第7章単位円:正弦関数と余弦関数The Unit Circle: Sine and Cosine Functions

Answer Key解答

Solution 1解答 1

Graph of a 240-degree angle with a counterclockwise rotation.

Solution 2解答 2

$\frac{3 π}{2}$

Solution 3解答 3

$−135 °$

Solution 4解答 4

$\frac{7 π}{10}$

Solution 5解答 5

$α = 150°$

Solution 6解答 6

$β = 60°$

Solution 7解答 7

$\frac{7 π}{6}$

Solution 8解答 8

$\frac{215 π}{18} = 37.525 units$

Solution 9解答 9

1.88

Solution 10解答 10

$\frac{3 π}{2}$ rad/s

毎秒 $\frac{3 π}{2}$ ラジアン

Solution 11解答 11

1655 kilometers per hour

毎時1655キロメートル

Solution 1解答 1

$\frac{7}{25}$

Solution 2解答 2

$\begin{matrix} sin t & = & \frac{33}{65}, & cos t & = & \frac{56}{65}, & tan t & = & \frac{33}{56}, \\ sec t & = & \frac{65}{56}, & csc t & = & \frac{65}{33}, & cot t & = & \frac{56}{33} \end{matrix}$

Solution 3解答 3

$\begin{matrix} sin (\frac{π}{4}) & = & \frac{\sqrt{2}}{2}, cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{π}{4}) = 1, \\ sec (\frac{π}{4}) & = & \sqrt{2}, csc (\frac{π}{4}) = \sqrt{2}, cot (\frac{π}{4}) = 1 \end{matrix}$

Solution 4解答 4

Solution 5解答 5

$adjacent = 10; opposite = 10 \sqrt{3};$ missing angle is $\frac{π}{6}$

$隣辺 = 10; 対辺 = 10 \sqrt{3};$ 。欠けている角は $\frac{π}{6}$ 。

Solution 6解答 6

About 52 ft

およそ52フィート

Solution 1解答 1

$\cos (t) = - \frac{\sqrt{2}}{2}, \sin (t) = \frac{\sqrt{2}}{2}$

Solution 2解答 2

$\cos (π) = - 1, \sin (π) = 0$

Solution 3解答 3

$\sin (t) = - \frac{7}{25}$

Solution 4解答 4

approximately 0.866025403

およそ0.866025403

Solution 5解答 5

$\frac{π}{3}$

Solution 6解答 6

ⓐ $cos (315°) = \frac{\sqrt{2}}{2}, sin (315°) = \frac{- \sqrt{2}}{2}$ ⓑ $cos (- \frac{π}{6}) = \frac{\sqrt{3}}{2}, \sin (- \frac{π}{6}) = - \frac{1}{2}$

Solution 7解答 7

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$

Solution 1解答 1

$\sin t = - \frac{\sqrt{2}}{2} \cos t = \frac{\sqrt{2}}{2}, \tan t = - 1, s e c t = \sqrt{2}, \csc t = - \sqrt{2}, \cot t = - 1$

Solution 2解答 2

$\sin \frac{π}{3} = \frac{\sqrt{3}}{2}, \cos \frac{π}{3} = \frac{1}{2}, \tan \frac{π}{3} = \sqrt{3}, s e c \frac{π}{3} = 2, c s c \frac{π}{3} = \frac{2 \sqrt{3}}{3}, c o t \frac{π}{3} = \frac{\sqrt{3}}{3}$

Solution 3解答 3

$\begin{array}{l} \sin (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \cos (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{- 7 π}{4}) = 1, \\ \sec (\frac{- 7 π}{4}) = \sqrt{2}, \csc (\frac{- 7 π}{4}) = \sqrt{2}, \cot (\frac{- 7 π}{4}) = 1 \end{array}$

Solution 4解答 4

$- \sqrt{3}$

Solution 5解答 5

$- 2$

Solution 6解答 6

$\sin t$

Solution 7解答 7

$\begin{array}{l} \cos t = - \frac{8}{17}, \sin t = \frac{15}{17}, \tan t = - \frac{15}{8} \\ \csc t = \frac{17}{15}, \cot t = - \frac{8}{15} \end{array}$

Solution 8解答 8

$\begin{array}{l} \sin t = - 1, \cos t = 0, \tan t = Undefined \\ \sec t = Undefined, \csc t = - 1, \cot t = 0 \end{array}$

Solution 9解答 9

$\sec t = \sqrt{2}, \csc t = \sqrt{2}, \tan t = 1, \cot t = 1$

Solution 10解答 10

$\approx - 2.414$

7.1 Section Exercises7.1 節末問題

Solution 1解答 1

Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.

