类11 NCERT解决方案-第3章函数-练习3.1

问题1.找到与以下度数度量相对应的弧度数：

(i)25°(ii)-47°30′(iii)240°(iv)520°

(i) As we know that

180° = π radian

So, 1° = π/180° radian

Then, 25° = (π/180°) × 25°

                 = 5π/36 radians

Hence, 25° equals to 5π/36 radians. 

(ii) As we know that

180° = π radian

So, 1° = π/180°

And 60′ = 1°

        30′ = (1/2)°

So, -47°30′ = -47 (1/2)°

-47(1/2)° = (π/180) × (-95/2) = (-19π/72) radian. 

Hence, -47°30′ is equals to -19π/72 radian. 

(iii) As we know that

180° = π radian

1° = π/180° radian

So 240° = (π/180°) × 240°

              = 4π/3 radians

Hence, 240° equals to 4π/3 radians. 

(iv) As we know that

180° = π radian

1° = π/180° radian

So 520° = (π/180°) × 520°

             = 26 π/9 radians

Hence, 520° equals to 26 π/9 radians. 

问题2.找到与以下弧度度量相对应的度数度量(使用π= 22/7)

(i)11/16(ii)-4(iii)5π/ 3(iv)7π/ 6

(i) 11/16 radian = (11/16) (180°/π)  {as 180° = π radian, then 1 radian = 180°/π}

                              = (11/16) × (180° × 7/22)

                              = (11 × 180° × 7/16 × 22)

                             = 315/8°

                             = 39 (3/8)°

                            = 39(3/8)°

                            = 39° + (3/8)°

Again 1° = 60′

So (3/8)° = 60′ × (3/8)

                    = 22 (1/2)’

                    = 22 (1/2)’ 

                    = 22′ + 1/2′

Again 1′ = 60″

                   = (1/2)’ = 30″

So 39 (3/8)° = 39° 22′ 30″

Hence, 11/16 radian results to 39° 22′ 30″.

(ii) -4 radian = -4 × (180°/π) {as 180° = π radian, then 1 radian = 180°/π}. 

                     = -4 ×180° × 7/22

                    = -229° (1/11) 

                    = -229 (1/11)°= -229° + (1/11)°

Again(1/11)° = (1/11) × 60′. {as 1° = 60′} 

                   = 5(5/11)’

 Also, 5 (5/11)’ = 5′ + (5/11)’

   (5/11)’ = (5/11) × 60″ {as 1′ = 60″}

              = 27″

So, -229(1/11) = -229° 5’27”

Hence, -4 radian results to -229° 5′ 27″.

(iii) 5π/3 radian = (5 π/3) × (180/π) {as 180° = π radian, then 1 radian =180°/π}. 

                              = (5 × 180/3)°

                             = 300°

Hence, 5π/3 results to 300°.

(iv) 7π/6 radian = (7π/6) × (180°/π)  {as 180° = π radian, then 1 radian =180°/π}.

                             = (7 × 180/6)°

                            = 210°

Hence, 7π/6 radian results to 210°.

问题3.轮子在一分钟内可旋转360圈。一秒钟可以旋转多少弧度?

解决方案：

Given that

Total revolutions made by the wheel in one  minute is 360. 

1 second = 360/6 = 60

We know that

When a  wheel revolves once it covers  2π radian of distance. 

In one minute, it will turn an angle of 360 × 2π radian = 720 π radian

In one second, it will turn an angle of 720 π radian/60 = 12 π radian {as 1 minute = 60 seconds} 

Hence, in one second, the wheel turns an angle of 12π radian.

问题4.找到半径为100 cm的圆的中心与长度为22 cm的圆弧对向的角度的度数度量(使用π= 22/7)

解决方案：

Given that 

The radius of circle (r) = 100 cm. 

Length of the arc (l) = 22 cm. 

Let us consider the angle subtended by the arc is θ. 

Also, we know that θ = l/r

The angle subtended (θ) = 22/100 radian

For finding the degree measure we have to multiply 180°/π with radian measure

So, θ = (22/100) × (180/π) 

θ = (22/100) × (180 × 7/22) 

θ = (22 × 180 × 7/22 × 100) 

θ = 126/10 degree

θ = 12 (3/5) degree

We know that 1° = 60′

                 (3/5)° = 60′ × (3/5)

                           = 36′

So 12 (3/5)° = 12° 36′

Hence, the degree measure of the angle subtended at the Centre of a circle is 12° 36′

问题5.在直径40厘米的圆圈中，和弦的长度为20厘米。找到和弦的小弧线的长度。

解决方案：

Given that

Diameter of circle (d) = 40 cm

Radius (r) = d/2 = 40/2 = 20 cm

Let us consider AB as the chord of circle having length 20 cm, and Centre at O. 

It forms a triangle OAB, 

Having Radius = OA = OB = 20 cm

Also, chord AB = 20 cm

Hence, In ΔOAB OA = OB = AB. (equilateral triangle.)

So angle subtend = (π/3) radian

We know that θ = l/r (where θ = angle subtended by the arc 

                                               l = length of arc

                                               r = radius) 

Putting values of r and θ we get

π/3 = l/20

So.         

l = 20 π/3

Hence, length of the arc is 20π/3 cm. 

问题6.如果在两个圆中，相同长度的圆弧在中心处对角分别为60°和75°，则求出它们的半径比。

解决方案：

Given that 

Angle subtend by 1st arc (θ₁) = 60

Angle subtend by 2nd arc (θ₂) = 75

We know that θ = l/r

For 1st arc θ₁ = l₁/r₁

For 2nd arc θ₂ = l₂/r₂

θ₁/θ₂ = (l₁/r₁)/(l₂/r₂) 

θ₁/θ₂ = (l/r₁)/(l/r₂)  {here l₁ = l₂ = l} 

θ₁/θ₂ = r₂/r₁

60/75 = r₂/r₁

r₂/r₁ = 4/5

r₁/r₂ = 5/4

Hence, ratio of their radius is 5:4. 

问题7.如果摆长为75厘米，并且笔尖描述了一个长度为弧度的弧度，则以弧度为单位找到摆动的角度

(i)10厘米(ii)15厘米(iii)21厘米

解决方案：

(i) Given that

Length of an arc (l) = 10 cm

Radius which represents length of pendulum(r) = 75

As We know that θ = l/r

So θ = 10/75 = 2/15 rad

Hence, θ = 2/15 rad

(ii) Given that

Length of an arc (l) = 15 cm

Radius which represents length of pendulum (r) = 75

As We know that θ = l/r

So θ = 15/75 = 1/5 rad

Hence, θ = 1/5 rad

(iii) Given that

Length of an arc (l) = 21 cm

Radius which represents length of pendulum(r) = 75

As We know that θ = l/r

So θ = 21/75 = 7/25 rad

Hence, θ = 7/25 radian