Multiple Choice (theta) means the symbol 0 with the dash in it.
1.)Which expression is equivalent to tan(theta)-(sec(theta))/(sin(theta))?
A.)-cot(theta)
B.)cot(theta)
C.)tan(theta)-cot(theta)
D.)tan(theta)-sec^2(theta)
2.)Find the exact value of:cos 375 degrees
A.)(sq root 6- sq root 2)/4
B.)(sq root 6 + sq root 2)/4
C.)(sq root 2 - sq root 6)/4
D.)-(sq root 2 - sq root 6)/4
3.)Which expression is equivalent to: cos ((theta)+pie/2).
A.)cos (theta)
B.)-cos(theta)
C.)sin(theta)
D.)-sin(theta)
4.)Find the exact value of sin 2(theta) if cos(theta)=(- sq root5)/2 and 180 degrees<(theta)<270 degrees.
A.)-1/9
B.)(-4 sq root 5)/9
C.)1/9
D.)(4 Sq root 5)/9
5.)Find the exact value of sin (theta)/2 if cos(theta)=2/3 and 270 degrees<(theta)<360 degrees.
A.)1/3
B.)-1/3
C.)sq root 6/6
D.)(- sq root 6)/6
6.)Find the exact value of cos105 degrees by using a half-angle formula.
A.)(sq root2-sq root 3)/2
B.)-(sq root 2- sq root 3)/2
C.)-(sq root 2+ sq root 3)/2
D.)(sq root 2+ sq root 3)/2
7.)Find the solution of sin2 (theta)=cos (theta) if (theta degrees)<=(theta)<180 degrees.
A.)30 degrees and 90 degrees
B.) 30 degrees and 150 degrees
C.)30 degrees, 90 degrees, 150 degrees
D.)0 degrees, 90 degrees, and 150 degrees
8.)An insect population P in a certain area fluctuates with the seasons. it is estimated that P=17,000+4500sin(pie t)/52, where t is given in weeks. Determine the number of weeks it would take for the population to initially reach 20,000.
A.)12 weeks
B.)692 weeks
C.)38 weeks
D.)42 weeks
looks like a test to me...
1.)A
2.)B
3.)D
4.)D
5.)C
6.)B
7.)C
8.)A
1.) To simplify the expression tan(theta)-(sec(theta))/(sin(theta)), we can rewrite sec(theta) as 1/cos(theta).
Therefore, the expression becomes tan(theta)-1/cos(theta))/(sin(theta)).
To combine the two fractions, we need a common denominator. The common denominator can be found by multiplying cos(theta) and sin(theta).
Therefore, the expression becomes (tan(theta) - 1/cos(theta))(sin(theta)/sin(theta)).
Expanding this expression, we have (sin(theta)*tan(theta) - sin(theta))/sin(theta).
Now, we can simplify further by distributing sin(theta), resulting in sin(theta)*tan(theta) - sin(theta).
And finally, we can factor out sin(theta) from both terms, giving us sin(theta)(tan(theta)-1).
Therefore, the expression is equivalent to A.) -cot(theta).
2.) To find the exact value of cos 375 degrees, we can use the unit circle.
75 degrees is equivalent to 360 degrees - 375 degrees. Since the cosine function is an even function, cos(75 degrees) = cos(360 degrees - 75 degrees) = cos(285 degrees).
Now, using the unit circle, we find that cos(285 degrees) = (sqrt(6) + sqrt(2))/4.
Therefore, the exact value of cos 375 degrees is B.) (sqrt(6) + sqrt(2))/4.
3.) To find the expression equivalent to cos(theta + pi/2), we can use the identity cos(theta + pi/2) = -sin(theta).
Therefore, the expression is equivalent to B.) -cos(theta).
4.) To find the exact value of sin 2(theta), we can use the double angle identity for sine: sin(2(theta)) = 2sin(theta)cos(theta).
Given cos(theta) = (-sqrt(5))/2, we can find sin(theta) using the Pythagorean identity: sin(theta) = sqrt(1 - cos^2(theta)).
So, sin(theta) = sqrt(1 - ((-sqrt(5))/2)^2) = sqrt(1 - 5/4) = sqrt(4/4 - 5/4) = sqrt(-1/4) = i/2.
Now, substituting the values in the double angle identity, we have sin(2(theta)) = 2(i/2)(-sqrt(5))/2 = (i(-sqrt(5)))/2 = -(sqrt(5)i)/2.
Therefore, the exact value of sin 2(theta) is B.) (-4sqrt(5))/9.
5.) To find the exact value of sin(theta/2), we can use the half-angle identity for sine: sin(theta/2) = sqrt((1 - cos(theta))/2).
Given cos(theta) = 2/3, we can substitute this value into the half-angle identity.
sin(theta/2) = sqrt((1 - 2/3)/2) = sqrt((3 - 2)/6) = sqrt(1/6) = sqrt(6)/6.
Therefore, the exact value of sin(theta/2) is C.) sqrt(6)/6.
6.) To find the exact value of cos 105 degrees using a half-angle formula, we can use the half-angle identity for cosine: cos(theta/2) = sqrt((1 + cos(theta))/2).
Since cos 210 degrees = cos(360 degrees - 210 degrees) = cos 150 degrees = -sqrt(3)/2, we can substitute this value into the half-angle identity.
cos(105 degrees) = sqrt((1 - sqrt(3)/2)/2) = sqrt((2 - sqrt(3))/4) = (sqrt(2) - sqrt(3))/2.
Therefore, the exact value of cos 105 degrees is A.) (sqrt(2) - sqrt(3))/2.
7.) To solve sin^2(theta) = cos(theta) for theta, we can use the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.
Substituting the given equation, we have cos^2(theta) + cos(theta) - 1 = 0.
Factoring this quadratic equation, we get (cos(theta) - 1)(cos(theta) + 1) = 0.
Setting each factor equal to zero, we have two cases:
1) cos(theta) - 1 = 0, which gives cos(theta) = 1. This occurs at theta = 0 degrees.
2) cos(theta) + 1 = 0, which gives cos(theta) = -1. This occurs at theta = 180 degrees.
Therefore, the solution for sin^2(theta) = cos(theta) is D.) 0 degrees, 90 degrees, and 150 degrees.
8.) To determine the number of weeks it would take for the insect population to initially reach 20,000, we can set the population equation equal to 20,000 and solve for t.
17,000 + 4500*sin(pi*t/52) = 20,000
Adding -17,000 to both sides, we have 4500*sin(pi*t/52) = 3,000.
Dividing both sides by 4500, we get sin(pi*t/52) = 3/4.
Taking the inverse sine of both sides, we have pi*t/52 = arcsin(3/4).
Multiplying both sides by 52/pi, we get t = (52/pi)*arcsin(3/4).
Using a calculator to approximate arcsin(3/4), we find it is approximately 0.8481.
Substituting this value into the equation, we have t = (52/pi)*0.8481 = 42 weeks (approximately).
Therefore, the number of weeks it would take for the population to initially reach 20,000 is D.) 42 weeks.