1 Trigonometry Geometry Mathematics Trig and Geometry cos sin 45 o 30 o 150 o 330 o III IIIIV This is the unit circle… It axes are sine and cosine All lines drawn here have a length of 1 and an angle equal to the angle we are working with. The height along the sin- axis is the sine of the angle. The distance to the right on the cos-axis is the cosine of the angle. Here are some other examples. Note that the angle always goes from the positive cos-axis counterclockwise. Also note that the cosine is negative if the line is drawn to the left on the cos-axis. Why is the sine negative here? We often speak of four quadrants. The first quadrant has positive cosines and sines. The second quadrant has negative cosines and positive sines. The third quadrant has negative cosines and sines. The fourth quadrant has positive cosines and negative sines.

2 Trigonometry Geometry Mathematics Trig and Geometry Given a right triangle, the trigonometric functions for either non-right angle are given by the following… θ hypotenuse ( h ) opposite ( o ) adjacent ( a ) The value of the angle can also be determine by using any two of the sides. For example,

3 Trigonometry Geometry Mathematics Trig and Geometry Here is an example of how to use it… θ =36.87 o h =5 o =3 a =4 The value of the angle can also be determine by using any two of the sides. For example, Note: This is NOT drawn to scale!

4 Trigonometry Geometry Mathematics Trig and Geometry Here are some useful angle relations… a a a b a a a a b b b b b b a b c A C a c B b a a

5 Trigonometry Geometry Mathematics Trig and Geometry For example… a b

6 Trigonometry Geometry Mathematics Trig and Geometry Here are some basic geometric and trigonometric formulae which we will use often in this and the next class… Circumference of a Circle Area of a Circle Surface Area of a Sphere Volume of a Sphere Surface Area of a Cylinder (not including end faces) Volume of a Cylinder Trigonometric Formulae Quadratic Formula