Solution 3解答 3

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

角が正か負かが向きを決める。正の角は反時計回りに、負の角は時計回りに描く。

Solution 5解答 5

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

線速度は、弧の距離を時間と比べて計算する量である。角速度は、弧の角を時間と比べて計算する量である。

Solution 7解答 7

Graph of a circle with an angle inscribed.

Solution 9解答 9

Graph of a circle with a 135 degree angle inscribed.

Solution 11解答 11

Graph of a circle with a 2pi/3 radians angle inscribed.

Solution 13解答 13

Graph of a circle with 5pi/6 radians angle inscribed.

Solution 15解答 15

Graph of a circle with a –pi/10 radians angle inscribed.

Solution 17解答 17

$240°$

Graph of a circle showing the equivalence of two angles.

Solution 19解答 19

$\frac{4 π}{3}$

Solution 21解答 21

$\frac{2 π}{3}$

Solution 23解答 23

$\frac{7 π}{2} \approx 11.00 {in}^{2}$

Solution 25解答 25

$\frac{81 π}{20} \approx 12.72 {cm}^{2}$

Solution 27解答 27

$20°$

Solution 29解答 29

$60°$

Solution 31解答 31

$−75°$

Solution 33解答 33

$\frac{π}{2}$ radians

$\frac{π}{2}$ ラジアン

Solution 35解答 35

$−3 π$ radians

$−3 π$ ラジアン

Solution 37解答 37

$π$ radians

$π$ ラジアン

Solution 39解答 39

$\frac{5 π}{6}$ radians

$\frac{5 π}{6}$ ラジアン

Solution 41解答 41

$\frac{5.02 π}{3} \approx 5.26$ miles

$\frac{5.02 π}{3} \approx 5.26$ マイル

Solution 43解答 43

$\frac{25 π}{9} \approx 8.73$ centimeters

$\frac{25 π}{9} \approx 8.73$ センチメートル

Solution 45解答 45

$\frac{21 π}{10} \approx 6.60$ meters

$\frac{21 π}{10} \approx 6.60$ メートル

Solution 47解答 47

104.7198 cm²

Solution 49解答 49

0.7697 in²

Solution 51解答 51

$250°$

Solution 53解答 53

$320°$

Solution 55解答 55

$\frac{4 π}{3}$

Solution 57解答 57

$\frac{8 π}{9}$

Solution 59解答 59

$1320$ rad/min $210.085$ RPM

毎分 $1320$ ラジアン、 $210.085$ RPM

Solution 61解答 61

$7$ in./s, 4.77 RPM , $28.65$ deg/s

毎秒 $7$ インチ、4.77 RPM、毎秒 $28.65$ 度

Solution 63解答 63

$1, 809, 557.37 mm/min =$
$30.16 m/s$

Solution 65解答 65

$5.76$ miles

$5.76$ マイル

Solution 67解答 67

$120°$

Solution 69解答 69

794 miles per hour

毎時794マイル

Solution 71解答 71

2,234 miles per hour

毎時2,234マイル

Solution 73解答 73

11.5 inches

11.5インチ

7.2 Section Exercises7.2 節末問題

Solution 1解答 1

A right triangle with side opposite, adjacent, and hypotenuse labeled.

Solution 3解答 3

The tangent of an angle is the ratio of the opposite side to the adjacent side.

