第10章三角法のさらなる応用Further Applications of Trigonometry

Review Exercises復習問題

Non-right Triangles: Law of Sines直角三角形でない三角形: 正弦定理

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

次の各問では、 $α$ が辺 $a, β$ と、 $b$ が辺 $γ$ と、 $c$ がそれぞれ向かい合うとせよ。できるなら各三角形を解け。各答えは小数第一位まで求めよ。

1. $β = 50°, a = 105, b = 45$

2. $α = 43.1°, a = 184.2, b = 242.8$

3. Solve the triangle.

3. この三角形を解け。

Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.

4. Find the area of the triangle.

4. この三角形の面積を求めよ。

A triangle. One angle is 75 degrees with opposite side unknown. The adjacent sides to the 75 degree angle are 8 and 11.

5. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in Figure 1. Find the distance of the plane from point $A$ and the elevation of the plane.

5. ある操縦士がまっすぐな幹線道路の上を飛んでいる。図1に示すとおり、2.1km離れた二つの里程標への俯角が25°と49°だと分かる。点 $A$ からこの飛行機までの距離と、この飛行機の高度を求めよ。

Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.

Non-right Triangles: Law of Cosines直角三角形でない三角形: 余弦定理

6. Solve the triangle, rounding to the nearest tenth, assuming $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c : a = 4, b = 6, c = 8.$

6. $α$ が辺 $a, β$ と、 $b$ が辺 $γ$ と、 $c : a = 4, b = 6, c = 8.$ がそれぞれ向かい合うとして、この三角形を小数第一位まで求めて解け。

7. Solve the triangle in Figure 2, rounding to the nearest tenth.

7. 図2の三角形を、小数第一位まで求めて解け。

A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.

8. Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

8. 長さ8.3、6.6、9.1の辺を持つ三角形の面積を求めよ。

9. To find the distance between two cities, a satellite calculates the distances and angle shown in Figure 3 (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

9. 二つの都市のあいだの距離を求めるため、ある人工衛星が図3(縮尺は正しくない)の距離と角を計算する。二つの都市のあいだの距離を求めよ。答えは小数第一位まで求めよ。

Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.

Polar Coordinates極座標

10. Plot the point with polar coordinates $(3, \frac{π}{6}) .$

10. 極座標 $(3, \frac{π}{6})$ の点を打て。

11. Plot the point with polar coordinates $(5, - \frac{2 π}{3})$

11. 極座標 $(5, - \frac{2 π}{3})$ の点を打て。

12. Convert $(6, - \frac{3 π}{4})$ to rectangular coordinates.

12. $(6, - \frac{3 π}{4})$ を直交座標に換えよ。

13. Convert $(- 2, \frac{3 π}{2})$ to rectangular coordinates.

13. $(- 2, \frac{3 π}{2})$ を直交座標に換えよ。

14. Convert $(7, - 2)$ to polar coordinates.

14. $(7, - 2)$ を極座標に換えよ。

15. Convert $(- 9, - 4)$ to polar coordinates.

15. $(- 9, - 4)$ を極座標に換えよ。

For the following exercises, convert the given Cartesian equation to a polar equation.

次の各問について、与えられたデカルトの式を極方程式に換えよ。

16. $x = - 2$

17. $x^{2} + y^{2} = 64$

18. $x^{2} + y^{2} = - 2 y$

For the following exercises, convert the given polar equation to a Cartesian equation.

次の各問について、与えられた極方程式をデカルトの式に換えよ。

19. $r = 7 cos θ$

20. $r = \frac{- 2}{4 \cos θ + \sin θ}$

For the following exercises, convert to rectangular form and graph.

次の各問について、直交形式に換えてグラフを描け。

21. $θ = \frac{3 π}{4}$

22. $r = 5 \sec θ$

Polar Coordinates: Graphs極座標: グラフ

For the following exercises, test each equation for symmetry.

