Trigonometric Functions
Section 5.2 Trigonometric Functions of Real Numbers
The Trigonometric Functions
EXAMPLE: Use the Table below to find the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2
1
EXAMPLE: Use the Table below to find the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2 Solution: (a) From the Table, we see that the terminal point determined by √ t = √ is P (1/2, 3/2). Since the coordinates are x = 1/2 and π/3 y = 3/2, we have √ √ π 3 3/2 √ π 1 π sin = cos = tan = = 3 3 2 3 2 3 1/2 √ √ π 3 2 3 π π 1/2 csc = = sec = 2 cot = √ 3 3 3 3 3 3/2 (b) The terminal point determined by π/2 is P (0, 1). So π π 1 π 0 π cos = 0 csc = = 1 cot = = 0 sin = 1 2 2 2 1 2 1 But tan π/2 and sec π/2 are undefined because x = 0 appears in the denominator in each of their definitions. π . 4 Solution: √ From the Table above, we see that √ terminal point determined by t = π/4 is the √ √ P ( 2/2, 2/2). Since the coordinates are x = 2/2 and y = 2/2, we have √ √ √ π 2 2 2/2 π π sin = =1 cos = tan = √ 4 2 4 2 4 2/2 √ π √ π π √ 2/2 csc = 2 sec = 2 cot = √ =1 4 4 4 2/2 EXAMPLE: Find the six trigonometric functions of each given real number t =
2
Values of the Trigonometric Functions
EXAMPLE: π π (a) cos > 0, because the terminal point of t = is in Quadrant I. 3 3 (b) tan 4 > 0, because the terminal point of t = 4 is in Quadrant III. (c) If cos t < 0 and sin t > 0, then the terminal point of t must be in Quadrant II. EXAMPLE: Determine the sign of each function. 7π (b) tan 1 (a) cos 4 Solution: (a) Positive (b) Positive EXAMPLE: Find each value. 2π π (a) cos (b) tan − 3 3 19π 4
(c) sin
3
EXAMPLE: Find each value. π 19π 2π (b) tan − (c) sin (a) cos 3 3 4 Solution: (a) Since 2π 3π − π 3π π π = = − =π− 3 3 3 3 3 the reference number for 2π/3 is π/3 (see Figure (a) below) and the terminal point of 2π/3 is in Quadrant II. Thus cos(2π/3) is negative and
(b) The reference number for −π/3 is π/3 (see Figure (b) below).
The Trigonometric Functions
EXAMPLE: Use the Table below to find the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2
1
EXAMPLE: Use the Table below to find the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2 Solution: (a) From the Table, we see that the terminal point determined by √ t = √ is P (1/2, 3/2). Since the coordinates are x = 1/2 and π/3 y = 3/2, we have √ √ π 3 3/2 √ π 1 π sin = cos = tan = = 3 3 2 3 2 3 1/2 √ √ π 3 2 3 π π 1/2 csc = = sec = 2 cot = √ 3 3 3 3 3 3/2 (b) The terminal point determined by π/2 is P (0, 1). So π π 1 π 0 π cos = 0 csc = = 1 cot = = 0 sin = 1 2 2 2 1 2 1 But tan π/2 and sec π/2 are undefined because x = 0 appears in the denominator in each of their definitions. π . 4 Solution: √ From the Table above, we see that √ terminal point determined by t = π/4 is the √ √ P ( 2/2, 2/2). Since the coordinates are x = 2/2 and y = 2/2, we have √ √ √ π 2 2 2/2 π π sin = =1 cos = tan = √ 4 2 4 2 4 2/2 √ π √ π π √ 2/2 csc = 2 sec = 2 cot = √ =1 4 4 4 2/2 EXAMPLE: Find the six trigonometric functions of each given real number t =
2
Values of the Trigonometric Functions
EXAMPLE: π π (a) cos > 0, because the terminal point of t = is in Quadrant I. 3 3 (b) tan 4 > 0, because the terminal point of t = 4 is in Quadrant III. (c) If cos t < 0 and sin t > 0, then the terminal point of t must be in Quadrant II. EXAMPLE: Determine the sign of each function. 7π (b) tan 1 (a) cos 4 Solution: (a) Positive (b) Positive EXAMPLE: Find each value. 2π π (a) cos (b) tan − 3 3 19π 4
(c) sin
3
EXAMPLE: Find each value. π 19π 2π (b) tan − (c) sin (a) cos 3 3 4 Solution: (a) Since 2π 3π − π 3π π π = = − =π− 3 3 3 3 3 the reference number for 2π/3 is π/3 (see Figure (a) below) and the terminal point of 2π/3 is in Quadrant II. Thus cos(2π/3) is negative and
(b) The reference number for −π/3 is π/3 (see Figure (b) below).