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cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1119 Given the polar equation of a cardioid, sketch its graph. 1. Check equation for the three types of symmetry. 2. Find the zeros. Set r = 0. 3. Find the maximum value of the equation according to the maximum value of the trigonometric expre... |
limaçons are sometimes referred to as dimpled b ≥ 2. limaçons when 1 < b < 2 and convex limaçons when a a Formulas for One-Loop Limaçons The formulas that produce the graph of a dimpled one-loop limaçon are given by r = a ± bcos θ and r = a ± bsin θ a b < 2. All four graphs are shown in Figure 10.75. where a > 0, b > ... |
, as the maximum value of the sine function is 1 when θ =, we will substitute θ = π 2 into the equation π 2 and solve for r. Thus, r = 1. Make a table of the coordinates similar to Table 10.4 2π 3 5π 6 π 7π 6 4π 3 3π 2 5π 3 11π 6 2π 4 2.5 1.4 1 1.4 2.5 4 5.5 6.6 7 6.6 5.5 4 Table 10.4 The graph is shown in Figure 10.76... |
of the equation is symmetric about the polar axis. Next, finding the zeros reveals that when r = 0, θ = 1.98. The maximum |r| is found when cos θ = 1 or when θ = 0. Thus, the maximum is found at the point (7, 0). Even though we have found symmetry, the zero, and the maximum, plotting more points will help to define th... |
� = Substitute 2θ back in for u. So, the point ⎛ ⎝0, ⎞ ⎠ π 4 is a zero of the equation. Now let’s find the maximum value. Since the maximum of cos u = 1 when u = 0, the maximum cos 2θ = 1 when 2θ = 0. Thus, r 2 = 4cos(0) r 2 = 4(1) = 4 r = ± 4 = 2 We have a maximum at (2, 0). Since this graph is symmetric with respect ... |
line θ = and the pole. π 2 Now we will find the zeros. First make the substitution u = 4θ. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1127 0 = 2cos 4θ 0 = cos 4θ 0 = cos u cos− = 4θ = The zero is θ =. The point ⎛ ⎝0, π 8 π 8 ⎞ ⎠ is on the ... |
the line θ =. Next, find the zeros and maximum. π 2 We will want to make the substitution u = 5θ. 0 = 2sin(5θ) 0 = sin u sin−1 0 = 0 u = 0 5θ = 0 θ = 0 The maximum value is calculated at the angle where sin θ is a maximum. Therefore, ⎛ r = 2sin ⎝5 ⋅ ⎞ ⎠ π 2 r = 2(1) = 2 Thus, the maximum value of the polar equation is... |
at the pole at the point (0, 0). While the graph hints of symmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value, unless the domain is restricted. Create a table such as Table 10.9. θ r π 4 π 2 π 3π 2 7π 4 2π 0.785 1.57 3.14 4.71 5.50 6.28 Table 10.9 Notice ... |
equations? Describe the shapes of the graphs of cardioids, 243. limaçons, and lemniscates. What part of the equation determines the shape of the 244. graph of a polar equation? Graphical 261. r = 3 + 2sin θ 262. r = 7 + 4sin θ 263. r = 4 + 3cos θ 264. r = 5 + 4cos θ 265. r = 10 + 9cos θ 266. r = 1 + 3sin θ 267. r = 2 ... |
r = 2sin θtan θ, a cissoid 286. r = 2 1 − sin2 θ, a hippopede On a graphing utility, graph each polar equation. 302. Explain the similarities and differences you observe in the graphs. r1 = 3θ r2 = 2θ r3 = θ Extensions For the following exercises, draw each polar equation on the same set of polar axes, and find the po... |
⎣0, 16π⎤ ⎛ graph r = sin ⎝ 16 5 ⎦. Describe θ⎞ ⎠ on the 299. On a r = sin θ + graphing 3 ⎛ ⎛ ⎝sin ⎝ ⎞ θ⎞ ⎠ ⎠ 5 2 on [0, 4π]. utility, graph and sketch 300. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs. r1 = 3sin(3θ) r2 = 2sin(3θ) r3 = sin(3θ) 301.... |
