m 11th class cbse trigonometric functions important questions

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TRIGONOMETRIC FUNCTIONS

An angle is the amount of rotation of a revolving line with respect to a fixed line. If the rotation is in clockwise direction the angle is negative and it is positive if the

rotation is in the anti-clockwise direction. Two types of conventions for measuring angles, i.e., (i) Sexagesimal system (ii)

Circular system. In sexagesimal system, the unit of measurement is degree. If the rotation from the initial to terminal side is (1/360) th of a revolution, the

angle is said to have a measure of 1°. The classifications in this system are as follows:

In circular system of measurement, the unit of measurement is radian. One radian is the angle subtended, at the centre of a circle, by an arc equal in

length to the radius of the circle. The length s of an arc PQ of a circle of radius r is given by s = r, where is the

angle subtended by the arc PQ at the centre of the circle measured in terms of radians.

Relation between degree and radian

The signs of trigonometric functions in different quadrants have been given as:

Domain and range of trigonometric functions:

VMC/ Maths 1 TRIGONOMETRY

CBSE NOTES Maths

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Trigonometric ratios of some standard angles

DEGREE 0 O 30 O 45 O 60 O 90 O 120 O 135 O 150 O 180 O

Radian 0

Sin 0 1 0

Cos 1 0 – 1

Tan 0 1 ∞ – 1 0

Degree 210 o 225 o 240 o 270 o 300 o 315 o 330 o 360 o

Radian

Sin – 1 0

Cos 0 1

Tan 1 – ∞ – 1 0


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3.1.6 Allied or related angles: The angles are called allied or related angles and are called

coterminal angles. For general reduction, we have the following rules. The value of any trigonometric function for

is numerically equal to

(a) The value of the same function if n is an even integer with algebraic sign of the function as per the quadrant in which angles lie.

(b) Corresponding cofunction of if n is an odd integer with algebraic sign of the function for the quadrant in which it lies. Here sine and cosine; tan and cot; sec and cosec are cofunctions of each other.

Functions of negative angles Let be any angle. Then

Some formulae regarding compound angles

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)


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(xvi)

(xvii)

(xviii)

(xix)

(xx)

(xxi)

(xxii)

(xxiii)

(xxiv)

(xv)

Trigonometric equations:

Equations involving trigonometric functions of a variables are called trigonometric equations.

Equations are called identities, if they are satisfied by all values of the unknown angles for which the

functions are defined.

The solutions of a trigonometric equations for which 0 ≤ θ < 2 π are called principal solutions.

The expression involving integer n which gives all solutions of a trigonometric equation is called the

general solution.

General Solution of Trigonometric Equations

(i) If for some angle , then

gives general solution of the given equation

(ii) If for some angle , then

gives general solution of the given equation

(iii) If then

gives general solution for both equations


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PROBLEMS FROM NCERT BOOK

1. Convert 6 radians into degree measure.

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

3. Find the values of other five trigonometric functions is

(a) lies in third quadrant. (b) lies in fourth quadrant.

4. Find the values of the trigonometric functions

(a) (b)

5. Prove that:

6. Show that

7. Prove that:

8. Prove that

9. Prove that:

10. Prove that:

11. Prove that:

12. Prove that:

13. Solve:

14. Solve:

15. Find the general solution:

16. If where x and y both lie in second quadrant, find the value of .

17. Prove that:

18. Find the value of

19. If find the value of .

20. Prove that:

21. Find and , if in quadrant II


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TRIGONOMETRY SUPPLEMENTARY MATERIAL

Theorem 1 (sine formula):

In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC

Theorem 2 (Cosine formula):

Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C, respectively, then

or

or

or

22. In triangle ABC, prove that:

In triangle ABC, prove that

23.

24.

25.

26.

27.

SOME OTHER IMPORTANT QUESTIONS

28. Prove:


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29. If prove that

30. Prove that:

31. Prove that:

32. Prove that:


(a) (b) (c) (d) (e)

34. Prove that:

35. Solve:


37. If are the solutions of the equation a , then show that

38. Solve:

39. Prove:

40. Prove:


42. If , then find the value of

43. Prove that: .

44. Show that:

45. Prove that

46. Prove that:

47. Prove that

48. Prove that

49. Draw the graph of tan x in

50. Solve the equation:


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SOLUTIONS1.

2. Let the radii of the two circles r1 and r2. Let an arc of length/subtend an angle of 60º at the centre of the circle of radius r1, while let an arc of length/subtend an angle of 75º at the centre of the circle of radius r2.

Now, 60º = radian and 75º = radian

In a circle of radius r unit, if an arc of length/unit subtends an angle radian at the centre, then

Thus, the ratio of the radii is 5 : 4

3. (a)

, ,

,

Since x lies in the 3rd quadrant, the value of sec x will be negative.

,


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(b)

Since x lies in the 4th quadrant, the value of sin x will be negative.

,

,

4. (a)

=

=

(b)

5.

6.


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7.

8. L.H.S.=

9.

=

=

=

=

10.

11. L.H.S. =

= =

= = =

= = R.H.S.


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12. L.H.S. =

=

=

=

= =

= = R.H.S.

13.

14.

15.


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Therefore, the general solution is

16.

17.


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18.

19.

Therefore or (x lies in 3rd quadrant)

Hence

20.


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21. Here, x is in quadrant II.

i.e.,

Therefore, are all positive.

It is given that

As x is in quadrant II, cos x is negative.

22. In triangle ABC, prove that:


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Similarly (ii) and (iii) can be proved.

23.

24.

, ,


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25.

L.H.S. =

= =

= =

=

= R.H.S.

26.

L.H.S. =

=

=

=

=

= 0

27. L.H.S. =

=

Multiply and divide by 2,


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= = R.H.S.

28.

Divide numerator and denominator by cos11o,

29.

30.

31.


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32.

33. (a) Let = 18º

Divide by

(b)

(c)


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,

(d)

34. Prove that:

35. Solve:



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37. If are the solutions of the equation

Given that

Since and are the roots of the equation, so

and

Therefore,

38.

39.


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40.

41.

put A = 22º

Let

,


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42.

= = =

43. L.H.S. = =

= =

= = =

= R.H.S.

44. L.H.S. =

=

= R.H.S.

45. L.H.S. = =

= R.H.S.


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46. Prove that:

L.H.S. =

Multiply & divide by

= R.H.S.

47. Prove that

L.H.S. =

= R.H.S.


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48. Prove that

49. Graph of tan x in

50.

Divide both sides by 2, we get

, which is not wanted.

Therefore solution is


m 11th class cbse trigonometric functions important questions

Documents

trigonometric ratios

related angles

coterminal angles

unknown angles

general reduction

sexagesimal system

circular system of measurement

integer n