1. Give exact value of tan(theta) if cos(theta)=-2/7 and pi/2<theta<pi.

2. Find the domain, range, period, and phase shift of

y=3sin(2x-4)+1

3. The city of cape town, South America, has latitude 33.92 degrees south and longitude 18.37 degrees east. The circumference of a meridian is almost 40,000 km. Find the distance between cape town and the south pole, to the nearest km.

4. convert 4pi/5 radians to degrees

5. Convert 324 degrees to radians exactly

6. In a circle of diameter 22 inches, a sector is formed by a central angle of 16 degrees. Find the area of the sector to the nearest hundreth of a square inch.

Please help with these I have no clue what im doing.

1. To find the exact value of tan(theta), we can use the identity: tan(theta) = sin(theta)/cos(theta).

Since we are given cos(theta) = -2/7, we can solve for sin(theta) using the Pythagorean identity: sin(theta) = √(1 - cos^2(theta)).
Substituting the given value of cos(theta), we get sin(theta) = √(1 - (-2/7)^2) = √(1 - 4/49) = √(45/49) = √45/7.

Therefore, tan(theta) = sin(theta)/cos(theta) = (√45/7) / (-2/7) = -√45/2.

2. Domain: The domain of a sine function is all real numbers.
Range: The range of y = 3sin(2x-4) + 1 is [-2, 4].
Period: The period of a sine function is given by the formula 2π/b, where b is the coefficient of x in the argument of sine. In this case, the period is 2π/2 = π.
Phase Shift: The phase shift is given by the formula c/b, where c is the constant term in the argument of sine. In this case, the phase shift is 4/2 = 2 units to the right.

3. To find the distance between Cape Town and the South Pole, we can use the formula:

Distance = (Latitude of Cape Town - Latitude of South Pole) x (Circumference of Meridian) / 360

Latitude of South Pole = -90 degrees
Distance = (33.92 - (-90)) x 40,000 / 360
Distance = 123.92 x 111.111 = 13,761.62 km (rounded to the nearest km).

4. To convert radians to degrees, we use the fact that π radians is equivalent to 180 degrees.

4π/5 radians = (4/5) * π * (180/π) degrees = 144 degrees.

5. To convert degrees to radians, we use the fact that 180 degrees is equivalent to π radians.

324 degrees = 324 * (π/180) radians = (36/5)π radians.

6. The area of a sector in a circle is given by the formula: (θ/360) * π * r^2, where θ is the central angle in degrees and r is the radius.

Given: Diameter = 22 inches, Radius = 11 inches, Central angle = 16 degrees.

Area of sector = (16/360) * π * (11)^2 = (4/9) * π * 121 = 53.33 square inches (rounded to the nearest hundredth).

Of course! I'll help you with each question step by step.

1. To find the value of tan(theta), we can use the relation between sine, cosine, and tangent: tan(theta) = sin(theta) / cos(theta).

Given that cos(theta) = -2/7, we need to find sin(theta) to calculate tan(theta). To do this, we can use the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.

Since we are given that pi/2 < theta < pi, we can determine that theta is in the second quadrant, where sine is positive. Now, solve for sin(theta):

sin^2(theta) + cos^2(theta) = 1
sin^2(theta) + (-2/7)^2 = 1
sin^2(theta) + 4/49 = 1
sin^2(theta) = 1 - 4/49
sin^2(theta) = 45/49
sin(theta) = sqrt(45/49) (taking the positive square root since we're in the second quadrant)

Now that we have sin(theta), we can find tan(theta):

tan(theta) = sin(theta) / cos(theta)
tan(theta) = (sqrt(45/49)) / (-2/7)
tan(theta) = -7sqrt(45)/2sqrt(49)
tan(theta) = (-7/2)sqrt(45)/7
tan(theta) = -sqrt(45)/2

Therefore, the exact value of tan(theta) is -sqrt(45)/2.

2. Let's analyze the equation y = 3sin(2x-4) + 1 to determine the domain, range, period, and phase shift.

The domain of a sine function is all real numbers, so x can take any value.

The range of a sine function is between -1 and 1. Since we are multiplying the entire function by 3 and shifting it up by 1 unit, the range will be between 1 and 4.

The period of a sine function is given by the formula 2π / b, where b is the coefficient of x inside the sin function. In this case, the period is 2π / 2 = π.

The phase shift is given by the formula c / b, where c is the constant inside the parenthesis of the sin function. In this case, the phase shift is 4 / 2 = 2 units to the right.

Therefore, the domain is all real numbers, the range is [1, 4], the period is π, and the phase shift is 2 units to the right.

3. To find the distance between Cape Town and the South Pole, we can use the formula for the circumference of a circle, which is C = 2πr, where r is the radius.

Given that the circumference of a meridian is approximately 40,000 km, we can find the radius using the formula:

C = 2πr
40,000 km = 2πr
r = 40,000 km / (2π)

Now, we need to calculate the distance between Cape Town and the South Pole. We can use the latitude difference between the two locations, since the distance is equal to the arc length of the meridian.

The latitude of Cape Town is 33.92 degrees south. The South Pole has a latitude of 90 degrees south. The difference in latitude is 90 - 33.92 = 56.08 degrees.

To find the arc length, we can use the formula:

arc length = circumference * (angle / 360 degrees)

arc length = (40,000 km / (2π) ) * (56.08 / 360)

Calculating this, we find the distance between Cape Town and the South Pole to the nearest kilometer.

4. To convert radians to degrees, we use the fact that π radians is equal to 180 degrees.

Given 4π/5 radians, we can convert it to degrees using the following formula:

Degrees = (radians * 180) / π

Degrees = (4π/5) * (180 / π)

Simplifying, we get:

Degrees = (4 * 180) / 5
Degrees = 72 degrees

Therefore, 4π/5 radians is equal to 72 degrees.

5. To convert degrees to radians, we use the fact that 180 degrees is equal to π radians.

Given 324 degrees, we can convert it to radians using the following formula:

Radians = (degrees * π) / 180

Radians = (324 * π) / 180

Simplifying, we get:

Radians = 6π / 5

Therefore, 324 degrees is equal to 6π/5 radians.

6. To find the area of a sector in a circle, we need the measure of the central angle (in radians) and the radius of the circle.

Given a diameter of 22 inches, we can find the radius by dividing it by 2:

radius = diameter / 2
radius = 22 / 2
radius = 11 inches

The central angle of the sector is given as 16 degrees. Since the question asks for the area to the nearest hundredth of a square inch, we will convert the degrees to radians exactly (using the formula from question 5).

16 degrees = (16 * π) / 180 radians

Now, we can calculate the area of the sector using the formula:

Area of sector = (θ/2) * r^2

Area of sector = ((16 * π) / 180) / 2 * (11^2)

Calculating this, we find the area of the sector to the nearest hundredth of a square inch.

I hope that helps! Let me know if you have any further questions.