Precalculus: Graphical, Numerical, Algebraic 8th Edition solutions

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=5 / 12\), find \(\cos 2 \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=8 / 15\), find \(\cos 2 \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=1 / \sqrt{3}\), find \(\cos 2 \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=1 / \sqrt{5}\), find \(\cos 2 \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=0\), find \(\alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=\sqrt{3}\), find \(\alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=3 / 4\), find \(\cos \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=3 / \sqrt{7}\), find \(\cos \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=3 / \sqrt{11}\), find \(\sin \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use trigonometric identities and assume \(0 \leq \alpha<\pi / 2\) Given that \(\cot 2 \alpha=45 / 28\), find \(\sin \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(x^{2}+y^{2}-6 x+10 y+18=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(4 x^{2}+y^{2}+24 x-2 y+21=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(y^{2}-8 x-8 y+8=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(x^{2}-4 y^{2}+6 x-40 y+91=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(-4 x y+16=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(2 x y+6=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(x y-y-8=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(2 x^{2}-5 x y+y=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(2 x^{2}-x y+3 y^{2}-3 x+4 y-6=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(-x^{2}+3 x y+4 y^{2}-5 x-10 y-20=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(2 x^{2}-4 x y+8 y^{2}-10 x+4 y-13=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 112, solve for y, and use a function grapher to graph the conic. \(2 x^{2}-4 x y+2 y^{2}-5 x+6 y-15=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1316, write an equation in standard form for the conic shown.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1316, write an equation in standard form for the conic shown.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1316, write an equation in standard form for the conic shown.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1316, write an equation in standard form for the conic shown.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1720, using the point P(x, y) and the translation information, find the coordinates of P in the translated \(x^{\prime} y^{\prime}\)-coordinate system. P(x, y) = (2, 3), h = -2, k = 4

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1720, using the point P(x, y) and the translation information, find the coordinates of P in the translated \(x^{\prime} y^{\prime}\)-coordinate system. P(x, y) = (-2, 5), h = -4, k = -7

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1720, using the point P(x, y) and the translation information, find the coordinates of P in the translated \(x^{\prime} y^{\prime}\)-coordinate system. \(P(x,\ y)=(6,\ -3),\ h=1,\ k=\sqrt{5}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1720, using the point P(x, y) and the translation information, find the coordinates of P in the translated \(x^{\prime} y^{\prime}\)-coordinate system. \(P(x,\ y)=(-5,\ -4),\ h=\sqrt{2},\ k=-3\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(4 y^{2}-9 x^{2}-18 x-8 y-41=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(2 x^{2}+3 y^{2}+12 x-24 y+60=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(x^{2}+2 x-y+3=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(3 x^{2}-6 x-6 y+10=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(9 x^{2}+4 y^{2}-18 x+16 y-11=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(16 x^{2}-y^{2}-32 x-6 y-57=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(y^{2}-4 y-8 x+20=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(2 x^{2}-4 x+y^{2}-6 y=9\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(2 x^{2}-y^{2}+4 x+6=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2130, identify the type of conic, write the equation in standard form, translate the conic to the origin, and sketch it in the translated coordinate system. \(y^{2}-2 y+4 x-12=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Translation Formulas Use the geometric relationships illustrated in Figure 8.35 to explain the translation formulas \(x=x^{\prime}+h\) and \(y=y^{\prime}+k\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Translation Formulas Prove that if \(x=x^{\prime}+h\) and \(y=y^{\prime}+k\), then \(x=x^{\prime}-h\) and \(y=y^{\prime}-k\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3336, using the point P(x, y) and the rotation information, find the coordinates of P in the rotated \(x^{\prime} y^{\prime}\)-coordinate system. \(P(x,\ y)=(-2,\ 5),\ \alpha=\pi/4\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3336, using the point P(x, y) and the rotation information, find the coordinates of P in the rotated \(x^{\prime} y^{\prime}\)-coordinate system. \(P(x,\ y)=(6,\ -3),\ \alpha=\pi/3\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3336, using the point P(x, y) and the rotation information, find the coordinates of P in the rotated \(x^{\prime} y^{\prime}\)-coordinate system. \(P(x,\ y)=(-5,\ -4),\ \cot2\alpha=-3/5\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3336, using the point P(x, y) and the rotation information, find the coordinates of P in the rotated \(x^{\prime} y^{\prime}\)-coordinate system. \(P(x,\ y)=(2,\ 3),\ \cot2\alpha=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 37 40, identify the type of conic, and rotate the coordinate axes to eliminate the xy-term. Write and graph the transformed equation. xy = 8

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 37 40, identify the type of conic, and rotate the coordinate axes to eliminate the xy-term. Write and graph the transformed equation. 3xy + 15 = 0