角の正接は、対辺と隣辺の比である。

Solution 5解答 5

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

たとえば、角の正弦はその余角の余弦に等しく、角の余弦はその余角の正弦に等しい。

Solution 7解答 7

$\frac{π}{6}$

Solution 9解答 9

$\frac{π}{4}$

Solution 11解答 11

$b = \frac{20 \sqrt{3}}{3}, c = \frac{40 \sqrt{3}}{3}$

Solution 13解答 13

$a = 10,000, c = 10,000.5$

Solution 15解答 15

$b = \frac{5 \sqrt{3}}{3}, c = \frac{10 \sqrt{3}}{3}$

Solution 17解答 17

$\frac{5 \sqrt{29}}{29}$

Solution 19解答 19

$\frac{5}{2}$

Solution 21解答 21

$\frac{\sqrt{29}}{2}$

Solution 23解答 23

$\frac{5 \sqrt{41}}{41}$

Solution 25解答 25

$\frac{5}{4}$

Solution 27解答 27

$\frac{\sqrt{41}}{4}$

Solution 29解答 29

$c = 14, b = 7 \sqrt{3}$

Solution 31解答 31

$a = 15, b = 15$

Solution 33解答 33

$b = 9.9970, c = 12.2041$

Solution 35解答 35

$a = 2.0838, b = 11.8177$

Solution 37解答 37

$a = 55.9808, c = 57.9555$

Solution 39解答 39

$a = 46.6790, b = 17.9184$

Solution 41解答 41

$a = 16.4662, c = 16.8341$

Solution 43解答 43

188.3159

Solution 45解答 45

200.6737

Solution 47解答 47

498.3471 ft

498.3471フィート

Solution 49解答 49

1060.09 ft

1060.09フィート

Solution 51解答 51

27.372 ft

27.372フィート

Solution 53解答 53

22.6506 ft

22.6506フィート

Solution 55解答 55

368.7633 ft

368.7633フィート

7.3 Section Exercises7.3 節末問題

Solution 1解答 1

The unit circle is a circle of radius 1 centered at the origin.

単位円とは、原点を中心とする半径1の円である。

Solution 3解答 3

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $t,$ formed by the terminal side of the angle $t$ and the horizontal axis.

動径が一致する角とは、同じ動径を持つ角である。基準角とは、その角 $t$ の動径と横軸がなす、最も小さい鋭角 $t$ の大きさである。

Solution 5解答 5

The sine values are equal.

正弦の値は等しい。

Solution 7解答 7

Solution 9解答 9

Solution 11解答 11

$\frac{\sqrt{3}}{2}$

Solution 13解答 13

$\frac{1}{2}$

Solution 15解答 15

$\frac{\sqrt{2}}{2}$

Solution 17解答 17

Solution 19解答 19

-1

Solution 21解答 21

$\frac{\sqrt{3}}{2}$

Solution 23解答 23

$60°$

Solution 25解答 25

$80 °$

Solution 27解答 27

$45°$

Solution 29解答 29

$\frac{π}{3}$

Solution 31解答 31

$\frac{π}{3}$

Solution 33解答 33

$\frac{π}{8}$

Solution 35解答 35

$60°,$ Quadrant IV, $sin (300°) = - \frac{\sqrt{3}}{2}$ , $\cos (300°) = \frac{1}{2}$

$60°$ 、第IV象限、 $sin (300°) = - \frac{\sqrt{3}}{2}$ 、 $\cos (300°) = \frac{1}{2}$

Solution 37解答 37

$45°,$ Quadrant II, $sin (135°) = \frac{\sqrt{2}}{2}$ , $\cos (135°) = - \frac{\sqrt{2}}{2}$

$45°$ 、第II象限、 $sin (135°) = \frac{\sqrt{2}}{2}$ 、 $\cos (135°) = - \frac{\sqrt{2}}{2}$

Solution 39解答 39

$60°,$ Quadrant II, $sin (120°) = \frac{\sqrt{3}}{2}$ , $\cos (120°) = - \frac{1}{2}$

$60°$ 、第II象限、 $sin (120°) = \frac{\sqrt{3}}{2}$ 、 $\cos (120°) = - \frac{1}{2}$

Solution 41解答 41

$30°,$ Quadrant II, $sin (150°) = \frac{1}{2}$ , $\cos (150°) = - \frac{\sqrt{3}}{2}$

$30°$ 、第II象限、 $sin (150°) = \frac{1}{2}$ 、 $\cos (150°) = - \frac{\sqrt{3}}{2}$

Solution 43解答 43

$\frac{π}{6},$ Quadrant III, $sin (\frac{7 π}{6}) = - \frac{1}{2}$ , $\cos (\frac{7 π}{6}) = - \frac{\sqrt{3}}{2}$

$\frac{π}{6}$ 、第III象限、 $sin (\frac{7 π}{6}) = - \frac{1}{2}$ 、 $\cos (\frac{7 π}{6}) = - \frac{\sqrt{3}}{2}$

Solution 45解答 45

$\frac{π}{4},$ Quadrant II, $sin (\frac{3 π}{4}) = \frac{\sqrt{2}}{2}$ , $\cos (\frac{3 π}{4}) = - \frac{\sqrt[]{2}}{2}$

$\frac{π}{4}$ 、第II象限、 $sin (\frac{3 π}{4}) = \frac{\sqrt{2}}{2}$ 、 $\cos (\frac{3 π}{4}) = - \frac{\sqrt[]{2}}{2}$

Solution 47解答 47

$\frac{π}{3},$ Quadrant II, $sin (\frac{2 π}{3}) = \frac{\sqrt{3}}{2}$ , $\cos (\frac{2 π}{3}) = - \frac{1}{2}$

