#$&*
course mth 164
4:00 pm on June 4
Sec 5.3 (6.3 starting with 8th edition)Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator.
6. Sec 540`theta deg
= Sec(0 + 2 pi)= Sec theta
= sec 180
= -1
10. sin 9`pi / 4
=sin 9 pi/4= sin (pi/4 + 4pi)
= sin pi/4= sqrt 2/2
12. csc 9 `pi /2
Csc 9pi/2= csc (pi/2 + 2 pi)
Csc pi/2= 1
18. sin `theta < 0, cos `theta > 0
Sin ‘theta < 0
= sin theta= y < 0
= cos ‘theta= x > 0
= Quadrant IV
20. cos `theta > 0, tan `theta > 0
Cos theta > 0= x > 0
Tan ‘theta > 0=
Cos will be positive and tan will also be positive = so it will be in Quadrant I
24. csc `theta > 0, cos `theta < 0
Csc theta > 0 (will be positive)
Cos’ theta < 0= x <0 (will be negative)
= Quadrant II
in the next problem sin `theta and cos `theta are given. Find the exact value of each of the four remaining trigonometric functions.
30. sin `theta = 2`sqrt(2) / 3 , cos `theta = -1/3
Tan ‘theta = sin ‘theta/cos ‘theta= 2 ‘sqrt 2/3 / -1/3= tan= -2
@& (2 sqrt(2) / 3) / (-1/3) = (2 sqrt(2) / 3) * (-3 / 1) = -2 sqrt(2).*@
Csc ‘theta= 1/sin’theta= 1/ 2’sqrt 2/3= ½
@& 1 / (2 sqrt(2) / 3) = 3 / (2 sqrt(2)) = 3/2 * sqrt(2) / 2 = 3 sqrt(2) / 4.*@
Sec ‘theta= 1/cos ‘theta= 1/ -1/3= -3
Cot ‘theta= 1/tan ‘theta= 1/-2= -1/2
find the exact value of each of the remaining trigonometric functions of `theta.
36. sin `theta = -5/13, `theta in quadrant 3
If theta is in quadrant III, it will be negative, except for tan and cot ‘theta> 0
???I am having trouble solving these problems where you find the remaining trigonometric funcions of theta. On 36, it gives you sin, but I do not know how to solve for cosine to get the answers for the rest of the functions.???
Csc= 1/y= 1/(-5/13)= -2
@& The magnitude of sin(theta) is 5 / 13 if theta is an angle of a triangle whose hypotenuse is 13 and whose leg opposite theta is 5. (If you prefer, you can think of a circular model, with a circle of radius 13, and a y coordinate of 5).
Applying the Pythagorean Theorem to this triangle we find that the other leg is sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12. (On the circular model the x coordinate would be 12).
It follows that the cosine of this angle is 12/13, the tangent is 5 / 12, etc..
In the third quadrant all the values would have the same magnitude, but some would be negative.
Sketch the triangle described above, and/or the circular model. Then see if you can get the other remaining value for this problem, as well as the next three problems.*@
40. sin `theta = -2/3, `pi < `theta < 3`pi/2
42. cos `theta = -1/4, tan `theta > 0
48. sec `theta = -2, tan `theta > 0
use the even- odd properties to find the exact value of each expression. Do not use a calculator.
50. cos (-30 deg )
Cos (=30)= -cos 30= - sqrt 3/2
@& cos(theta) is an even function so cos(-theta) = cos(theta) = +sqrt(3) / 2.*@
54. csc (-30 deg)
Csc (-30)= - csc 30 = -2
60. sin (-`pi/3)
Sin (-pi/3)= - sin pi/3= - sqrt 3/2
66. csc (-`pi/3)
Csc (-pi/3)= -csc pi/3= -2 sqrt 3/3
@& Good. The preceding three functions are in fact all odd functions.*@
find the exact value of each expression. Do not use a caculator.
70. tan (-6`pi) + cos (9`pi/4)
0 + cos (0 + 2 pi)= cos pi/4
= sqrt 2/2
72. cos (-17`pi/4) - sin (-3 `pi/2)
= cos (pi/4 + 4 pi) = - cos pi/4= - sqrt 2/2
=sin (pi/2 + 2 pi)= - sin pi/2= -1
= - sqrt 2/2 -1= -2
@& sqrt(2) / 2 - 1 isn't -2. *@
78. cot 20 deg - cos 20 deg/sin20 deg
???
80. If cos `theta = 0.2, find the value of cos `theta + cos ( `theta+2`pi) + cos (`theta+4`pi).
??? cannot figure 80 and 84 out???
@& How is the value of cos(theta + 2 pi) related to the value of cos(theta)?
What insight does this give you into the question?*@
84. What is the domain of the cosine function?
All real numbers
90. What is the range of the cosine function?
All real numbers from -1,1 (inclusive)
96. If the cosine function even, odd or neither? Is its graph symmetric? With respect to what?
Cosine f(0)= cos ‘theta, the cosine range is all real numbers from -1 to 1, cosine is an even function, The graph of an even function is symmetric with respect to the y axis.
100. Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what?
the secant only takes values above 1 or below -1 and it has a period of 2 , cosecant is odd with respect to x axis
@& The period is 2 pi, which is what I think you intended to say.*@
102. If f(x) = cos x and f(a) = 1/4, find the exact values of:
• f(-a) = -1/4
@& This would be so if the cosine was an odd function, but it's even.*@
110. Use the periodic and even- odd properties to show that the range of the cotangent function is the set of all real numbers
Periodic property= cot( 0 + pi)= cot ‘theta
@& cos(theta + pi) = cot(theta)*@
Even-odd property= cot (-‘theta)= -cot ‘theta
The domain of the cotangent function is all real numbers, except integer multiples of pi(180 deg)
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