Exercise 3.1

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Q1. Find the radian measures corresponding to the following degree measures:

(i) 25°     (ii) - 47°30' (iii) 240°    (iv)  520°

Solution:

(i) 25°

Q3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Q4. Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use π = 22/7).

Q5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

Q6. If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Q7. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length

(i) 10 cm (ii) 15 cm (iii) 21 cm

  

 

Exercise 3.2

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Find the values of other five trigonometric functions in Exercises 1 to 5.

∵ x lies in third quadrant

∴ tan and cot are positive in the third quadrant but remain all trigonometry ratios are negative;

So we have all trigonometry ratios,

∵ x lies in second quadrant

∴ sin and cosec are positive in the second quadrant but remain all trigonometry ratios are negative;

So we have all trigonometry ratios,

∵ x lies in third quadrant

∴ tan and cot are positive in the third quadrant but remain all trigonometry ratios are negative;

So we have all trigonometry ratios,

∵ x lies in fourth quadrant

∴ cos and sec are positive in the fourth quadrant but remain all trigonometry ratios are negative;

So we have all trigonometry ratios,

∵ x lies in second quadrant

∴ sin and cosec are positive in the second quadrant but remain all trigonometry ratios are negative;

So we have all trigonometry ratios,

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Exercise 3.3

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Q5.  Find the value of

          (i) sin 75°                     (ii) tan 15°

Solution:

(i) sin 75° = sin (45° + 30°)      [∵ sin(x + y) = sin x cos y + cos x sin y ]

                 = sin 45° cos 30°  + cos 45° sin 30°  

(ii) tan 15°

Solution:

(ii)  tan 15° = tan (45° - 30°)

Q10. sin(n + 1)x sin(n + 2)x + cos(n + 1)x cos(n + 2)x = cos x

Solution:

LHS = sin(n + 1)x sin(n + 2)x + cos(n + 1)x cos(n + 2)x

Or       cos(n + 2)x cos(n + 1)x + sin(n + 2)x sin(n + 1)x 

Let be the A = (n + 2)x, B = (n + 1)x

Now we have,

LHS = cos A cos B + sin A sin B    

       =  cos ( A - B)      [∵ sin(x + y) = sin x cos y + cos x sin y ]

       = cos [(n + 2)x - (n + 1)x]  

       = cos [nx + 2x - (nx + x) ]

       = cos [nx + 2x - nx - x ]

       = cos x

LHS = RHS 

Q12. sin² 6x - sin² 4x = sin 2x sin 10x

Solution:

Sin² A - sin²B = sin(A + B) sin(A - B)

LHS = sin² 6x - sin² 4x

        = sin (6x + 4x) sin(6x - 4x)

        = sin 10x sin 2x

        = sin 2x sin 10x

LHS = RHS 

Q13. cos² 2x - cos² 6x = sin 4x sin 8x

Solution:

LHS = cos² 2x - cos² 6x

        = (1 - sin² 2x) - (1 - sin² 6x)

        = (1 - sin² 2x - 1 + sin² 6x)

        = - sin² 2x + sin² 6x

        = sin² 6x - sin² 2x

        = sin (6x + 2x) sin(6x -2x)

        = sin 10x sin 4x    

Q24. cos 4x = 1 – 8 sin²  x cos² x

Solution:

LHS = cos 2(2x)

       = cos 2A   [ Let be A = 2x ]

       = 1 - 2 sin² A

       = 1 - 2 sin² 2x    [Putting A = 2x ]

       = 1 - 2 [sin 2x]²

       = 1 - 2 [2 sin x cos x ]²

       = 1 - 2 [4 sin² x cos² x]

       = 1 - 8 sin² x cos² x

LHS = RHS 

Q25.  cos 6x = 32 cos⁶ x – 48cos⁴ x + 18 cos² x – 1

Solution:

LHS = cos 6x  = cos 3(2x)

        = cos 3A    [Let be A = 2x]

        = 4 cos³ A - 3 cos A

        = 4 cos³ 2x - 3 cos 2x    [Putting A = 2x]

        = 4 [cos 2x]³ - 3 [cos 2x]

        = 4 [2cos² x – 1]³ - 3 [2cos² x – 1]

        = 4 [(2cos² x)³ - 1³ - 3(2cos² x)² (1) + 3 (2cos² x)(1)²  ] - 3 [2cos² x – 1]

        = 4 [8 cos⁶ x - 1 - 12 cos² x + 6 cos² x] - 6 cos² + 3

        = 32 cos⁶ x - 4 - 48 cos² x + 24 cos² x - 6 cos² + 3

        = 32 cos⁶ x - 48 cos² x + 18 cos² x - 1

LHS = RHS 

 

 

Exercise 3.4

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Find the solution for each of the following equations

5. cos4x  = cos 2x                     

6. cos3x  + cos x  - cos  2x = 0

7. sin2x  + cos x  = 0

8. sec² 2x = 1 – tan 2x

9. sin x  + sin 3x  + sin 5x   = 0

Miscellaneous Exercise on Chapter - 3

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Prove that;