Honors Geometry Section 10.3 Trigonometry on the Unit Circle

Trigonometry has been used by many cultures for over 4,000 years Trigonometry has been used by many cultures for over 4,000 years! On page 870 of your text is a table of approximate values for the sine, cosine and tangent of angles between 0° and 90°. These are the same approximate values your calculator will give you. You may wonder where these values come from? One way of finding some of these values is to use the unit circle, a circle with its center _______________ of the coordinate plane and with a radius of ___. at the origin (0, 0)

Let’s put a 30° angle in the coordinate plane so that its vertex is at the origin, one side lies on the positive x-axis and the second side lies in quadrant I. Call the point where the second side intersects the unit circle point A.

Draw a line from point A perpendicular to the x-axis Draw a line from point A perpendicular to the x-axis. A 30-60-90 triangle is formed. Which side of this triangle do we know the length of? ___________ What is its length? ____ What are the lengths of the two legs in this 30-60-90 triangle? hypotenuse 1

Use the lengths of the sides in the 30-60-90 triangle, to find the exact value of:   sin 30°= cos 30°= tan 30°=

Let’s do the same with a 45° angle Let’s do the same with a 45° angle. Use the lengths of the sides in the 45-45-90 triangle, to find the exact value of sin 45°= cos 45°= tan 45°=

We can extend the unit circle to include angles greater that 90° and negative angles by discussing the angle of rotation.   An angle of rotation has its vertex at the origin and one side on the positive x-axis. The side on the x-axis is called the _____ side of the angle. The second side of the angle, called the _______side, is obtained by doing a rotation either counterclockwise for ________angles or clockwise for ________ angles. initial terminal positive negative

Unit Circle Definition of Sine and Cosine: Let (the Greek letter theta) be an angle of rotation. Sin is equal to the ____________of the point where the terminal side intersects the unit circle and cos is equal to the ____________ of this point. Tan equals y-coordinate x-coordinate

Examples: Give the exact value of the sine, cosine and tangent for each angle measure.

Examples: Give the exact value of the sine, cosine and tangent for each angle measure.