22 Publicación CCI No. 560, Incoterms 2000
3.4.12.2 DESCRIPCIÓN GENERAL
Introduction:
What is Trigonometry?
The word trigonometry is derived from the Greek words ‘tri’ means 3, ‘gon’ means sides and metron means measure. Trigonometry is the study of relationships between the sides and angles of a triangle.
Use of learning trigonometry:
Trigonometry is used in astronomy, surveying geography, physics and navigation. The captains of ships use trigonometry to calculate the distances from far off islands, sea shores, cliffs and other ships in the ocean.
For learning trigonometry we should know the trigonometrical ratios.
Trigonometric Ratios:
¾ Let us take a right triangle ABC as shown in Fig.
¾ Here, ∠CAB (or, in brief, angle A) is an acute angle.
¾ Note the position of the side BC with respect to angle A. It faces ∠A.
¾ We call it the side opposite to angle A. AC is the hypotenuse of the right triangle and the side AB is a part of ∠A.
¾ So, we call it the side adjacent to angle A.
Introduction
Basic Identities
Basic Ratios Standard Angles Range of θ
Table Method Triangles Method
Prove Identities Verify if Identity Solve for θ Complementary Angles
Direct
Applications Prove Identities Find Value
Side adjacent to ∠A
Side opposite to ∠A Hypotenuse
A B C
Chapter 08: Introduction of Trigonometry 95
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 95 Note: If we consider ∠C then side AB faces ∠C. Hence AB is the side
opposite to ∠C and BC is the side adjacent to ∠C. Remember the hypotenuse remains unchanged.
¾ You have studied the concept of ‘ratio’ in your earlier classes.
¾ We now define certain ratios involving the sides of a right triangle and call them trigonometric ratios.
¾ The trigonometric ratios of the angle A in right triangle ABC are defined as follows:
sine of ∠A = respectively. Note that the ratios cosec A, sec A and cot A are respectively, the reciprocals of the ratios sin A, cos A and tan A.
¾ So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.
Remark : Note that the symbol sin A is used as an abbreviation for ‘the sine of the angle A’. sin A is not the product of ‘sin’ and A. ‘sin’ separated from A has no meaning.
Similarly, cos A is not the product of ‘cos’ and A. Similar interpretations follow for other trigonometric ratios also.
¾ Now, if we take a point P on the hypotenuse AC or a point Q on AC extended, of the right triangle ABC and draw PM perpendicular to AB and QN perpendicular to
AB extended
¾ By AA similarity criterion. you will see that the triangles PAM and CAB are similar.
¾ Therefore, by the property of similar triangles, the corresponding sides of the triangles are proportional.
¾ So, we have
96 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
¾ This shows that the trigonometric ratios of angle A in ΔPAM do not differ from those of angle A in ΔCAB.
¾ In the same way, we conclude that the value of sin A (and also of other trigonometric ratios) remains the same in ΔQAN also.
¾ From our observations, it is now clear that the values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.
Note: Since the hypotenuse is the longest side in a right triangle, the value of sin A or cos A is always less than 1 (or, in particular, equal to 1).
SOLVED EXAMPLES 8.1:
1) Given tan A = 3
4, find the other trigonometric ratios of the angle A.
Sol: Let us first draw a right ΔABC
Now, by using the Pythagoras Theorem, We have AC2 = AB2 + BC2 = (4k)2 + (3k)2 = 25k2 So, AC = 5k
Now, we can write all the trigonometric ratios using their definitions.
sin A =
Chapter 08: Introduction of Trigonometry 97
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 97 3) If sec θ =
UNSOLVED EXERCISE 8.1:
CW Exercise:
1) Complete the table given below with other trigonometric ratios of Angle θ from the given trigonometric ratio:
sin θ cos θ tan θ cot θ sec θ cosec θ
98 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
13, calculate all other trigonometric ratios.
4) If cos A = 1 / 3, determine the value of sin A sec A + tan A cosec A.
5) The adjacent figure shows a right angle triangle ∆PQR where PQ = 6 cm and PR = 10 cm. If ∠PRQ = θ, determine the value of
14) State whether the following are true or false. Justify your answer.
i) The value of tan A is always less than 1.
ii) sec A = 5
12 for some value of angle A.
iii) cos A is the abbreviation used for the cosecant of angle A.
iv) cot A is the product of cot and A.
