Find the value of sec theta for angle theta in standard position if the point at (–2, –4) lies on its terminal side.

If 0° lessthanorequalto x lessthanorequalto 360°, solve the equation sec x = –2.

The terminal side of an angle theta in standard position coincides with the line 4x + y = 0 in Quadrant II. Find sec theta to the nearest thousandth.

To find the value of sec theta for an angle theta in standard position, we need to use the coordinates of the point on its terminal side.

1) Given that the point is (–2, –4), we can calculate the hypotenuse of the right triangle formed by the x and y coordinates.
Hypotenuse = sqrt((-2)^2 + (-4)^2) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5)

2) Now we can find the value of sec theta. Secant is the reciprocal of cosine, so we can use the x-coordinate and the hypotenuse to calculate it.
Sec theta = hypotenuse / x-coordinate = (2*sqrt(5)) / (-2) = -sqrt(5)

Therefore, the value of sec theta is -sqrt(5).

Next,

3) To solve the equation sec x = -2, we can apply the definition of secant:
Sec x = 1 / cos x = -2

Therefore, 1 / cos x = -2

4) Multiplying both sides by cos x, we get:
1 = -2cos x

5) Dividing both sides by -2, we have:
cos x = -1/2

6) The cosine function is negative in the second and third quadrants. The reference angle with positive cosine is 60 degrees in the first quadrant (180 - 60 = 120 degrees). So, there are two possible solutions: 120 degrees and 240 degrees.

Therefore, the solutions for x in the equation sec x = -2 are x = 120 degrees and x = 240 degrees.

Finally,

7) The line 4x + y = 0 intersects the y-axis at the point (0, 0) and has a slope of -4.
Since the terminal side coincides with the line, the angle theta will form a triangle with a vertical leg of 4 and a horizontal leg of 1.
The hypotenuse can be found using the Pythagorean theorem:
sqrt(4^2 + 1^2) = sqrt(16 + 1) = sqrt(17)

8) To find sec theta, we can use the x-coordinate and the hypotenuse:
Sec theta = hypotenuse / x-coordinate = sqrt(17) / 1 = sqrt(17)

Therefore, the value of sec theta to the nearest thousandth is sqrt(17).

To find the value of sec(theta) for an angle, we first need to understand what sec(theta) represents. Sec(theta) is the reciprocal of the cosine function, so we need to find the cosine of the angle and then take its reciprocal.

1. In the first question, we are given that the point (-2, -4) lies on the terminal side of the angle in standard position. To find sec(theta), we need to find the cosine of theta. The cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. Since we have the coordinates of a point on the terminal side, we can use these coordinates to form a right triangle.

Let's denote the two legs of the right triangle as x and y. In this case, x = -2 and y = -4. To find the hypotenuse, we can use the Pythagorean theorem: h^2 = x^2 + y^2.

Substituting the values, we get h^2 = (-2)^2 + (-4)^2 = 4 + 16 = 20. Taking the square root, we find that the hypotenuse, h, is sqrt(20) ≈ 4.472.

Now we can find the cosine of theta. Cos(theta) = adjacent/hypotenuse = x/h = -2/4.472 ≈ -0.447.

Finally, we take the reciprocal to find sec(theta). Sec(theta) = 1/cos(theta) = 1/-0.447 ≈ -2.236.

Therefore, sec(theta) is approximately -2.236.

2. In the second question, we are given the equation sec(x) = -2 and asked to solve for x. To solve this equation, we need to take the inverse of the secant function (arcsecant) to isolate x.

Sec(x) = -2 can be rewritten as 1/cos(x) = -2. Taking the reciprocal, we get cos(x) = -1/2.

We know that the cosine function is negative in the second and third quadrants. In the second quadrant, the reference angle (the angle between the terminal side and the x-axis) that has a cosine of -1/2 is 120 degrees (or pi - pi/3 radians).

However, we need to consider all solutions between 0 and 360 degrees. Since the cosine function has a period of 2pi (360 degrees), we can add or subtract multiples of 2pi to the angle.

So, the solutions are x = pi - pi/3 + 2npi and x = pi + pi/3 + 2npi, where n is an integer. Simplifying, we have x = 2pi/3 + 2npi and x = 4pi/3 + 2npi, where n is an integer.

Therefore, the solutions to sec(x) = -2 are x = 2pi/3 + 2npi and x = 4pi/3 + 2npi, where n is an integer.

3. In the third question, we are given the equation of the line that coincides with the terminal side of the angle in standard position. We need to find the secant of the angle.

The line 4x + y = 0 has a slope of -4/1 = -4. In Quadrant II, the cosine function is negative. Since the slope is negative, we can conclude that the tangent of the angle is positive.

The tangent of an angle is equal to the slope of the line passing through the origin and a point on the terminal side. Since we have the coordinates of a point on the terminal side (0, 0), the tangent of the angle is equal to the y-coordinate divided by the x-coordinate. Let's denote the tangent of the angle as t.

t = y/x = -4/1 = -4.

Now, we know that sec(theta) is the reciprocal of cosine, which is the reciprocal of 1/t. So, sec(theta) = 1/t = 1/(-4) = -1/4.

Hence, sec(theta) is approximately -0.250 to the nearest thousandth.

the terminal side of an angle in standard position coincides with the line y=x in quadrant I. find tan to the nearest thousandth