7-5 The Other Trigonometric Functions Objective: To find values of the tangent, cotangent, secant, and cosecant functions and to sketch the functions’ graphs

The Other Trigonometric Functions (0, r) (-r, 0) (0, -r) y x P (x, y) (r, 0) r Besides the sine and cosine functions, there are some other trigonometric functions.

Other Trigonometric Functions tangent cotangent secant cosecant

we can write these other four functions in terms of sin  and cos .

Reciprocals Secant and cosine are reciprocals. Cosecant and sine are reciprocals. Cotangent and tangent are reciprocals. As for the “sec” and “csc” functions, as a way to help keep them straight I think, the "s" doesn't go with "s" and the "c" doesn't go with "c" so if we want secant, it won't be the one that starts with an "s" so it must be the reciprocal of cosine. (have to just remember that tangent & cotangent go together but this will help you with sine and cosine).

The domain of the cosecant function is the set of all real numbers except integral multiples of  (180 o ). The domain of the cosine function is the set of all real numbers. The domain of the tangent function is the set of all real numbers except odd multiples of  /2 (90 o ). The domain of the secant function is the set of all real numbers except odd multiples of  /2 (90 o ). The Domain of the Trigonometric Functions The domain of the cotangent function is the set of all real numbers except integral multiples of  (180 o ). The domain of the sine function is the set of all real numbers.

Fill in your trig table

The Special Values of All Trigonometric Functions

Signs of functions in quadrants IIIIIIIV sin csc ++-- cos sec +--+ tan cot +-+-

The Sign of All Trigonometric Functions A ll I S ine II III T angent IV C osine A good way to remember this chart is that ASTC stands for All Students Take Calculus.

Find the value of each expression with a calculator a)Tan 185˚ b)Cot 155˚ c)Csc (-1) d)Sec 11 a) b) c) d) Degree Mode Radian Mode

x Example 1: Find the six trig functions of 330 o. Second, find the reference angle, 360 o – 330 o = 30 o [Solution] First draw the 330 o angle. To compute the trig functions of the 30 o angle, draw the “special” triangle or recall from the table. Determine the correct sign for the trig functions of 330 o. Only the cosine and the secant are “+”. A S TC 330 o 30 o

[Solution] The six trig functions of 330 o are: Example 1: Find the six trig functions of 330 o.

y x Example 2: Find the six trig functions of. First determine the location of. With a denominator of 3, the distance from 0 to radians is cut into thirds. Count around the Cartesian coordinate system beginning at 0 until we get to. We can see that the reference angle is, which is the same as 60. Therefore, we will compute the trig functions of using the 60 angle of the special triangle.

0 radians Problem 3: Find the sin. All that’s left is to find the correct sign. And we can see that the correct sign is “-”, since the sin is always “-” in the 3 rd quadrant. A S TC We will first draw the angle by counting in a negative direction in units of. We can see that is the reference angle and we know that is the same as 30. So we will draw our 30 triangle and see that the sin 30 is. 1 2 Answer: sin =

A S TC Example 2: Find the six trig functions of. y x Before we write the functions, we need to determine the signs for each function. Remember “All Students Take Calculus”. Since the angle,, is located in the 3 rd quadrant, only the tangent and cotangent are positive. All the other functions are negative..

0 radians Problem 7: Find the exact value of cos. We will first draw the angle to determine the quadrant. A S TC We know that is the same as 45, so the reference angle is 45. Using the special triangle we can see that the cos of 45 or is. Note that the reference angle is. We see that the angle is located in the 3rd quadrant and the cosine is negative in the 3 rd quadrant. cos =

Practice Exercises 1.Find the value of the sec 360 without using a calculator. 2.Find the exact value of the tan Find the exact value of sin. 4.Find the tan 270 without using a calculator. 5.Find the exact value of the csc. 6.Find the exact value of the cot (-225 ). 7.Find the exact value of the sin. 8.Find the exact value of the cos. 9.Find the value of the cos(- ) without using a calculator. 10.Find the exact value of the sec 315.

Key For The Practice Exercises 1.sec 360 = 1 2.tan 420 = 3.sin = 4.tan 270 is undefined 5.csc = 6.cot (-225 ) = -1 7.sin = 8.cos = 9.cos(- ) = sec 315 =

Example 3: Given that tan  = –3/4, find the values of the other five trigonometric functions. [Solution] Since tan  = –3/4 < 0, so  is an 2 nd or 4 th quadrant angle. If  is an 2 nd quadrant angle, we can draw a diagram as shown at the right. Then:

Example 3: Given that tan  = –3/4, find the values of the other five trigonometric functions. [Solution] If  is a 4 th quadrant angle, we can draw a diagram as shown at the right. Then: (4, -3) y x 5  -3 4

If and -90˚< <90˚, find the values of the other five trigonometric functions. Since sin 0. x² + y² = r² x = √17² - 15² = 8

Assignment P. 285 # 2,4,6, 13-18, 20, Quiz tomorrow sine, cosine, & tangent Test Wednesday

Tangent Graph Unit circle at 90˚ would be (0,1) so tan would be 1/0. Is this possible?

Tangent Graph in Radians

The Secant Graph Draw secant function by graphing the cosine function. Note the vertical asymptotes at odd multiples of ¶/2

The Tangent Graph The domain of the tangent function is the set of all real numbers except odd multiples of  /2 (90 o ).

The Tangent Graph Vertical Asymptote:  = k  +  /2, where k  Z

The Cotangent Graph Vertical Asymptote:  = k , where k  Z

The Secant Graph

Vertical Asymptote:  = k  +  /2, where k  Z tan and sec have the same Vertical Asymptote:  = k  +  /2, where k  Z

The Cosecant Graph Vertical Asymptote:  = k , where k  Z cot and csc have the same Vertical Asymptote:  = k , where k  Z