$\frac{π}{3}$ 、第II象限、 $sin (\frac{2 π}{3}) = \frac{\sqrt{3}}{2}$ 、 $\cos (\frac{2 π}{3}) = - \frac{1}{2}$

Solution 49解答 49

$\frac{π}{4},$ Quadrant IV, $sin (\frac{7 π}{4}) = - \frac{\sqrt{2}}{2}, cos (\frac{7 π}{4}) = \frac{\sqrt{2}}{2}$

$\frac{π}{4}$ 、第IV象限、 $sin (\frac{7 π}{4}) = - \frac{\sqrt{2}}{2}, cos (\frac{7 π}{4}) = \frac{\sqrt{2}}{2}$

Solution 51解答 51

$\frac{\sqrt{77}}{9}$

Solution 53解答 53

$- \frac{\sqrt{15}}{4}$

Solution 55解答 55

$(−10, 10 \sqrt{3})$

Solution 57解答 57

$(–2.778, 15.757)$

Solution 59解答 59

$[–1, 1]$

Solution 61解答 61

$\sin t = \frac{1}{2}, \cos t = - \frac{\sqrt{3}}{2}$

Solution 63解答 63

$\sin t = - \frac{\sqrt{2}}{2}, \cos t = - \frac{\sqrt{2}}{2}$

Solution 65解答 65

$\sin t = \frac{\sqrt{3}}{2}, \cos t = - \frac{1}{2}$

Solution 67解答 67

$\sin t = - \frac{\sqrt{2}}{2}, \cos t = \frac{\sqrt{2}}{2}$

Solution 69解答 69

$\sin t = 0, \cos t = - 1$

Solution 71解答 71

$\sin t = - 0.596, \cos t = 0.803$

Solution 73解答 73

$\sin t = \frac{1}{2}, \cos t = \frac{\sqrt{3}}{2}$

Solution 75解答 75

$\sin t = - \frac{1}{2}, \cos t = \frac{\sqrt{3}}{2}$

Solution 77解答 77

$\sin t = 0.761, \cos t = - 0.649$

Solution 79解答 79

$\sin t = 1, \cos t = 0$

Solution 81解答 81

−0.1736

Solution 83解答 83

0.9511

Solution 85解答 85

−0.7071

Solution 87解答 87

−0.1392

Solution 89解答 89

−0.7660

Solution 91解答 91

$\frac{\sqrt{2}}{4}$

Solution 93解答 93

$- \frac{\sqrt{6}}{4}$

Solution 95解答 95

$\frac{\sqrt{2}}{4}$

Solution 97解答 97

$\frac{\sqrt{2}}{4}$

Solution 99解答 99

Solution 101解答 101

$(0, –1)$

Solution 103解答 103

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

37.5秒、97.5秒、157.5秒、217.5秒、277.5秒、337.5秒

7.4 Section Exercises7.4 節末問題

Solution 1解答 1

Yes, when the reference angle is $\frac{π}{4}$ and the terminal side of the angle is in quadrants I and III. Thus, a $x = \frac{π}{4}, \frac{5 π}{4},$ the sine and cosine values are equal.

はい。基準角が $\frac{π}{4}$ で、角の動径が第I象限と第III象限にあるときである。よって $x = \frac{π}{4}, \frac{5 π}{4}$ で正弦と余弦の値が等しい。

Solution 3解答 3

Substitute the sine of the angle in for $y$ in the Pythagorean Theorem $x^{2} + y^{2} = 1.$ Solve for $x$ and take the negative solution.

ピタゴラスの定理 $x^{2} + y^{2} = 1.$ の $y$ に、その角の正弦を代入する。 $x$ について解き、負の解をとる。

Solution 5解答 5

The outputs of tangent and cotangent will repeat every $π$ units.

