Question

PLEASE HELP ME with this

Answer 1

An angle in standard position has one side on the x-axis, with the vertex of the angle at the origin (0, 0) of the coordinate plane (see picture 2).

We know that the terminal side of the angle is in quadrant IV. The quadrants are numbered counterclockwise starting with the top right quadrant (see picture 1). That means our angle will look something similar to the angle displayed in picture 2. The angle will create a right triangle with the x axis that looks like the blue triangle in picture 3 (not drawn to scale), with angle A as one of the angles of the triangle. Drawing this diagram will help you visualize the relationship of the sides in the following steps.

We are given that
`sec(A) = (4)/(3)`. Using your reciprocal identities, remember that
`sec(A) = (1)/(cos(A))`. Also remember that
`cos(A) = (adjacent)/(hypotenuse)` (a helpful mnemonic to remember sine, cosine, and tangent would be SOHCAHTOA). That means:

`sec(A) = (1)/(cos(A)) \\ sec(A) = (1)/((adjacent)/(hypotenuse))\\ sec(A) = (hypotenuse)/(adjacent)`.

Since
`sec(A) = (4)/(3) = (hypotenuse)/(adjacent)`, that means the hypotenuse of your triangle is 4 and the adjacent length to angle A is 3 (seen in green in picture 3).

Use the Pythagorean Theorem to find the length of the third side that is opposite the angle A. Remember that the Pythagorean Theorem states that
`a^(2) + b^(2) = c^(2)`, where a and b are the length of the legs of the triangle and c is the length of the hypotenuse. We know that one of the legs, let's call it a, equals 3. We also know the hypotenuse c = 4. Plug those values into the Pythagorean Theorem and solve for b, your other leg:

`3^(2) + b^(2) = 4^(2)\\ b^(2) = 16 - 9\\ b^(2) = 7\\ b = √(7)`

Now you know b, the length of your other leg, is √7 (seen in purple in picture 3).

You're looking for cot(A). Once again, use your reciprocal identities to remember that
`cot(A) = (1)/(tan(A))`. Since
`tan(A) = (opposite)/(adjacent)`, that means
`cot(A) = (1)/((opposite)/(adjacent)) = (adjacent)/(opposite)`.

Look at picture 3/your diagram. The adjacent side to the angle A is 3 and the opposite side to your angle A is √7. Plug those values into your equation for cot(A) to get the value of cot(A) and simplify the radical:

`cot(A) = (adjacent)/(opposite)\\ cot(A) = (3)/( √(7)) \\ cot(A) = (3)/( √(7)) * (√(7))/(√(7)) = (3√(7))/(7)`

The value of cot(A) =
`(3√(7))/(7)`.

Answer 2

There are a couple of ways you could do this. One way is to see what the Sec(x) can be related to in terms of either the tangent or cotangent.

x^2 + y^2 = c^2
x^2/y^2 + 1 = c^2/y^2
c = the hypotenuse
x = a line segment on the x axis.
y = a line segment on the y axis
cot^(theta) + 1 = csc^2(theta) This is not going to help much.

x^2 + y^2 = c^2
1 + y^2/x^2 = c^2 / x^2
1 + tan^2(theta) = sec^2(theta) There we go. This is much better.
1 + tan^2(theta) = (4/3)^2
1 + tan^2(theta) = 16/9
tan^2(theta) = 16/9 - 1 = 16/9 - 9/9 = 7/9
tan(theta) = sqrt(7) / sqrt(9)
tan(theta) = sqrt(7) / 3
But Cot(theta) = 1/ tan (theta) = 1/sqrt(7)/3 = 3/sqrt(7)

You might want to rationalize the denominator 3*sqrt(7) / [sqrt(7)*sqrt(7)]
cot(theta) = 3*sqrt(7) / 7

Since the angle is in the 4th quadrant, cot(theta) = - 3*sqrt(7) / 7 <<<< answer