Section 5.1 Trigonometric Functions of Real Numbers In calculus and in the sciences many of the applications of the trigonometric functions require that the inputs be real numbers, rather than angles. By making this small but crucial change in our viewpoint, we can define the trigonometric functions in such a way that the inputs are real numbers. The definitions of the trig functions, and the identities that we have already met (and will meet later) will remain the same, and will be valid whether the inputs are angles or real numbers. Definition of the Trigonometric Functions of Real Numbers Let Pxy(, ) be a point on the unit circle xy22 1 whose arc length from the point (1,0) is t . Then the six trigonometric functions of the real number t is defined as: 1 sin(ty ) csc( t ) ( y 0) y 1 cos(tx ) sec( t ) ( x 0) x yx tan(tty ) cot( ) ( 0) xy 1 Here are some identities you need to know: Reciprocal Identities Pythagorean Identities sin(t) sin22 (tt ) cos ( ) 1 tan(t) cos(t) 1tan()sec()22tt cos(t) 1cot()csc()22tt cot(t) sin(t) 1 sec(tt ) , cos( ) 0 cos(t ) 1 csc(tt ) , sin( ) 0 sin(t ) 1 cot(tt ) , tan( ) 0 tan(t ) 2 1 Example 1: If tan(t ) and sin(t ) 0, find sec(t ) . 5 1 Example 2: Given that cos(t) and tan(t ) 0 ; find sin(t ) and tan(t ). 4 3 Opposite Angle Identities The positive direction from the point (1,0) is the counterclockwise direction and the negative direction is the clockwise direction. sin(tt ) sin( ) cos(tt ) cos( ) tan(tt ) tan( ) cot(tt ) cot( ) sec(tt ) sec( ) csc(tt ) csc( ) These identities tell us that: Sine, tangent, cotangent and cosecant are ODD functions; Cosine and Secant are EVEN functions. These identities hold true both for angles and real numbers. 4 Example 3: Use the opposite-angle identities to find: 2 a. sin 3 5 b. cos 6 3 c. tan cot 44 5 12 Example 4: Given: sin(x ) , find the value of: sin(x ) cos(x ) . 13 6 POPPER FOR SECTION 5.1 Question#1: Which of the following statements is/are true? I. 1tan()sec()22tt II. cos(x ) cos(x ) III. sin(x )csc(x ) 1 A) I only B) II only C) III only D) I and III only E) I, II and III F) None of these 7 Periodic functions: A function is periodic if there is some number p such that f (x p) f (x) . The smallest such number is called the “period” of this function. Periodicity of Sine and Cosine Functions The circumference of the unit circle is 2 . Thus, if we start at a point P and travel for 2 units, we come back to that point P. The period for sine and cosine functions is 2 . sin(tt 2 ) sin( ) cos(tt 2 ) cos( ) More generally: sin(tk 2 ) sin( t ) cos(tk 2 ) cos( t ) for all real numbers t and all integers k. 8 Periodicity of Tangent and Cotangent Functions The tangent and cotangent functions are also periodic functions. However, these functions repeat themselves after each . sin(tt ) sin( ) tan(tt ) tan( ) cos(tt ) cos( ) So tan(t ) tan(t) more generally tan(t k ) tan(t) cot(t ) cot(t) cot(t k ) cot(t) for all real numbers t and all integers k. 9 Remark: The period for the sine and cosine functions is 2 while the period for the tangent and cotangent functions is . Since secant and cosecant functions are the reciprocal functions, they will follow the same periodicity rules as sine and cosine. sec(t 2k) sec(t) csc(t 2k) csc(t) for all real numbers t and all integers k. 10 We will use the identities and periodicity to evaluate trig functions of real numbers. Periodicity sin(t 2k ) sin(t) tan(t k ) tan(t) cos(t 2k ) cos(t) cot(t k ) cot(t) sec(t 2k) sec(t) csc(t 2k) csc(t) (for all real numbers t and all integers k.) 15 Example 4: Evaluate tan 4 25 Example 5: Evaluate cos 6 11 19 Example 6: Evaluate sin 3 20 Example 7: Evaluate sin 3 12 19 21 cos tan 34 Example 8: Evaluate 25 cos(8 )sin 6 14 17 tan 21 4 Example 9: Evaluate cot 4 17 cos(11 )sin 2 15 POPPER FOR SECTION 5.1 41 Question#2: Evaluate: cos(19 ) sin tan 64 A) 1 B) 5/2 C) 1/2 D) -1 E) -1/2 F) None of these 16 Simplifying Trigonometric Expressions Using Identities Example 1: Simplify: cot(tt )sec( ) 17 Example 2: Simplify: cos(t) cos(t) tan2 (t) 18 sin 6 csc 2 Example 3: Simplify: 1tan2 5 19 5 Example 4: Given cos(x ) and 0 x , evaluate the following 13 2 expression: sinxx sin 2 Remark: Recall the fact about “complementary angles”: sinx cos(x ) 2 cosx sin(x ) 2 tanx cot(x ) 2 20 4 Example 5: Given sin(x ) and 0 x , evaluate the following 5 2 expression: sin xxx 11 cos 9 tan 17 21.