次の各問について、各式の対称性を調べよ。

23. $r = 4 + 4 \sin θ$

24. $r = 7$

25. Sketch a graph of the polar equation $r = 1 - 5 \sin θ .$ Label the axis intercepts.

25. 極方程式 $r = 1 - 5 \sin θ$ のグラフの概形を描け。座標軸との交点に印を付けよ。

26. Sketch a graph of the polar equation $r = 5 \sin (7 θ) .$

26. 極方程式 $r = 5 \sin (7 θ)$ のグラフの概形を描け。

27. Sketch a graph of the polar equation $r = 3 - 3 \cos θ$

27. 極方程式 $r = 3 - 3 \cos θ$ のグラフの概形を描け。

Polar Form of Complex Numbers複素数の極形式

For the following exercises, find the absolute value of each complex number.

次の各問について、各複素数の絶対値を求めよ。

28. $- 2 + 6 i$

29. $4 - 3 i$

Write the complex number in polar form.

この複素数を極形式で書け。

30. $5 + 9 i$

31. $\frac{1}{2} - \frac{\sqrt{3}}{2} i$

For the following exercises, convert the complex number from polar to rectangular form.

次の各問について、複素数を極形式から直交形式に換えよ。

32. $z = 5 cis (\frac{5 π}{6})$

33. $z = 3 cis (40°)$

For the following exercises, find the product $z_{1} z_{2}$ in polar form.

次の各問について、積 $z_{1} z_{2}$ を極形式で求めよ。

34. $z_{1} = 2 cis (89°)$

$z_{2} = 5 cis (23°)$

35. $z_{1} = 10 cis (\frac{π}{6})$

$z_{2} = 6 cis (\frac{π}{3})$

For the following exercises, find the quotient $\frac{z_{1}}{z_{2}}$ in polar form.

次の各問について、商 $\frac{z_{1}}{z_{2}}$ を極形式で求めよ。

36. $z_{1} = 12 cis (55°)$

$z_{2} = 3 cis (18°)$

37. $z_{1} = 27 cis (\frac{5 π}{3})$

$z_{2} = 9 cis (\frac{π}{3})$

For the following exercises, find the powers of each complex number in polar form.

次の各問について、各複素数の累乗を極形式で求めよ。

38. Find $z^{4}$ when $z = 2 cis (70°)$

38. $z = 2 cis (70°)$ のとき $z^{4}$ を求めよ。

39. Find $z^{2}$ when $z = 5 cis (\frac{3 π}{4})$

39. $z = 5 cis (\frac{3 π}{4})$ のとき $z^{2}$ を求めよ。

For the following exercises, evaluate each root.

次の各問について、各根の値を求めよ。

40. Evaluate the cube root of $z$ when $z = 64 cis (210°) .$

40. $z = 64 cis (210°)$ のとき、 $z$ の三乗根の値を求めよ。

41. Evaluate the square root of $z$ when $z = 25 cis (\frac{3 π}{2}) .$

41. $z = 25 cis (\frac{3 π}{2})$ のとき、 $z$ の平方根の値を求めよ。

For the following exercises, plot the complex number in the complex plane.

次の各問について、複素平面にその複素数を打て。

42. $6 - 2 i$

43. $- 1 + 3 i$

Parametric Equations媒介変数表示

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

次の各問について、媒介変数 $t$ を消して、媒介変数表示をデカルトの式に書き直せ。

44. ${\begin{array}{l} x (t) = 3 t - 1 \\ y (t) = \sqrt{t} \end{array}$

45. ${\begin{cases} x (t) = - \cos t \\ y (t) = 2 \sin^{2} t \end{cases}$

46. Parameterize (write a parametric equation for) each Cartesian equation by using $x (t) = a \cos t$ and $y (t) = b \sin t$ for $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1.$

46. $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1.$ について $x (t) = a \cos t$ と $y (t) = b \sin t$ を使って、各デカルトの式を媒介変数で表せ(媒介変数表示を書け)。

47. Parameterize the line from $(- 2, 3)$ to $(4, 7)$ so that the line is at $(- 2, 3)$ at $t = 0$ and $(4, 7)$ at $t = 1.$

