The given point lies on the terminal side of an angle in standard position. Find the values of the six trigonometric functions of .

1.(3, 4)

SOLUTION:Use the values of x and y to find r.

Use x = 3, y = 4, and r = 5 to write the six trigonometric ratios.

2.(6, 6)

SOLUTION:Use the values of x and y to find r.

Use x = , y = 6, and r = towritethesixtrigonometricratios.

3.(4, 3)

SOLUTION:Use the values of x and y to find r.

Use x = , y = , and r = 5 to write the six trigonometric ratios.

4.(2, 0)

SOLUTION:Use the values of x and y to find r.

Use x = 2, y =0, and r = 2 to write the six trigonometric ratios.

5.(1, 8)

SOLUTION:Use the values of x and y to find r.

Use x = 1, y = , and r = towritethesixtrigonometricratios.

6.(5, 3)

SOLUTION:Use the values of x and y to find r.

Use x = 5, y = , and r = towritethesixtrigonometricratios.

7.(8, 15)

SOLUTION:Use the values of x and y to find r.

Use x = , y = 15, and r = 17 to write the six trigonometric ratios.

8.(1, 2)

SOLUTION:Use the values of x and y to find r.

Use x = , y = , and r = towritethesixtrigonometricratios.

Find the exact value of each trigonometric function, if defined. If not defined, write undefined.

9.sin

SOLUTION:

Theterminalsideof instandardpositionliesonthepositivey-axis. Choose a point P(0, 1) on the terminal side

of the angle because r = 1.

10.tan 2

SOLUTION:

The terminal side of instandardpositionliesonthepositivex-axis. Choose a point P(1, 0) on the terminal side of the angle because r = 1.

11.cot (180)

SOLUTION:

The terminal side of instandardpositionliesonthenegativex-axis. Choose a point P( , 0) on the terminalside of the angle because r = 1.

12.csc 270

SOLUTION:

The terminal side of instandardpositionliesonthenegativey-axis. Choose a point P(0, ) on the terminal side of the angle because r = 1.

13.cos (270)

SOLUTION:

The terminal side of instandardpositionliesonthepositivey-axis. Choose a point P(0, 1) on the terminal side of the angle because r = 1.

14.sec 180

SOLUTION:

The terminal side of instandardpositionliesonthenegativex-axis. Choose a point P( , 0) on the terminal side of the angle because r = 1.

15.tan

SOLUTION:

The terminal side of in standard position lies on the negative x-axis. Choose a point P( , 0) on the terminal side of the angle because r = 1.

16.

SOLUTION:

The terminal side of in standard position lies on the negative y-axis. Choose a point P(0, ) on the terminal

side of the angle because r = 1.

Sketch each angle. Then find its reference angle.17.135

SOLUTION:

The terminal side of 135 lies in Quadrant II. Therefore, its reference angle is ' = 180 135 or 45.

18.210

SOLUTION:

The terminal side of 210 lies in Quadrant III. Therefore, its reference angle is ' = 210 180 or 30.

19.

SOLUTION:

The terminal side of liesinQuadrantII.Therefore,itsreferenceangleis ' = .

20.

SOLUTION:

A coterminal angle is 2 or , which lies in Quadrant IV. So, the reference angle is ' is 2 or

.

21.405

SOLUTION:

A coterminal angle is 405 + 360(2) or 315. The terminal side of 315 lies in Quadrant IV, so its reference angleis 360 315 or 45.

22.75

SOLUTION:

A coterminal angle is 75 + 360 or 285. The terminal side of 285 lies in Quadrant IV, so its reference angle is 360 285 or 75.

23.

SOLUTION:

Theterminalsideof liesinQuadrantII.Therefore,itsreferenceangleis ' = .

24.

SOLUTION:

A coterminal angle is + 2(1) or Theterminalsideof lies in Quadrant I, so the reference angle is

Find the exact value of each expression.

25.cos

SOLUTION:

Because the terminal side of lies in Quadrant III, the reference angle ' is or . In Quadrant III, cos

is negative.

26.tan

SOLUTION:

Because the terminal side of lies in Quadrant III, the reference angle ' is or . In Quadrant III, tan

is positive.