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Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 99
Trigonometric Ratio of some specific angles:
Trigonometric Ratio of 0° and 90°:
z Let ∠ XAY = θ be an acute angle and let P be a point on its terminal side AY.
z Draw perpendicular PM from P on AX.
z In Δ AMP, we have AP
= PM θ
sin AP
= AM θ
cos and
AM
= PM θ tan
z It is evident from Δ AMP that as θ becomes smaller and smaller, line segment PM also becomes smaller and smaller; and finally when θ become 0°; the point P will coincide with M.
z Consequently, we have PM = 0 and AP = AM
∴ sin 0° = 0 0,cos0 1
=
=
=
°
=
= AP
AP AP AM AP
AP
PM and, tan0°= = 0 =0
AP AP PM
z Thus, we have sin 0° = 0, cos 0° = 1 and tan 0° = 0
z From Δ AMP, it is evident that as θ increase, line segment AM becomes smaller and smaller and finally when θ becomes 90° the point M will coincide with A.
z Consequently, we have AM = 0, AP = PM
∴ sin 90° = = =1 PM PM AP
PM and cos 90° = = 0 =0 AP AP AM
z Thus, we have sin 90° = 1 and cos 90° = 0
Remark: It is evident from the above discussion that tan 90° =
0 PM
PM =AM is not defined.
Similarly, sec 90°, cosec 0°, cot 0° are not defined.
Trigonometric Rations of 30° and 60°:
z Consider an equilateral triangle ABC with each side of length 2a. Since each angle of an equilateral triangle is of 60°.
z Therefore, each angle of Δ ABC is of 60°. Let AD be perpendicular from A on BC.
z Since the triangle is equilateral.
z Therefore, AD is the bisector of ∠A and D is the mid–point of BC.
∴ BD = DC = a and ∠BAD = 30°
z Thus, in Δ ABD, ∠D is a right angle, hypotenuse AB = 2a and BD = a.
z So by Pythagoras theorem, we have AB2 = AD2 + BD2
⇒ (2a)2 = AD2 + a2 ⇒ AD2 = 4a2 – a2 ⇒ AD = 3 a
Trigonometric ratios of 30°:
z In right triangle ADB, we have
Base = AD = 3 , Perpendicular = BD = a, a Hypotenuse = AB = 2a and ∠DAB = 30°
A θ M
P Y
X
B D a
30° 30°
60 60
A
C
2a 2a
a
30°
60° B D
A
C
100 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
Trigonometric ratios of 60°:
z In right triangle ADB, we have
Trigonometric Ratios of 45 °:
z Consider a right triangle ABC with right angle at B such that ∠A = 45°.
Chapter 08: Introduction of Trigonometry 101
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 101 z Following table gives the value of various trigonometric rations of 0°, 30°, 45°, 60° and 90°
for ready reference.
T. ratios
↓ 0° 30° 45° 60° 90°
sin θ 0
2 1
2 1
2
3 1
cos θ 1
2 3
2 1
2
1 0
tan θ 0
3
1 1 3 Not Defined
cosec θ Not Defined 2 2
3
2 1
sec θ 1
3
2 2 2 Not Defined
cot θ Not Defined 3 1
3
1 0
¾ Certain angles like 0°, 30°, 45°, 60° and 90° are considered standard angles and ratios of these are expected to be memorised.
Short–cut way to remember the table:
No Ratio / Operation 0° 30° 45° 60° 90°
1 Write numbers from 0 to 4 in each column 0 1 2 3 4
2 Divide each number by 4 0
4 1
2 1
4
3 1
3 Take Square Root for each number (these are the values for sin θ) 0
2 1
2 1
2
3 1
4 sin θ (copy the values from row 3) 0
2 1
2 1
2
3 1
5 cos θ (invert values from row 4) 1
2 3
2 1
2
1 0
6 tan θ (row 3 value ÷ row 4 value) 0
3
1 1 3 defined Not
7 cot θ (invert values from row 6) definedNot 3 1
3
1 0
8 cosec θ (1 ÷ row 4 values) definedNot 2 2
3
2 1
9 sec θ (invert values from row 8) 1
3
2 2 2 definedNot
θ 0° 30° 45° 60° 90°
sin θ 4 0
4 1
4 2
4 3
4 4
102 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
Draw two triangles as shown below and with these values, all trigonometric ratios for all standard angles can be easily computed.
