Trigonometric Function Of Any AnglePresentation Transcript
Trigonometric Functions of Any Angle
Definitions of Trigonometric Functions of Any Angle
Let is be any angle in standard position, and let P = ( x , y ) be a point on the terminal side of . If is the distance from (0, 0) to ( x , y ), the six trigonometric functions of are defined by the following ratios.
Text Example Let P = (-3, -4) be a point on the terminal side of . Find each of the six trigonometric functions of . Solution The situation is shown below. We need values for x , y , and r to evaluate all six trigonometric functions. We are given the values of x and y . Because P = (-3, -4) is a point on the terminal side of , x = -3 and y = -4. Furthermore, r x = -3 y = -4 P = (-3, -4) x y -5 5 -5 5
Text Example Cont.
Now that we know x , y , and r , we can find the six trigonometric functions of .
The bottom row shows the reciprocals of the row above.
The Signs of the Trigonometric Functions x y Quadrant II Sine and cosecant positive (-,+) Quadrant I All functions positive (+,+) Quadrant III tangent and cotangent positive (-,-) Quadrant IV cosine and secant positive (+,-)
Example: Evaluating Trigonometric Functions Given tan = -2 / 3 and cos > 0, find cos and csc . Solution Because the tangent is negative and the cosine is positive, lies in quadrant IV. This will help us to determine whether the negative sign in tan = -2 / 3 should be associated with the numerator or the denominator. Keep in mind that in quadrant IV, x is positive and y is negative. Thus, In quadrant IV, y is negative. x = 3 y = -2 P = (3, -2) x y -5 5 -5 5 r = 13 Thus, x = 3 and y = -2. Furthermore, Now that we know x , y and r , find cos and csc .
Definition of a Reference Angle
Let be a non-acute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle ´ (prime) formed by the terminal side of and the x-axis.
Find the reference angle , for the following angle: =315º
´ =360º - 315º = 45º
a b a b P ( a , b )
Using Reference Angles to Evaluate Trigonometric Functions
The values of a trigonometric functions of a given angle, , are the same as the values for the trigonometric functions of the reference angle, ´, except possibly for the sign . A function value of the acute angle, ´, is always positive. However, the same functions value for may be positive or negative.
A Procedure for Using Reference Angles to Evaluate Trigonometric Functions
The value of a trigonometric function of any angle is found as follows:
Find the associated reference angle, ´ , and the function value for ´ .
Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.
Example: Using Reference Angles to Evaluate Trigonometric Functions Use reference angles to find the exact value of each of the following trigonometric functions. a. sin 135° x y 135° 45° more more x y 4 / 3 / 3 x y / 3 - / 3