Plane Plotting a complex number a + bi is similar to plotting a real number, except that the horizontal axis represents the real part of the number, a, and the vertical axis represents the imaginary part of the number, bi. Given a complex number a + bi, plot it in the complex plane. 1. Label the horizontal axis as the... |
Finding the Absolute Value of a Complex Number Given z = 3 − 4i, find |z|. Solution Using the formula, we have This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1137 |z| = x2 + y2 |z| = (3)2 + (−4)2 |z| = 9 + 16 |z| = 25 |z| = 5 The absolute valu... |
polar form, we have to calculate r first. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1139 r = x2 + y2 r = 02 + 42 r = 16 r = 4 Next, we look at x. If x = rcos θ, and x = 0, then θ = the complex number z = 0 + 4i can be written as z = 4 ⎛ ⎛... |
�� ⎠ ⎠ π 6 Solution We begin by evaluating the trigonometric expressions. ⎛ cos ⎝ π 6 ⎞ ⎠ = 3 2 ⎛ and sin ⎝ π 6 ⎞ ⎠ = 1 2 After substitution, the complex number is z = 12 ⎛ ⎝ 3 2 + 1 2 i⎞ ⎠ We apply the distributive property: z = 12 ⎛ ⎝ 3 2 = (12) 3 2 i⎞ ⎠ + 1 2 + (12)1 2 i The rectangular form of the given point in co... |
⎡ ⎣cos⎛ z1 z2 = r1 r2 z1 z2 = r1 r2 cis⎛ ⎝θ1 + θ2 ⎝θ1 + θ2 ⎞ ⎠ ⎞ ⎠ + isin⎛ ⎝θ1 + θ2 ⎤ ⎞ ⎦ ⎠ Notice that the product calls for multiplying the moduli and adding the angles. Example 10.41 Finding the Product of Two Complex Numbers in Polar Form Find the product of z1 z2, given z1 = 4(cos(80°) + isin(80°)) and z2 = 2(cos... |
2 ⎞ ⎠, z2 ≠ 0 Notice that the moduli are divided, and the angles are subtracted. Given two complex numbers in polar form, find the quotient. 1. Divide r1 r2. 2. Find θ1 − θ2. 3. Substitute the results into the formula: z = r(cos θ + isin θ). Replace r with r1 r2, and replace θ with θ1 − θ2. 4. Calculate the new trigono... |
’s Theorem applies to complex numbers written in polar form, we must first write (1 + i) in polar form. Let us find r. r = x2 + y2 r = (1)2 + (1)2 r = 2 Then we find θ. Using the formula tan θ = y x gives 1144 Chapter 10 Further Applications of Trigonometry tan θ = 1 1 tan θ = 1 π θ = 4 Use De Moivre’s Theorem to evalu... |
the nth Root of a Complex Number Evaluate the cube roots of z = 8 ⎛ ⎛ ⎝cos ⎝ 2π 3 ⎛ ⎞ ⎠ + isin ⎝ ⎞ ⎞ ⎠. ⎠ 2π 3 Solution We have cos ⎜ ⎣ ⎝ 2π 3 3 + 2kπ 3 ⎛ ⎞ ⎜ ⎟ + isin ⎝ ⎠ 2π 3 3 + 2kπ cos ⎝ 2π 9 + 2kπ 3 ⎛ ⎞ ⎠ + isin ⎝ 2π 9 + 2kπ 3 ⎤ ⎞ ⎦ ⎠ There will be three roots: k = 0, 1, 2. When k = 0, we have This content is ava... |
2(1)π 3 ⎞ ⎠ ⎛ ⎝ 3 3 + 2π 2(1)π = 2π 3 3 3 3 = 2π 9 = 8π 9 ⎞ ⎛ 1 ⎠ + ⎝ 3 + 6π 9 10.29 Find the four fourth roots of 16(cos(120°) + isin(120°)). Access these online resources for additional instruction and practice with polar forms of complex numbers. • The Product and Quotient of Complex Numbers in Trigonometric Form (... |
= 4cis ⎝ π 2 ⎞ ⎛ ⎠; z2 = 2cis ⎝ ⎞ ⎠ π 4 For the following exercises, find z1 z2 in polar form. 340. z1 = 21cis(135°); z2 = 3cis(65°) 341. z1 = 2cis(90°); z2 = 2cis(60°) For the following exercises, write the complex number in polar form. 342. z1 = 15cis(120°); z2 = 3cis(40°) 323. 2 + 2i 324. 8 − 4i 325. − 1 2 − 1 2 i ... |
16 Chapter 10 Further Applications of Trigonometry 1147 351. ⎛ Find z3 when z = 3cis ⎝ ⎞ ⎠. 5π 3 Use the polar to rectangular feature on the graphing 371. calculator to change 2cis(45°) to rectangular form. For the following exercises, evaluate each root. 352. Evaluate the cube root of z when z = 27cis(240°). Use the ... |
.94. At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only fo... |
