the point P(x, -2/3) lies on the unit circle and is in quadrant 4.
a) sketch the unit circle showing the point P and the principal statndard position angel theta drawn to the terminal arm containing point P.
b) determine the exact value of x.
c) determine the exact value of the six trigonometric ratios for theta.
d) determine all coterminal angles to theta in the interval -4pi is less than or equal to theta and less than or equal to 4pi
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a) The unit circle with point P and the principal standard position angle theta drawn to the terminal arm containing point P is shown in the figure below.
b) The exact value of x is -2/3.
c) The exact values of the six trigonometric ratios for theta are:
sin(theta) = -2/3
cos(theta) = -1/3
tan(theta) = -2
csc(theta) = -3/2
sec(theta) = -3
cot(theta) = 2
d) The coterminal angles to theta in the interval -4pi is less than or equal to theta and less than or equal to 4pi are:
theta + 2pi, theta + 4pi, theta - 2pi, theta - 4pi.
a) To sketch the unit circle, which is a circle with a radius of 1 centered at the origin (0, 0), we need to plot the point P(x, -2/3) in quadrant 4. In this case, the x-coordinate will be positive, and the y-coordinate will be negative. We can label this point as P on the circle.
Next, we draw the principal standard position angle, which is the angle formed between the positive x-axis and the terminal arm containing point P. We can label this angle as θ.
b) To determine the exact value of x, we can use the Pythagorean theorem. Since the point P lies on the unit circle, the distance between the origin and P will be 1. Using the coordinates of point P, we have:
x^2 + (-2/3)^2 = 1^2
x^2 + 4/9 = 1
x^2 = 1 - 4/9
x^2 = 9/9 - 4/9
x^2 = 5/9
To get the exact value of x, we can take the square root of both sides:
x = ±√(5/9)
Since point P is in quadrant 4, the x-coordinate will be positive, so we take the positive square root:
x = √(5/9)
c) To determine the exact value of the six trigonometric ratios for θ, we can use the coordinates of point P(x, -2/3). The six trigonometric ratios are:
sin θ = y / r = (-2/3) / 1 = -2/3
cos θ = x / r = √(5/9) / 1 = √5/3
tan θ = y / x = (-2/3) / √(5/9) = -(2√5)/5
csc θ = 1 / sin θ = 1 / (-2/3) = -3/2
sec θ = 1 / cos θ = 1 / (√5/3) = √3/√5
cot θ = 1 / tan θ = 1 / (-(2√5)/5) = -5/(2√5)
d) To determine all coterminal angles to θ in the given interval of -4π ≤ θ ≤ 4π, we can add or subtract integer multiples of 2π to θ:
θ + 2πn, where n is an integer.
For example, if we add 2π to θ, we get a new angle:
θ + 2π = θ + 2π(1) = θ + 2π
Similarly, if we subtract 2π from θ, we get another new angle:
θ - 2π = θ - 2π(1) = θ - 2π
We can continue adding or subtracting 2π to θ to find all the coterminal angles within the given interval.
a) To sketch the unit circle and point P, start by drawing a circle with a radius of 1 unit. Label the origin of the circle as O(0,0). Since the point P is in the fourth quadrant, plot it on the x-axis with a y-coordinate of -2/3. Label this point P(x, -2/3). Next, draw a line from the origin O to point P. This line represents the terminal arm. Label the angle formed between the positive x-axis and the terminal arm as theta.
b) To determine the exact value of x, we can use the Pythagorean theorem. Since the point lies on the unit circle, the distance from the origin to point P is equal to 1. Therefore, we can form the equation: x^2 + (-2/3)^2 = 1^2. Solving this equation will give you the exact value of x.
c) To determine the exact value of the six trigonometric ratios for theta, we can use the coordinates of point P. The six trigonometric ratios are:
1. sine (sin(theta)): sin(theta) = opposite/hypotenuse = y-coordinate/1 = -2/3
2. cosine (cos(theta)): cos(theta) = adjacent/hypotenuse = x-coordinate/1 = x (which we determined in part b)
3. tangent (tan(theta)): tan(theta) = sin(theta)/cos(theta) = (-2/3)/x
4. cosecant (csc(theta)): csc(theta) = 1/sin(theta) = 1/(-2/3) = -3/2
5. secant (sec(theta)): sec(theta) = 1/cos(theta) = 1/x
6. cotangent (cot(theta)): cot(theta) = 1/tan(theta) = 1/((-2/3)/x)
d) To determine all coterminal angles to theta in the interval -4π ≤ θ ≤ 4π, you can add or subtract multiples of 2π from the original angle theta. To keep it within the given interval, subtract multiples of 2π until you reach an angle in the desired interval. For example, you can subtract 8π from theta to get a coterminal angle within the given interval. Similarly, you can add 8π to theta to obtain another coterminal angle. Repeat this process, subtracting or adding 2π to theta, until you have covered the entire interval.