47. $t = 0$ のとき $(- 2, 3)$ に、 $t = 1.$ のとき $(4, 7)$ にあるように、 $(- 2, 3)$ から $(4, 7)$ への直線を媒介変数で表せ。

Parametric Equations: Graphs媒介変数表示: グラフ

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

次の各問について、各媒介変数表示の組の値の表を作り、その式のグラフを描き、向きを入れよ。そしてデカルトの式を書け。

48. ${\begin{array}{l} x (t) = 3 t^{2} \\ y (t) = 2 t - 1 \end{array}$

49. ${\begin{array}{l} x (t) = e^{t} \\ y (t) = - 2 e^{5 t} \end{array}$

50. ${\begin{array}{l} x (t) = 3 \cos t \\ y (t) = 2 \sin t \end{array}$

51. A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

51. ある球が初速毎秒80フィート、水平に対して40°の角で打ち出される。球は地上4フィートの高さで放たれる。

ⓐFind the parametric equations to model the path of the ball.ⓑWhere is the ball after 3 seconds?ⓒHow long is the ball in the air?

Vectorsベクトル

For the following exercises, determine whether the two vectors, $u$ and $v,$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2},$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4} .$

次の各問について、 $u$ が始点 $P_{1}$ 、終点 $P_{2}$ を持ち、 $v$ が始点 $P_{3}$ 、終点 $P_{4}$ を持つとして、二つのベクトル $u$ と $v$ が等しいかどうかを判定せよ。

52. $P_{1} = (- 1, 4), P_{2} = (3, 1), P_{3} = (5, 5)$ and $P_{4} = (9, 2)$

52. $P_{1} = (- 1, 4), P_{2} = (3, 1), P_{3} = (5, 5)$ と $P_{4} = (9, 2)$

53. $P_{1} = (6, 11), P_{2} = (- 2, 8), P_{3} = (0, - 1)$ and $P_{4} = (- 8, 2)$

53. $P_{1} = (6, 11), P_{2} = (- 2, 8), P_{3} = (0, - 1)$ と $P_{4} = (- 8, 2)$

For the following exercises, use the vectors $u = 2 i - j, v = 4 i - 3 j,$ and $w = - 2 i + 5 j$ to evaluate the expression.

次の各問について、ベクトル $u = 2 i - j, v = 4 i - 3 j$ と $w = - 2 i + 5 j$ を使って式の値を求めよ。

54. u − v

55. 2v − u + w

For the following exercises, find a unit vector in the same direction as the given vector.

次の各問について、与えられたベクトルと同じ向きの単位ベクトルを求めよ。

56. a = 8i − 6j

57. b = −3i − j

For the following exercises, find the magnitude and direction of the vector.

次の各問について、そのベクトルの大きさと向きを求めよ。

58. $〈 6, −2 〉$

59. $〈 −3, −3 〉$

For the following exercises, calculate $u \cdot v .$

次の各問について、 $u \cdot v$ を計算せよ。

60. u = −2i + j and v = 3i + 7j

60. u = −2i + j かつ v = 3i + 7j

61. u = i + 4j and v = 4i + 3j

61. u = i + 4j かつ v = 4i + 3j

62. Given v $= ⟨ −3, 4 ⟩$ draw v, 2v, and $\frac{1}{2}$ v.

62. v $= ⟨ −3, 4 ⟩$ が与えられたとき、v、2v、 $\frac{1}{2}$ v を描け。

63. Given the vectors shown in Figure 4, sketch u + v, u − v and 3v.

63. 図4に示されたベクトルについて、u + v、u − v、3v の概形を描け。

Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude.

64. Given initial point $P_{1} = (3, 2)$ and terminal point $P_{2} = (- 5, - 1),$ write the vector $v$ in terms of $i$ and $j .$ Draw the points and the vector on the graph.

64. 始点 $P_{1} = (3, 2)$ と終点 $P_{2} = (- 5, - 1)$ が与えられたとき、ベクトル $v$ を $i$ と $j$ で書け。点とベクトルをグラフに描け。