equivalent pair of equations in three variables, x, y, and t. One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time. When we graph parametric equations, we can observe the individual behaviors of x and of y. T... |
2 −2 y(−2) = (−2)2 − 1 = 3 −1 −1 y(−1) = (−1)(0) = (0)2 − 1 = − 1 y(1) = (1)2 − 1 = 0 y(2) = (2)2 − 1 = 3 y(3) = (3)2 − 1 = 8 y(4) = (4)2 − 1 = 15 Table 10.10 See the graphs in Figure 10.96. It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as t increases. Figure 10... |
(−1) = 1 − (−1)(0) = 1 − 0 = 1 y(1) = 1 − (1)2 = 0 y(2) = 1 − (2)2 = − 3 y(3) = 1 − (3)2 = − 8 Table 10.11 The graph of y = 1 − t 2 is a parabola facing downward, as shown in Figure 10.97. We have mapped the curve over the interval [−3, 3], shown as a solid line with arrows indicating the orientation of the curve accor... |
: 1154 Chapter 10 Further Applications of Trigonometry x(t) = 2t − 5 y(t) = − t + 3 Using these equations, we can build a table of values for t, x, and y (see Table 10.12). In this example, we limited values of t to non-negative numbers. In general, any value of t can be used. t 0 1 2 3 4 x(t) = 2t − 5 y(t) = − t + 3 x... |
t. We substitute the resulting expression for t into the second equation. This gives one equation in x and y. Example 10.48 Eliminating the Parameter in Polynomials Given x(t) = t 2 + 1 and y(t) = 2 + t, eliminate the parameter, and write the parametric equations as a Cartesian equation. Solution We will begin with th... |
Figure 10.100 Example 10.50 Eliminating the Parameter in Logarithmic Equations Eliminate the parameter and write as a Cartesian equation: x(t) = t + 2 and y(t) = log(t). Solution Solve the first equation for tx − 2)2 = t Square both sides. Then, substitute the expression for t into the y equation. y = log(t) y = log(x... |
The graph for the equation is shown in Figure 10.101. x2 16 Figure 10.101 Analysis Applying the general equations for conic sections (introduced in Analytic Geometry, we can identify x2 16 t = y2 9 the coordinates are (0, 3). This shows the orientation of the curve with increasing values of t. = 1 as an ellipse center... |
etric Equations for Curves Defined by Rectangular Equations Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the param... |
�� x(t) = t 5 ⎨ y(t) = t 10 ⎩ Explain how to eliminate a parameter given a set of 375. parametric equations. What is a benefit of writing a system of parametric 376. equations as a Cartesian equation? 377. What is a benefit of using parametric equations? Why are there many sets of parametric equations to 378. represent... |
t + 4 ⎧ ⎨ y(t) = 2sin2 t ⎩ 395. x(t) = t − 1 ⎧ ⎨ y(t) = t 2 ⎩ 396. x(t) = − t ⎧ ⎨ y(t) = t 3 + 1 ⎩ 397. x(t) = 2t − 1 ⎧ ⎨ y(t) = t 3 − 2 ⎩ For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. 398. x(t) = 2t − 1 ⎧ ⎨ y(t) = t + 4 ⎩ 399. x(t) = 4 − t ⎧ ⎨ y(t) = 3t... |
t = 1. Parameterize the line from (−1, 5) to (2, 3) so that 412. the line is at (−1, 5) at t = 0, and at (2, 3) at t = 1. Parameterize the line from (4, 1) to (6, −2) so that 413. the line is at (4, 1) at t = 0, and at (6, −2) at t = 1. Technology For the following exercises, use the table feature in the the graphs gr... |
at an angle of approximately 45° to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using par... |
restrictions on the domain. The arrows indicate direction according to increasing values of t. The graph does not represent a function, as it will fail the vertical line test. The graph is drawn in two parts: the positive values for t, and the negative values for t. 10.36 Sketch the graph of the parametric equations x... |
