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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
VMC/ Maths 2 TRIGONOMETRY
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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)
VMC/ Maths 3 TRIGONOMETRY
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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
VMC/ Maths 4 TRIGONOMETRY
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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
VMC/ Maths 5 TRIGONOMETRY
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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:
VMC/ Maths 6 TRIGONOMETRY
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29. If prove that
30. Prove that:
31. Prove that:
32. Prove that:
33. Find the value of
(a) (b) (c) (d) (e)
34. Prove that:
35. Solve:
36. Find the value of
37. If are the solutions of the equation a , then show that
38. Solve:
39. Prove:
40. Prove:
41. Find the value of
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.
,
VMC/ Maths 8 TRIGONOMETRY
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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.
VMC/ Maths 10 TRIGONOMETRY
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12. L.H.S. =
=
=
=
= =
= = R.H.S.
13.
14.
15.
VMC/ Maths 11 TRIGONOMETRY
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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.
VMC/ Maths 13 TRIGONOMETRY
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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:
VMC/ Maths 14 TRIGONOMETRY
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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,
VMC/ Maths 16 TRIGONOMETRY
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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:
36. Find the value of
VMC/ Maths 19 TRIGONOMETRY
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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.
VMC/ Maths 20 TRIGONOMETRY
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40.
41.
put A = 22º
Let
,
VMC/ Maths 21 TRIGONOMETRY
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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.
VMC/ Maths 22 TRIGONOMETRY
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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.
VMC/ Maths 23 TRIGONOMETRY
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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
VMC/ Maths 24 TRIGONOMETRY