30° – 60° – 90° Triangle 45° – 45° – 90° Triangle
SOLVED EXAMPLES 8.2:
1) Find the value of the following expressions.
i) cos 60° cos 45° – sin 60° sin 45°
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Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 103
= 2
UNSOLVED EXERCISE 8.2:
CW Exercise:
1) Find the value of the following expressions.
i) sin 45° cos 30° – cos 45° sin 30° ii)
6) Determine the value of required trigonometric ratio from the relation given below i) sin 15° from sin(A – B) = sin A cos B – cos A sin B
ii) cos 75° from cos(A + B) = cos A cos B – sin A sin B
7) If θ is an acute angle and tan θ + cot θ = 2, find the value of tan7θ + cot7θ.
8) For x > y, find the acute angles x and y if sin (x + 2y) = √3/2 and cos (x + 4y) = 0.
9) In Δ ABC, right-angled at B, AB = 5 cm and ∠ ACB = 30° (Fig.).
Determine the lengths of the sides BC and AC.
10) In Δ PQR, right-angled at Q (Fig.), PQ = 3 cm and PR = 6 cm.
104 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
1) Find the value of the following expressions.
i) cos 45° cos 30° + sin 45° sin 30° ii) cos 45° cos 60° – sin 45° sin 60°
3) Verify the following.
i) For A = 30°, verify that: cos 2A = 2cos2A – 1 = 1 – 2sin2A
4) Find the value of unknown k in the following expressions.
i) cos2 45° + tan2 60° = 3(sin2 45° - tan2 30°) + k
9) Choose the correct option and justify your choice:
i)
13) An equilateral triangle is inscribed in a circle of radius 6 cm. Find its side.
Chapter 08: Introduction of Trigonometry 105
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 105 14) In an acute angled triangle ABC, if tan (A + B – C) = 1 and sec (B + C – A) = 2, find the value of
A, B and C.
15) Given that tan(θ + φ) =
φ θ
−
φ + θ
tan tan 1
tan
tan where θ and φ are acute angles.
Calculate θ + φ when tan θ = ½ and tan φ = 1/3.
Complementary Angles:
Definition:
z If the sum of two angles is equal to 90°, the angles are said to be complementary angles.
z In a right triangle, the two acute angles are complementary angles.
Theorem:
z If θ is an acute angle, then prove that
sin (90° – θ) = cos θ, cos(90° – θ) = sin θ, tan (90° – θ) = cot θ, cot(90° – θ) = tan θ, sec (90° – θ) = cosec θ and cosec (90° – θ) = sec θ
Proof:
z Consider a right triangle OPM, right angled at M as shown in fig.
Let MOP = θ, then ∠OPM = (90° – θ) For the reference angle θ, we have sin θ =
OP
PM, cos θ = OP
OM, tan θ = OM
PM , cosec θ = PM
OP, cot θ = PM
OM and sec θ = OM
OP –– (i) For the reference angle (90° – θ), we have
Base = PM, Perpendicular = OM and Hypotenuse = OP
∴ sin(90° – θ) = OP
OM , cos(90° – θ) = OP PM,
tan (90° – θ) = PM
OM , cosec (90° – θ) = OM OP ,
sec(90° – θ) = PM
OP and cot(90° – θ) = OM
PM ––– (ii) From (i) and (ii), we obtain,
sin (90° – θ) = cos θ, cos(90° – θ) = sin θ, tan (90° – θ) = cot θ, cot(90° – θ) = tan θ, sec(90° – θ) = cosec θ and cosec (90° – θ) = sec θ
Trigonometric Inter-relationships:
The complementary angles have a special property that inter - relates their trigonometry ratios as follows:
z sin θ = cos (90° – θ)
z cos θ = sin (90° – θ)
z tan θ = cot (90° – θ)
P
M x
x′ O
y′
y
Q
106 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume z cot θ = tan (90° – θ)
z cosec θ = sec (90° – θ)
z sec θ = cosec (90° – θ)
SOLVED EXAMPLES 8.3:
1) Without using trigonometric tables evaluate,
°
2) Prove the following identities without using the tables.