⎞ ⎠ = − 3 ⎛ y = 4sin ⎝ ⎞ ⎠ = − 2 7π 6 ⎛ x = 2cos ⎝ ⎞ ⎠ = − 1 4π 3 ⎛ y = 4sin ⎝ 4π 3 ⎞ ⎠ = − 2 3 ⎛ x = 2cos ⎝ ⎞ ⎠ = 0 3π 2 ⎛ y = 4sin ⎝ ⎞ ⎠ = − 4 3π 2 ⎛ x = 2cos ⎝ ⎞ ⎠ = 1 5π 3 ⎛ y = 4sin ⎝ 5π 3 ⎞ ⎠ = − 2 3 ⎛ x = 2cos ⎝ 11π 6 ⎞ ⎠ = 3 ⎛ y = 4sin ⎝ 11π 6 ⎞ ⎠ = − 2 x = 2cos(2π) = 2 y = 4sin(2π) = 0 Table 10.14 Figure 10.1... |
t x = 5cos t y = 2sin t 0 1 2 3 4 5 −1 −2 −3 −4 −5 x = 5cos(0) = 5 y = 2sin(0) = 0 x = 5cos(1) ≈ 2.7 y = 2sin(1) ≈ 1.7 x = 5cos(2) ≈ −2.1 y = 2sin(2) ≈ 1.8 x = 5cos(3) ≈ −4.95 y = 2sin(3) ≈ 0.28 x = 5cos(4) ≈ −3.3 y = 2sin(4) ≈ −1.5 x = 5cos(5) ≈ 1.4 y = 2sin(5) ≈ −1.9 x = 5cos(−1) ≈ 2.7 y = 2sin(−1) ≈ −1.7 x = 5cos(−... |
rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph. Figure 10.107 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1171 Example 10.57 Graphing Parametric Equations and... |
x gives horizontal distance, and the equation for y gives the vertical distance. Given a projectile motion problem, use parametric equations to solve. 1. The horizontal distance is given by x = ⎛ ⎝v0 cos θ⎞ ⎠t. Substitute the initial speed of the object for v0. 2. The expression cos θ indicates the angle at which the ... |
available for free at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1173 c. To calculate how long the ball is in the air, we have to find out when it will hit ground, or when y = 0. Thus, ⎝140sin(45∘)⎞ ⎠t + 3 y = − 16t 2 + ⎛ y = 0 t = 6.2173 Set y(t) = 0 and solve the quadratic. ... |
of parametric 425. equations. t x y −3 −2 −1 0 1 2 Why are 426. understanding projectile motion? parametric graphs important in Graphical For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph. 427. x(t) = t ⎧ ⎨ y(t 429. x(t) = 2 + t ⎧ ⎨ y(... |
�� y(t) = t + 2 ⎩ 436. x(t) = − t + 2 ⎧ ⎨ y(t) = 5 − |t| ⎩ 437. x(t) = 4sin t ⎧ ⎨ y(t) = 2cos t ⎩ 438. x(t) = 2sin t ⎧ ⎨ y(t) = 4cos t ⎩ 439. 440. ⎧ x(t) = 3cos2 t ⎨ y(t) = −3sin t ⎩ ⎧ x(t) = 3cos2 t ⎨ y(t) = −3sin2 t ⎩ 441. x(t) = sec t ⎧ ⎨ y(t) = tan t ⎩ 442. x(t) = sec t ⎧ ⎨ y(t) = tan2 t ⎩ 443. ⎧ x(t) = 1 e2t ⎨ y(t... |
domain [−π, 0], where a = 5 and 458. b = 4, and include the orientation. If a is 1 more than b, describe the effect the values 459. of a and b have on the graph of the parametric equations. 460. Describe the graph if a = 100 and b = 99. 1176 Chapter 10 Further Applications of Trigonometry What happens if b is 1 more t... |
at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1177 477. and then eliminate time to write height as a function of horizontal position. be 485. A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can equation y(t) ... |
478. domain [0, 2π]. Graph all three sets of parametric equations on the 479. domain [0, 4π]. Graph all three sets of parametric equations on the 480. domain ⎡ ⎣−4π, 6π⎤ ⎦. The graph of each set of parametric equations appears 481. to “creep” along one of the axes. What controls which axis the graph creeps along? 493.... |
refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in... |
. Figure 10.111 Properties of Vectors A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at (0, 0) and is ... |
�� a, b 〉, formed with the x-axis, or with the y-axis, depending on the application. For a position vector, the direction is found by tan θ = ⎞ ⎠, as illustrated in Figure 10.114. ⎠ ⇒ θ = tan− Figure 10.114 Two vectors v and u are considered equal if they have the same magnitude and the same direction. Additionally, if... |
