WARM UP For θ = 2812° find a coterminal angle between 0° and 360°. What is a periodic function? What are the six trigonometric functions? 292° A function with repeating values Sine, cosine, tangent, cotangent, secant, cosecant

Values of the Six Trigonometric Functions

OBJECTIVES Be able to find values of the six trigonometric functions approximately, by calculator, for any angle and exactly for certain special angles. In this section you will define four other trigonometric functions

KEY TERMS & CONCEPTS Cotangent Tangent Secant Cosecant Reciprocal properties Unit circle Ellipsis format Complementary angles

THE 6 TRIGONOMETRIC FUNCTIONS Sine and cosine have been defined for any angle as ratios of the coordinates (u, v) of a point on the terminal side of the angle and, equivalently, as ratios of the displacements in the reference angle. Four other ratios can be made using u, v and r. Their names are tangent, cotangent, secant, and cosecant.

THE 6 TRIGONOMETRIC FUNCTIONS The right triangle definition of tangent for an acute angle is extended to the ratio of v to u for a point on the terminal side of any angle. The cotangent, secant and cosecant functions are reciprocal of the tangent, cosine, and sine functions, respectively. The relationship between each pair of functions, such as cotangent and tangent, is called the reciprocal property of trigonometric functions.

THE 6 TRIGONOMETRIC FUNCTIONS When you write the functions in a column in the order sin θ, cos θ, tan θ, cot θ, sec θ, csc θ, the functions and their reciprocals have this pattern. sin θ cos θ tan θ cot θ sec θ csc θ reciprocals

DEFINITIONS Let (u, v) be a point r units from the origin on the terminal side of a rotating ray. If θ is the angle to the ray, in standard position, then the following definitions hold. Right Triangle Form Coordinate Form Note: The coordinates u and v are also the horizontal and vertical displacements of the point (u, v) in the reference triangle.

THE NAMES TANGENT AND SECANT To see why the names tangent and secant are used, look at the graph. The point (u, v) has been chosen on the terminal side of angle θ where r = 1 unit. The circle traced by (u, v) is called the unit circle (a circle with radius 1 unit). The value of sine is given by

TANGENT Hence the name tangent is used. Thus the sine of an angle is equal to the vertical coordinate of a point on the unit circle. Similarly, cos θ = u/1 = u, the horizontal coordinate of a point on the unit circle.

SECANT The hypotenuse of this larger reference triangle is part of a secant line, a line that cuts through the circle. In the larger triangle, Hence the name secant is used. By properties of similar triangles, for any point (u, v) on the terminal side and

EXAMPLE 1 Example 1 shows how you can find approximate values of all six trigonometric functions using your calculator. Evaluate by calculator the six trigonometric functions of 58.6°. Round to four decimal places. Solution: You can find sine, cosine, and tangent directly on the calculator. sin 58.6° = ≈ Note preferred usage of the ellipsis and the ≈ sign. cos 58.6° = ≈ tan 58.6° = ≈

EXAMPLE 1 CONT. The other three functions are the reciprocals of the sine, cosine, and tangent functions. Notice that the reciprocals follow the pattern described earlier.

EXACT VALUES BY GEOMETRY If you know a point on the terminal side of an angle, you can calculate the values of the trigonometric functions exactly. Example 2 shows you the step The terminal side of angle θ contains the point (−5, 2). Find exact values of the six trigonometric functions of θ. Use radicals if necessary, but no decimals.

EXAMPLE 2 Sketch the angle in standard position Pick a point on the terminal side, (-5, 2) in this instance and draw a perpendicular. Solution: You can find sine, cosine, and tangent directly on the calculator. Mark the displacements on the reference triangle, using the Pythagorean theorem.

EXAMPLE 2 CONT.

Note: You can use the proportions of the side lengths 30°-60°-90° triangle and the 45°-45°-90° triangle to find exact function values for angles whose reference angle measure is a multiple of 30° or 45°. The graphs show these proportions.

EXAMPLE 3 Find the exact values (no decimals) of the six trigonometric functions of 300° SOLUTION: Sketch an angle terminating in Quadrant IV and a reference triangle. Use the negative square root because v is negative. Simplify Use reciprocal property

EXAMPLE 3 CONT. Note: To avoid errors in placing the 1, 2, and √3 on the reference triangle, remember that the hypotenuse is the longest side of a right triangle and that 2 is greater than √3.

EXAMPLE 4 This example will show you how to find the function values for an angle that terminates on a quadrant boundary. Do the obvious simplification Example 4: Without using a calculator, evaluate the six trigonometric functions for an angle of 180°.

SOLUTION The graph shows a 180° angle in standard position. The terminal side falls on the negative side of the horizontal axis. Pick any point on the terminal side, such as (-3, 0). Note that although the u-coordinate of the point is negative, the distance r from the origin to the point is positive because it is the radius of a circle. The vertical coordinate, v, is 0. Use “vertical, radius” rather than “opposite, hypotenuse.” Do the obvious simplification

CH. 2.4 ASSIGNMENTS Textbook pg. 78 #2 through 40, every fourth one & 44.