suppose that the angle theta is in the second quadrant, and that sin(theta)=1/3. Find exact values for cos (theta) and csc (theta)

csc=1/sin=3

cos^2 Theta+sin^2 Theta=1
cos^2=1-sin^2=8/9
cosTheta=sqrt(4*2/9)=2/3 sqrt2 and in the second quadrant, it is negative (minus)

To find the values of cos(theta) and csc(theta) given that sin(theta) = 1/3 in the second quadrant, we can use the Pythagorean Identity and the definition of reciprocal trigonometric functions.

Step 1: Start with the equation sin(theta) = 1/3.
Step 2: In the second quadrant, sin(theta) is Positive, and cos(theta) will be Negative.
Step 3: Use the Pythagorean Identity to find cos(theta).
- sin^2(theta) + cos^2(theta) = 1
- (1/3)^2 + cos^2(theta) = 1
- 1/9 + cos^2(theta) = 1
- cos^2(theta) = 1 - 1/9
- cos^2(theta) = 8/9
- cos(theta) = ± sqrt(8/9)
- Since we are in the second quadrant, cos(theta) will be negative.
- cos(theta) = - sqrt(8/9)
Step 4: Use the definition of csc(theta) to find its value.
- csc(theta) = 1/sin(theta)
- csc(theta) = 1/(1/3)
- csc(theta) = 3/1
- csc(theta) = 3

Therefore, the exact values for cos(theta) in the second quadrant is -sqrt(8/9), and the exact value for csc(theta) is 3.

To find the values of cos(theta) and csc(theta), we need to use the given information that theta is in the second quadrant and sin(theta) = 1/3.

First, let's analyze the given information. In the second quadrant, the sine function is positive, and since sin(theta) = 1/3, this means that the y-coordinate of the point on the unit circle corresponding to theta is 1/3.

To find the value of cos(theta), we can use the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.

In this case, sin^2(theta) = (1/3)^2 = 1/9.

Substituting this into the Pythagorean identity, we get:

1/9 + cos^2(theta) = 1.

Rearranging the equation, we get:

cos^2(theta) = 1 - 1/9 = 8/9.

Taking the square root of both sides, we find:

cos(theta) = ±√(8/9).

Since theta is in the second quadrant, the x-coordinate of the point on the unit circle corresponding to theta is negative. Therefore, cos(theta) = - √(8/9).

Next, let's find the value of csc(theta). The reciprocal of sine is defined as csc(theta), so:

csc(theta) = 1/sin(theta).

Substituting sin(theta) = 1/3, we get:

csc(theta) = 1/(1/3) = 3/1 = 3.

Therefore, the exact values for cos(theta) and csc(theta) are:

cos(theta) = - √(8/9)

csc(theta) = 3