�� = 〈 −6, 5 〉 Since the position vectors are the same, v and u are the same. An alternative way to check for vector equality is to show that the magnitude and direction are the same for both vectors. To show that the magnitudes are equal, use the Pythagorean Theorem. This content is available for free at https://cnx.o... |
.117 Vector subtraction is similar to vector addition. To find u − v, view it as u + (−v). Adding −v is reversing direction of v and adding it to the end of u. The new vector begins at the start of u and stops at the end point of −v. See Figure 10.118 for a visual that compares vector addition and vector subtraction us... |
ar Multiplication Given vector v = 〈 3, 1 〉, find 3v, 1 2 v, and −v. Solution See Figure 10.120 for a geometric interpretation. If v = 〈 3, 1 〉, then 3v = 〈 3 ⋅ 3, 3 ⋅ 1 〉 = 〈 9v = 〈 −3, −1 〉 ⋅ 3, 1 2, 1 2 〉 ⋅ 1 〉 1186 Chapter 10 Further Applications of Trigonometry Figure 10.120 Analysis Notice that the vector 3v is t... |
have v1 with initial point (0, 0) and terminal point (2, 0). We also have v2 with initial point (0, 0) and terminal point (0, 3). v1 = 〈 2 − 0, 0 − 0 〉 = 〈 2, 0 〉 Therefore, the position vector is v2 = 〈 0 − 0, 3 − 0 〉 = 〈 0, 3 〉 v = 〈 2 + 0, 3 + 0 〉 = 〈 2, 3 〉 Using the Pythagorean Theorem, the magnitude of v1 is 2, ... |
j = 〈 0, 1 〉 and is directed along the positive vertical axis. See Figure 10.123. Figure 10.123 The Unit Vectors If v is a nonzero vector, then v |v| is a unit vector in the direction of v. Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the s... |
y2) j The position vector from (0, 0) to (a, b), where (x2 − x1) = a and (y2 − y1) = b, is written as v = ai + bj. This vector sum is called a linear combination of the vectors i and j. The magnitude of v = ai + bj is given as |v| = a2 + b2. See Figure 10.125. Figure 10.125 Example 10.68 Writing a Vector in Terms of i... |
ors Find the sum of v1 = 2i − 3 j and v2 = 4i + 5 j. Solution According to the formula, we have 1192 Chapter 10 Further Applications of Trigonometry v1 + v2 = (2 + 4)i + ( − 3 + 5) j = 6i + 2 j Calculating the Component Form of a Vector: Direction We have seen how to draw vectors according to their initial and terminal... |
vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in m... |
Vectors Find the angle between u = 〈 −3, 4 〉 and v = 〈 5, 12 〉. Solution Using the formula, we have This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 10 Further Applications of Trigonometry 1195 ⋅ ⋅ ⎛ ⎝ ⎞ ⎠ = v u |v| |u| = θ = cos−1 ⎛ u ⎝ |u| −3i + 4 j 5 ⎛ ⎝− 3 5 = − 15 65 ⎞ ⋅ 5 ⎠ + 13... |
7 = 0.04896 sin−1(0.04896) = 2.8° Therefore, the plane has a SE bearing of 140°+2.8°=142.8°. The ground speed is 212.7 miles per hour. Access these online resources for additional instruction and practice with vectors. • Introduction to Vectors (http://openstaxcollege.org/l/introvectors) • Vector Operations (http://ope... |
terms of i and j. For the following exercises, use the vectors u = i + 5j, v = −2i− 3j, and w = 4i − j. 510. Find u + (v − w) 511. Find 4v + 2u For the following exercises, use the given vectors to compute u + v, u − v, and 2u − 3v. 512. u = 〈 2, − 3 〉, v = 〈 1, 5 〉 513. u = 〈 −3, 4 〉, v = 〈 −2, 1 〉 Let v = −4i + 3j. ... |
�� 2, −5 〉 1198 Chapter 10 Further Applications of Trigonometry 525. 〈 −4, −6 〉 For the following exercises, use the vectors shown to sketch 2u + v. 526. Given u = 3i − 4j and v = −2i + 3j, calculate u ⋅ v. 536. 527. Given u = −i − j and v = i + 5j, calculate u ⋅ v. Given u = 〈 −2, 4 〉 and v = 〈 −3, 1 〉, 528. calculate... |