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 37 40, identify the type of conic, and rotate the coordinate axes to eliminate the xy-term. Write and graph the transformed equation. \(2 x^{2}+\sqrt{3} x y+y^{2}-10=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 37 40, identify the type of conic, and rotate the coordinate axes to eliminate the xy-term. Write and graph the transformed equation. \(3 x^{2}+2 \sqrt{3} x y+y^{2}-14=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 41 and 42, identify the type of conic, solve for y, and graph the conic. Approximate the angle of rotation needed to eliminate the xy-term. \(16 x^{2}-20 x y+9 y^{2}-40=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 41 and 42, identify the type of conic, solve for y, and graph the conic. Approximate the angle of rotation needed to eliminate the xy-term. \(4 x^{2}-6 x y+2 y^{2}-3 x+10 y-6=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(x^{2}-4 x y+10 y^{2}+2 y-5=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(x^{2}-4 x y+3 x+25 y-6=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(9 x^{2}-6 x y+y^{2}-7 x+5 y=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(-x y+3 y^{2}-4 x+2 y+8=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(8 x^{2}-4 x y+2 y^{2}+6=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(3 x^{2}-12 x y+4 y^{2}+x-5 y-4=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(x^{2}-3 y^{2}-y-22=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(5 x^{2}+4 x y+3 y^{2}+2 x+y=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(4 x^{2}-2 x y+y^{2}-5 x+18=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4352, use the discriminant \(B^{2}-4 A C\) to decide whether the equation represents a parabola, an ellipse, or a hyperbola. \(6 x^{2}-4 x y+9 y^{2}-40 x+20 y-56=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Revisiting Example 5 Using the results of Example 5, find the center, vertices, and foci of the hyperbola 2xy - 9 = 0 in the original coordinate system.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Revisiting Examples 3 and 6 Use information from Examples 3 and 6 (a) to prove that the point P(x, y) = (3.75, 9.375 where the graphs of \(\mathrm{Y} 1=(45-2 x+\sqrt{225-60 x}) / 4\) and \(\mathrm{Y} 2=(45-2 x-\sqrt{225-60 x}) / 4\) meet is not the vertex of the parabola; (b) to prove that th

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Rotation Formulas Prove \(x=x^{\prime} \cos \alpha-y^{\prime} \sin \alpha\) and \(y=x^{\prime} \sin \alpha+y^{\prime} \cos \alpha\) using the geometric relationships illustrated in Figure 8.37.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Rotation Formulas Prove that \(x^{\prime}=x \cos \alpha+y \sin \alpha\) and \(y^{\prime}=-x \sin \alpha+y \cos \alpha\), then \(x=x^{\prime} \cos \alpha-y^{\prime} \sin a\) and \(y=x^{\prime} \sin \alpha+y^{\prime} \cos \alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The graph of the equation \(A x^{2}+C y^{2}+D x+E y+F=0\) (A and C not both zero) has a focal axis aligned with the coordinate axes. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The graph of the equation \(x^{2}+y^{2}+D x+E y+F=0\) is a circle or a degenerate circle. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5962, solve the problem without using a calculator. Multiple Choice Which of the following is not a reason to translate the axes of a conic? (A) To simplify its equation (B) To eliminate the cross-product term (C) To place its center or vertex at the origin (D)To make

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5962, solve the problem without using a calculator. Multiple Choice Which of the following is not a reason to rotate the axes of a conic? (A) To simplify its equation (B) To eliminate the cross-product term (C) To place its center or vertex at the origin (D)To make it

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5962, solve the problem without using a calculator. Multiple Choice The vertices of \(9 x^{2}+16 y^{2}-18 x+64 y-71=0\) are (A) \((1\pm4,\ -2)\). (B) \((1\pm3,\ -2)\). (C) \((4\pm1,\ 3)\). (D) \((4\pm2,\ 3)\). (E) \((1,-2 \pm 3)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5962, solve the problem without using a calculator. Multiple Choice The asymptotes of the hyperbola xy = 4 are (A) y = x, y = -x. (B) \(y=2x,\ y=-\frac{x}{2}\). (C) \(y=-2x,\ y=\frac{x}{2}\). (D) \(y=4x,\ y=-\frac{x

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Axes of Oblique Conics The axes of conics that are not aligned with the coordinate axes are often included in the graphs of conics. (a) Recreate the graph shown in Figure 8.38 using a function grapher including the \(x^{\prime}-\) and \(y^{\prime}-\)-axes. What are the equations of these rotated

Precalculus: Graphical, Numerical, Algebraic 8th Edition
The Discriminant Determine what happens to the sign of \(B^{2}-4 A C\) within the equation \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) when (a) the axes are translated h units horizontally and k units vertically; (b) both sides of the equation are multiplied by the same nonzero

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity Working together, prove that the formulas for the coefficients \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\), \(D^{\prime}\), \(E^{\prime}\), and \(F^{\prime}\) in a rotated system given on page 615 are correct.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Identifying a Conic Develop a way to decide whether \(A x^{2}+C y^{2}+D x+E y+F=0\), with A and C not both 0, represents a parabola, an ellipse, or a hyperbola. Write an example to illustrate each of the three cases.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Rotational Invariant Prove that \(B^{\prime 2}-4 A^{\prime} C^{\prime}=B^{2}-4 A C\) when the xy-coordinate system is rotated through an angle \(\alpha\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Other Rotational Invariants Prove that each of the following are invariants under rotation: (a) F (b) A + C (c) \(D^{2}+E^{2}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Degenerate Conics Graph all of the degenerate conics listed in Table 8.2. Recall that degenerate cones occur when the generator and axis of the cone are parallel or perpendicular. (See Figure 8.2.) Explain the occurrence of all of the degenerate conics listed on the basis of cross sections of typical