正接と余接の出力は $π$ ごとに繰り返す。

Solution 7解答 7

$\frac{2 \sqrt{3}}{3}$

Solution 9解答 9

$\sqrt{3}$

Solution 11解答 11

$\sqrt{2}$

Solution 13解答 13

Solution 15解答 15

Solution 17解答 17

$\frac{\sqrt{3}}{3}$

Solution 19解答 19

$- \frac{2 \sqrt{3}}{3}$

Solution 21解答 21

$\sqrt{3}$

Solution 23解答 23

$- \sqrt{2}$

Solution 25解答 25

–1

Solution 27解答 27

-2

Solution 29解答 29

$- \frac{\sqrt{3}}{3}$

Solution 31解答 31

Solution 33解答 33

$\frac{\sqrt{3}}{3}$

Solution 35解答 35

–2

Solution 37解答 37

–1

Solution 39解答 39

$\sin t = - \frac{2 \sqrt{2}}{3}$ , $\sec t = - 3$ , $\csc t = - \frac{3 \sqrt{2}}{4}$ , $\tan t = 2 \sqrt{2}$ , $\cot t = \frac{\sqrt{2}}{4}$

Solution 41解答 41

$\sec t = 2,$ $\csc t = \frac{2 \sqrt{3}}{3},$ $\tan t = \sqrt{3},$ $\cot t = \frac{\sqrt{3}}{3}$

Solution 43解答 43

$- \frac{\sqrt{2}}{2}$

Solution 45解答 45

3.1

Solution 47解答 47

1.4

Solution 49解答 49

$\sin t = \frac{\sqrt{2}}{2}$ , $\cos t = \frac{\sqrt{2}}{2}$ , $\tan t = 1$ , $\cot t = 1$ , $\sec t = \sqrt{2}$ , $\csc t = \sqrt{2}$

Solution 51解答 51

$\sin t = - \frac{\sqrt{3}}{2}$ , $\cos t = - \frac{1}{2}$ $\tan t = \sqrt{3}$ , $\cot t = \frac{\sqrt{3}}{3}$ , $\sec t = - 2$ , $\csc t = - \frac{2 \sqrt{3}}{3}$

Solution 53解答 53

–0.228

Solution 55解答 55

–2.414

Solution 57解答 57

1.414

Solution 59解答 59

1.540

Solution 61解答 61

1.556

Solution 63解答 63

$\sin (t) \approx 0.79$

Solution 65解答 65

$\csc t \approx 1.16$

Solution 67解答 67

even

偶関数

Solution 69解答 69

even

偶関数

Solution 71解答 71

$\frac{\sin t}{\cos t} = \tan t$

Solution 73解答 73

13.77 hours, period: $1000 π$

13.77時間。周期: $1000 π$

Solution 75解答 75

3.46 inches

3.46インチ

Review Exercises復習問題

Solution 1解答 1

$45°$

Solution 3解答 3

$- \frac{7 π}{6}$

Solution 5解答 5

10.385 meters

10.385メートル

Solution 7解答 7

$60°$

Solution 9解答 9

$\frac{2 π}{11}$

Solution 11解答 11

This is an image of a graph of a circle with a negative angle inscribed.

Solution 13解答 13

This is an image of a graph of a circle with an angle inscribed.

Solution 15解答 15

1036.73 miles per hour

毎時1036.73マイル

Solution 17解答 17

$\frac{\sqrt{2}}{2}$

Solution 19解答 19

$\frac{\sqrt{3}}{3}$

Solution 21解答 21

$72°$

Solution 23解答 23

$a = \frac{10}{3}, c = \frac{2 \sqrt{106}}{3}$

Solution 25解答 25

$\frac{6}{11}$

Solution 27解答 27

$a = \frac{5 \sqrt{3}}{2}, b = \frac{5}{2}$

Solution 29解答 29

369.2136 ft

369.2136フィート

Solution 31解答 31

$\frac{\sqrt{2}}{2}$

Solution 33解答 33

$60°$

Solution 35解答 35

$\frac{\sqrt{3}}{2}$

Solution 37解答 37

all real numbers

実数全体

Solution 39解答 39

$\frac{\sqrt{3}}{2}$

Solution 41解答 41

$\frac{2 \sqrt{3}}{3}$

Solution 43解答 43

Solution 45解答 45

–2.5

Solution 47解答 47

$\frac{1}{3}$

Solution 49解答 49

cosine, secant

余弦、正割

Practice Test実力テスト

Solution 1解答 1

$150°$

Solution 3解答 3

6.283 centimeters

6.283センチメートル

Solution 5解答 5

$15°$

Solution 7解答 7

Solution 9解答 9

3.351 feet per second, $\frac{2 π}{75}$ radians per second

毎秒3.351フィート、毎秒 $\frac{2 π}{75}$ ラジアン

Solution 11解答 11

$a = \frac{9}{2}, b = \frac{9 \sqrt{3}}{2}$

Solution 13解答 13

$\frac{1}{2}$

Solution 15解答 15

real numbers

実数

Solution 17解答 17

Solution 19解答 19

$- \sqrt{2}$

Solution 21解答 21

–0.68

Solution 23解答 23

$\frac{π}{3}$