horizontal. Round to the nearest hundredth. Find the magnitude of the horizontal and vertical 553. components of a vector with magnitude 5 pounds pointed in a direction of 55° above the horizontal. Round to the nearest hundredth. For the following exercises, write the vector shown in component form. 540. 541. Given in... |
A man starts walking from home and walks 3 miles at 559. 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction? 1200 Chapter 10 Further Applications of Trigonometry 560. A woman starts walking from h... |
the northwest at 80 km/hr. The plane flies with an airspeed of 500 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane? 565. As part of a video game, the point (5, 7) is rotated counterclockwise about the origin through an angle of 35°. Find the new coordinates of thi... |
over the domain. argument the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis cardioid a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given as r = a ± bcos θ and r = a ± bsin θ, where a b = 1 con... |
to the point (x, y); also called the amplitude oblique triangle any triangle that is not a right triangle one-loop limaҫon a polar curve represented by r = a ± bcos θ and r = a ± bsin θ such that a > 0, b > 0, and a b > 1; may be dimpled or convex; does not pass through the pole parameter a variable, often representin... |
(initial point) and an end point (terminal point) vector addition the sum of two vectors, found by adding corresponding components KEY EQUATIONS Law of Sines Area for oblique triangles sin α a = a sin α = sin β b = b sin β = sin γ c c sin γ Area = 1 2 = 1 2 = 1 2 bcsin α acsin β absin γ Law of Cosines a2 = b2 + c2 − 2... |
angle measurements and lengths of sides in oblique triangles. • The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be relat... |
and then graph it in the rectangular plane. See Example 10.21, Example 10.22, and Example 10.23. 10.4 Polar Coordinates: Graphs • It is easier to graph polar equations if we can test the equations for symmetry with respect to the line θ =, the π 2 polar axis, or the pole. • There are three symmetry tests that indicate... |
Archimedes’ spiral is given by r = θ, θ ≥ 0. See Example 10.33. 10.5 Polar Form of Complex Numbers • Complex numbers in the form a + bi are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as the imaginary axi... |
solve one of the equations for t, and substitute the expression into the second equation. See Example 10.48, Example 10.49, Example 10.50, and Example 10.51. • Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for t in one of the equations, an... |
.63. • Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See Example 10.64 and Example 10.65. • Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis. See Exam... |
the plane. Figure 10.129 Non-right Triangles: Law of Cosines 579. Solve the triangle, rounding to the nearest tenth, assuming α is opposite side a, β is opposite side b, and γ s opposite side c : a = 4, b = 6, c = 8. 580. Solve the triangle in Figure 10.130, rounding to the nearest tenth. This content is available for... |
587. Convert (7, − 2) to polar coordinates. Write the complex number in polar form. 588. Convert (−9, − 4) to polar coordinates. For the following exercises, convert the given Cartesian equation to a polar equation. 589. x = − 2 590. x2 + y2 = 64 603. 5 + 9i 604. 1 2 − 3 2 i For the following exercises, convert the co... |
