1.) Find all solutions of the equation. Leave answers in trigonometric form.
x^2 + 1 - sqrt3i = 0
2.) Give the rectangular coordinates for the point.
(9, 2pi/3)
3.) The rectangular coordinates of a point are given. Express the point in polar coordinates with r L 0 and 0° K theta < 360°.
(1/4 , -sqrt3/4)
4.) For the given rectangular equation, give its equivalent polar equation.
8x - 7y = -10
I need help setting this up.
#1
in polar form,
-1+√3i = (2,2π/3)
So, the two square roots of that are
(√2,π/3) and (√2,4π/3)
#2,#3
just plug in
x = r cosθ
y = r sinθ
#4
8x - 7y = -10
7y - 8x = 10
7r sinθ - 8r cosθ = 10
r = 10/(7sinθ-8cosθ)
1.) To find the solutions of the equation x^2 + 1 - sqrt(3)i = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 0, and c = 1 - sqrt(3)i. Substituting these values into the quadratic formula, we get:
x = (0 ± √(0^2 - 4(1)(1 - sqrt(3)i))) / (2(1))
Simplifying further:
x = ± √(-4 + 4sqrt(3)i) / 2
Now, let's simplify the term inside the square root:
√(-4 + 4sqrt(3)i) = √(4(-1 + sqrt(3)i))
Since we know that √(-1) = i, we can rewrite the expression as:
√(4(-1 + sqrt(3)i)) = 2i√(1 - sqrt(3)i)
Now, let's simplify the square root term:
√(1 - sqrt(3)i) = √((√1)^2 - (√(sqrt(3)i))^2)
= √(1 - (sqrt(3)i)^2)
= √(1 - (-3))
= √4
= 2
Substituting this back into the expression for x:
x = ± (2i * 2) / 2
= ± 2i
Therefore, the solutions to the equation x^2 + 1 - sqrt(3)i = 0 are x = 2i and x = -2i.
2.) The rectangular coordinates for the point (9, 2pi/3) are represented by an ordered pair (x, y), where x = 9 and y = 2pi/3.
3.) To express the point (1/4, -sqrt(3)/4) in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°, we can use the following formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
In this case, x = 1/4 and y = -sqrt(3)/4. Substituting these values into the formulas, we get:
r = √[(1/4)^2 + (-sqrt(3)/4)^2]
= √[1/16 + 3/16]
= √4/16
= √1/4
= 1/2
θ = arctan((-sqrt(3)/4) / (1/4))
= arctan(-sqrt(3))
= -pi/3
Therefore, the point (1/4, -sqrt(3)/4) in polar coordinates is given by (r, θ) = (1/2, -pi/3).
4.) To convert the given rectangular equation 8x - 7y = -10 into its equivalent polar equation, we can use the following conversions:
x = r*cos(θ)
y = r*sin(θ)
Substituting these conversions into the rectangular equation, we get:
8(r*cos(θ)) - 7(r*sin(θ)) = -10
Simplifying further:
8r*cos(θ) - 7r*sin(θ) = -10
This is the equivalent polar equation for the given rectangular equation.
Sure! I'll explain how to find the solutions for each of the given problems.
1.) To find the solutions of the equation x^2 + 1 - sqrt(3)i = 0, you can solve it using the quadratic formula. The quadratic formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 1, b = 0, and c = 1 - sqrt(3)i.
Plugging these values into the quadratic formula, we get:
x = (-0 ± √(0^2 - 4(1)*(1 - sqrt(3)i))) / (2*1)
Simplifying further:
x = ± √(-4(1 - sqrt(3)i)) / 2
x = ± √(-4 + 4√3i) / 2
x = ± √(4√3i - 4) / 2
x = ± √(2√3i - 2)
So the solutions to the equation x^2 + 1 - sqrt(3)i = 0 are ± √(2√3i - 2) in trigonometric form.
2.) The point (9, 2π/3) can be expressed in rectangular coordinates as (9cos(2π/3), 9sin(2π/3)). Using the values of cosine and sine for 2π/3, which are -1/2 and √3/2 respectively, we have:
Rectangular coordinates = (9*(-1/2), 9*(√3/2))
= (-9/2, 9√3/2)
So the rectangular coordinates for the point (9, 2π/3) are (-9/2, 9√3/2).
3.) The given point (1/4 , -sqrt3/4) can be expressed in polar coordinates using the formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
Plugging in the values:
r = √( (1/4)^2 + (-sqrt3/4)^2 )
r = √(1/16 + 3/16)
r = √(4/16)
r = 1/2
θ = arctan((-sqrt3/4)/(1/4))
θ = arctan(-sqrt3)
θ ≈ -60°
So the polar coordinates for the point (1/4 , -sqrt3/4) are (1/2, -60°).
4.) To find the equivalent polar equation for the given rectangular equation 8x - 7y = -10, we can use the following conversion formula:
r = √(x^2 + y^2)
θ = arctan(y/x)
Plugging in the values, we have:
8x - 7y = -10
Solve this equation for x:
x = (7y - 10)/8
Now, substituting x in terms of y, we get:
r = √(((7y - 10)/8)^2 + y^2)
Simplifying this equation further will result in the equivalent polar equation.
I hope this helps! Let me know if you have any further questions.