Trigonometry Primer

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.

N S EW 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.

Motion in Two Dimensions A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.

The Sides of a Triangle  hypotenuse opposite adjacent

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

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

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

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

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

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

The Pythagorean Theorem  In a right triangle where c is the hypotenuse: a 2 + b 2 = c 2  Put another way: c = a 2 + b 2

The Pythagorean Theorem  Find the length of side b, given the other two sides.  a 2 + b 2 = c 2  (52.5 m) 2 + b 2 = (76.5 m) 2  2760 m 2 + b 2 = 5850 m 2  b 2 = 3090 m 2  b = 55.6 m 52.5 m b 76.5 m