t) = et ⎧ ⎨ y(t) = − 2e5 t ⎩ 623. x(t) = 3cos t ⎧ ⎨ y(t) = 2sin t ⎩ For the following exercises, find the powers of each complex number in polar form. 611. Find z4 when z = 2cis(70°) ⎛ 612. Find z2 when z = 5cis ⎝ ⎞ ⎠ 3π 4 624. A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the ... |
627. u − v Chapter 10 Further Applications of Trigonometry 1209 628. 2v − u + w For the following exercises, find a unit vector in the same direction as the given vector. 629. a = 8i − 6j 630. b = −3i − j the following exercises, For direction of the vector. find the magnitude and 631. 〈 6, −2 〉 632. 〈 −3, −3 〉 For th... |
642. Convert ⎛ ⎝2, π 3 ⎞ ⎠ to rectangular coordinates. y(t) = bsin t : x2 36 + y2 100 = 1. 643. Convert the polar equation to a Cartesian equation: x2 + y2 = 5y. 658. Graph the set of parametric equations and find the x(t) = − 2sin t ⎧ Cartesian equation: ⎨ y(t) = 5cos t ⎩. 659. A ball is launched with an initial velo... |
content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1211 11 | SYSTEMS OF EQUATIONS AND INEQUALITIES Figure 11.1 Enigma machines like this one, once owned by Italian dictator Benito Mussolini, were used by government and military officials for enciphering and deciphering top-secret communications durin... |
and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses. 11.1 | Systems of Linear Equations: Two Variables Learning Objectives In this section, you will: 11.1.1 Solve systems of equations by graphing. 11.1.2 Solve systems of equations by substitution... |
the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists. 2(4) + (7) = 15 True 3(4) − (7) = 5 True In addition to considering ... |
= y Solution Substitute the ordered pair (5, 1) into both equations. (5) + 3(1) = 8 8 = 8 True 2(5) − 9 = (1) 1=1 True The ordered pair (5, 1) satisfies both equations, so it is the solution to the system. Analysis We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered... |
, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system. Solving Systems of Equations by Substitution Solving a l... |
8) − 5(3) = 1 True True 11.3 Solve the following system of equations by substitution. x = y + 3 4 = 3x−2y Can the substitution method be used to solve any linear system in two variables? Yes, but the method works best if one of the equations contains a coefficient of 1 or –1 so that we do not have to deal with fraction... |
for x The solution to this system is ⎛ ⎝− 7 3 ⎞ ⎠., 2 3 Check the solution in the first equation. x + 2y = −1 ⎛ ⎝− 1 = −1 True Analysis We gain an important perspective on systems of equations by looking at the graphical representation. See Figure 11.6 to find that the equations intersect at the solution. We do not ne... |
3y = −16 5x−10y = 30 Solution One equation has 2x and the other has 5x. The least common multiple is 10x so we will have to multiply both equations by a constant in order to eliminate one variable. Let’s eliminate x by multiplying the first equation by −5 and the second equation by 2. Then, we add the two equations to... |
solution is ⎛ ⎝ 11 2 ⎞ ⎠. Check it in the other equation., 7 x 2 11 2 2 11 11.5 Solve the system of equations by addition. 2x + 3y = 8 3x + 5y = 10 Identifying Inconsistent Systems of Equations Containing Two Variables Now that we have several methods for solving systems of equations, we can use the methods to identif... |
of Dependent Equations Containing Two Variables Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution ... |
