Presentation on theme: "7.4 Trigonometric Functions of General Angles"— Presentation transcript:
17.4 Trigonometric Functions of General Angles MAT 204 SP097.4 Trigonometric Functions of General AnglesIn this section, we will study the following topics:Evaluating trig functions of any angleUsing the unit circle to evaluate the trig functions of quadrantal anglesFinding coterminal anglesUsing reference angles to evaluate trig functions.
3Trig Functions of Any Angle MAT 204 SP09Trig Functions of Any AngleIn 7.3, we looked at the definitions of the trig functions of acute angles of a right triangle. In this section, we will expand upon those definitions to include ANY angle.We will be studying angles that are greater than 90° and less than 0°, so we will need to consider the signs of the trig functions in each of the quadrants.We will start by looking at the definitions of the trig functions of any angle.
4Definitions of Trigonometric Functions of Any Angle Definitions of Trig Functions of Any AngleDefinitions of Trigonometric Functions of Any AngleLet be an angle in standard position with (x, y) a point on the terminal side of andyx(x, y)r
5Example*Let (-12, -5) be a point on the terminal side of . Find the exact values of the six trig functions of .r-5yx(-12, -5)-12First you must find the value of r:
9The Signs of the Trig Functions Since the radius is always positive (r > 0), the signs of the trig functions are dependent upon the signs of x and y. Therefore, we can determine the sign of the functions by knowing the quadrant in which the terminal side of the angle lies.
11All Students Take Calculus A trick to remember where each trig function is POSITIVE:All Students Take CalculusTranslation:A = All 3 functions are positive in Quad 1S= Sine function is positive in Quad 2T= Tangent function is positive in Quad 3C= Cosine function is positive in Quad 4ACTS*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tan is positive, but sine and cosine are negative; ...**Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, …
12Example Determine if the following functions are positive or negative: sin 210°cos 320°cot (-135°)csc 500°tan 315°
13Example*Given and , find the values of the five other trig function of .SolutionFirst, determine the quadrant in which lies. Since the cosine is negative and the cotangent is positive, we know that lies in Quadrant _____ .
14Example* (cont)Now we can find the values of the remaining trig functions:
15MAT 204 SP09Another ExampleGiven and , find the values of the five other trig functions of .
17Trig functions of Quadrantal Angles To find the sine, cosine, tangent, etc. of angles whose terminal side falls on one of the axes , we will use the unit circle.(1, 0)(0, 1)(-1, 0)(0, -1)Unit Circle:Center (0, 0)radius = 1x2 + y2 = 1
18Now using the definitions of the trig functions with r = 1, we have:
19Example* Find the value of the six trig functions for (0, 1) (-1, 0) MAT 204 SP09Example*Find the value of the six trig functions for(1, 0)(0, 1)(-1, 0)(0, -1)
20MAT 204 SP09ExampleFind the value of the six trig functions for
21In each of these illustrations, angles and are coterminal. Coterminal AnglesTwo angles in standard position are said to be coterminal if they have the same terminal sides. is a negative anglecoterminal to is a positive angle (> 360°)coterminal to In each of these illustrations, angles and are coterminal.
22Example of Finding Coterminal Angles You can find an angle that is coterminal to a given angle by adding or subtracting multiples of 360º or 2.Example:Find one positive and one negative angle that are coterminal to 112º.For a positive coterminal angle, add 360º : 112º + 360º = 472ºFor a negative coterminal angle, subtract 360º: 112º - 360º = -248ºNote: There are an infinite number of angles that are coterminal to 112 º.
23ExampleFind one positive and one negative coterminal angle of
26Reference AnglesThe values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles.I will use the notation to represent an angle’s reference angle.
28Example Find the reference angle for Solution yxBy sketching in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.
29More Examples Find the reference angles for the following angles.
30So what’s so great about reference angles? Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies.For example,In Quad 3, sin is negative45° is the ref angle
31Trig Functions of Common Angles Using reference angles and the special reference triangles, we can find the exact values of the common angles.To find the value of a trig function for any common angle Determine the quadrant in which the angle lies.Determine the reference angle.Use one of the special triangles to determine the function value for the reference angle.Depending upon the quadrant in which lies, use the appropriate sign (+ or –).
32More ExamplesGive the exact value of the trig function (without using a calculator).
33More ExamplesGive the exact value of the trig function (without using a calculator).