Theorem • In a right triangle where c is the hypotenuse: • a2 + b2 = c2 • Put another way: • c = a2 + b2

  13. 76.5 m b 52.5 m The Pythagorean Theorem • Find the length of side b, given the other two sides. • a2 + b2 = c2 • (52.5 m)2 + b2 = (76.5 m)2 • 2760 m2 + b2 = 5850 m2 • b2 = 3090 m2 • b = 55.6 m

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