the equation R = xp, where x = quantity and p = price. The revenue function is shown in orange in Figure 11.11. The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue ... |
function is P(x) = 0.7x−35,000. Analysis The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. See Figure 11.12. This content is available for free at https://cnx.org/content/col11758/1.5... |
000 a = 2,000 − c 25c + 50(2,000 − c) = 70,000 25c + 100,000 − 50c = 70,000 − 25c = −30,000 c = 1,200 Substitute c = 1,200 into the first equation to solve for a. 1,200 + a = 2,000 a = 800 We find that 1,200 children and 800 adults bought tickets to the circus that day. Meal tickets at the circus cost $4.00 for childre... |
+ y = 2 12x−4y = − = 16 = 11 y = 3 6. 7. 8. 9. 5x − y = 4 x + 6y = 2 and (4, 0) −3x − 5y = 13 − x + 4y = 10 and (−6, 1) 3x + 7y = 1 2x + 4y = 0 and (2, 3) −2x + 5y = 7 2x + 9y = 7 and (−1, 1) 10. x + 8y = 43 3x−2y = −1 and (3, 5) the following exercises, For substitution. solve each system by 11. 12. 13. 14. x + 3y = ... |
= − 21 12 6 y = 2 y = −.2x + 1.3y = −0.1 4.2x + 4.2y = 2.1 0.1x + 0.2y = 2 0.35x−0.3y = 0 Graphical For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. 41. This con... |
a total cost of production C = 12x + 30 and a revenue function R = 20x. Find the break-even point. A fast-food restaurant has a cost of production 57. C(x) = 11x + 120 and a revenue function R(x) = 5x. When does the company start to turn a profit? A cell phone factory has a cost of production function 58. C(x) = 150x ... |
many of each gender attended the conference? A jeep and BMW enter a highway running east-west at 68. the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 3... |
adult tickets were sold? Admission into an amusement park for 4 children and 2 77. adults is $116.90. For 6 children and 3 adults, the admission is $175.35. Assuming a different price for children and 1232 Chapter 11 Systems of Equations and Inequalities adults, what is the price of the child’s ticket and the price of... |
, y, z), which we call an ordered triple. A system in upper triangular form looks like the following: Ax + By + Cz = D Ey + Fz = G Hz = K The third equation can be solved for z, and then we back-substitute to find y and x. To write the system in upper triangular form, we can perform the following operations: 1. Interch... |
three planes intersect with each other, but not at a common point. (b) Two of the planes are parallel and intersect with the third plane, but not with each other. (c) All three planes are parallel, so there is no point of intersection. Example 11.12 Determining Whether an Ordered Triple Is a Solution to a System Deter... |
The second step is multiplying equation (1) by −2 and adding the result to equation (3). These two steps will eliminate the variable x. −2x + 4y − 6z = −18 (1) multiplied by − 2 2x − 5y + 5z = 17 (3) ____________________________________ − y − z = −1 (5) In equations (4) and (5), we have created a new two-by-two system... |
= 12,000 We form the second equation according to the information that John invested $4,000 more in mutual funds than he invested in municipal bonds. z = y + 4,000 The third equation shows that the total amount of interest earned from each fund equals $670. Then, we write the three equations as a system. 0.03x + 0.04y... |
ations Containing Three Variables Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and o... |
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