27.sin

SOLUTION:

Because the terminal side of lies in Quadrant II, the reference angle ' is or . In Quadrant II, sin is

positive.

28.cot (45)

SOLUTION:

A coterminal angle is 45 + 360 or 315. Because the terminal side of 315 lies in Quadrant IV, the reference angle ' is 360 315 or 45. Because tangent and cotangent are reciprocal functions and tan is negative in Quadrant IV, it follows that cot is also negative in Quadrant IV.

29.csc 390

SOLUTION:

A coterminal angle is 390 + 360 or 30, which lies in Quadrant I. So, the reference angle ' is 360 30 or 30. Because sine and cosecant are reciprocal functions and sin is positive in Quadrant I, it follows that csc is also positive in Quadrant I.

30.sec (150)

SOLUTION:

A coterminal angle is 150 + 360 or 210, which lies in Quadrant III. Because the terminal side of lies in Quadrant III. , the reference angle ' is 210 180 or 30. Because secant and cosine are reciprocal functions and cos is negative in Quadrant III, it follows that sec is also negative in Quadrant III.

31.tan

SOLUTION:

Because the terminal side of lies in Quadrant IV, the reference angle ' is or . In Quadrant IV, tan

is negative.

32.sin 300

SOLUTION:

Because the terminal side of lies in Quadrant IV, the reference angle ' is or . In Quadrant IV,sin is negative.

Find the exact values of the five remaining trigonometric functions of .33.tan = 2, where sin > 0 and cos > 0

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin and cos are positive, so must lie in Quadrant I. This means that both x and y are positive.

Because tan = or , use the point (1, 2) to find r.

Use x = 1, y = 2, and r = to write the five remaining trigonometric ratios.

34.csc = 2, where sin > 0 and cos < 0

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin is positive cos are negative, so must lie in Quadrant II. This means that x is negative and y is positive.

Because csc = or , use the point (x,1)andr = 2 to find x.

Use x = 1, y = 1, and r = 2to write the five remaining trigonometric ratios.

35.

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin is negative and cos is positive, so must lie in Quadrant IV. This means that x is positive and y is negative.

Because sin = or , use the point (x, ) and r= 5 to find x.

Use x = , y = , and r = 5 to write the five remaining trigonometric ratios.

36.

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin and cos are negative, so must lie in Quadrant III. This means that both x and y are negative.

Because cos = or , use the point ( , y) and r = 13 to find y .

Use x = , y = , and r = 13 to write the five remaining trigonometric ratios.

37.sec = , where sin < 0 and cos > 0

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin is negative and cos is positive, so mustlieinQuadrantIV.Thismeansthatx is positive and y is negative.

Because sec = or , use the point (1, y) and r = tofindy .

Use x = 1, y = , and r = to write the five remaining trigonometric ratios.

38.cot = 1, where sin < 0 and cos < 0

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin and cos are negative, so must lie in Quadrant III. This means that both x and y are negative.

Because cot = or , use the point ( , ) to find r.

Use x = , y = , and r = towritethefiveremainingtrigonometricratios.

39.tan = 1, where sin < 0

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin is negative and cos is positive, so must lie in Quadrant IV. This means that x is positive and y is negative.

Because tan = or , use the point ( , ) to find r.

Use x = , y = , and r = towritethefiveremainingtrigonometricratios.

40.

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of . You know that sin is positive and cos is negative, so must lie in Quadrant II. This means that x is negative and y is positive.

Because cos = or , use the point ( ,y)andr = 2 to find y. .

Use x = , y = , and r = 2 to write the five remaining trigonometric ratios.

41.CAROUSELZoe is on a carousel at the carnival. The diameter of the carousel is 80 feet. Find the position of her seat from the center of the carousel after a rotation of 210.

SOLUTION:

Let the center of the carousel represent the origin on the coordinate plane and Zoes position after the 210 rotationhave coordinates (x, y). The definitions of sine and cosine can then be used to find the values of x and y . The value

of r is 80 2 or 40. The seat rotates 210, so the reference angle is 210 180 or 30. Because the final position of the seat corresponds to Quadrant III, the sine and cosine of 210 are negative.

Therefore, the position of her seat relative to the center of the caro