i) sin 43° cos 47° + cos 43° sin 47° = 1
UNSOLVED EXERCISE 8.3:
CW Exercise:
1) Express the following in terms of trigonometric ratios of angles between 0° and 45°
i) tan 68° + sec 69° ii) cot 85° + cos 75°
2) Evaluate the following:
i) 2 +
Chapter 08: Introduction of Trigonometry 107
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 107 3) Evaluate the following: ⎟⎟⎠
2 – cos 0° + tan 15°.tan25°.tan60°.tan65°.tan75°.
5) What must be subtracted from the expression given below such that its value becomes zero tan 7° × tan 23° × tan 60° × tan 67° × tan 83° +
6) Evaluate the following:
i) cosec (65° + θ) – sec (25° – θ) – tan (55° – θ) + cot (35° + θ)
7) Prove the following identities without using the tables.
i) sec 70°sin 20°– cos 20°cosec 70° = 0 ii) sin(60° + θ) = cos (30° – θ)
1) Without using trigonometric tables evaluate the following.
i)
2) Prove the following identities.
i) sin(90° – θ) cos θ + cos(90° – θ) sin θ = 1
108 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
8) Find the value of x satisfying the equations given below:
i) x = sin 1° cos 1° sin 2° cos 2° sin 3° cos 3° ………… sin 180° cos 180°
An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved. Following are the identifies:
sin2 A + cos2 A = 1
Chapter 08: Introduction of Trigonometry 109
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 109 tan A =
Identical expressions of:
z sin2 A + cos2 A = 1 are 1 – sin2 A = cos2 A and 1 –cos2 A = sin2 A
z 1 + tan2 A = sec2 A are sec2 A – tan2 A = 1 and sec2 A – 1 = tan2 A
z 1 + cot2 A = cosec2 A are cosec2 A – cot2 A = 1 and cosec2 A – 1 = cot2 A
Using these identities, we can express each trigonometric ratio in terms of other trigonometric ratios. i.e., if any one of the ratios is known, we can also determine the values of other trigonometric ratios.
SOLVED EXAMPLES 8.4:
1) Prove the following identities,
i) + θ
1 [Multiplying numerator and denominator by (1+cos θ)]
= sin (1 cos )
110 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
[Multiplying numerator and denominator by (sec θ – tan θ)]
= θ− θ
2) Determine whether the following equations are identities.
i) + θ
Sol: The variable θ can take any value in the range 0° < θ ≤ 90° as cot θ is not defined for θ = 0°
Chapter 08: Introduction of Trigonometry 111
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 111
⇒ cos θ = 0 or cos θ = ½ θ = 90° or 60°
UNSOLVED EXERCISE 8.4:
CW Exercise:
1) Write all the other trigonometric ratios of ∠A in terms of sec A 2) Prove the following identities
i) (1 + tan2θ) (1 – sin θ) (1 + sin θ) = 1 ii) sec2θ−1cosec2θ = tanθ + cotθ 7) Prove the following identities.
i) (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ
112 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
1) If the ratio of the cosine of an angle to its sine is 8:15 find all the trigonometric ratios of that angle 2) Find the value of the following expressions.
i) 4(sin4 30° + cos4 60°) – 3(cos2 45° – sin2 90°)
6) Prove the following identities:
Chapter 08: Introduction of Trigonometry 113
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 113 i) (1 + sin θ) (1 – sin θ) =
8) Solve the following equations
114 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume 9) Without using trigonometric tables evaluate the following.
i) 12) Prove the following identities, where the angles involved are acute angles for which the
expressions are defined.
i)
Chapter 08: Introduction of Trigonometry 115
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 115
iv)
b) Prove the following identities
i) (sin2 63° + sin2 27°) + (cos2 73° – sin2 17°) = 1
14) Determine whether the following equations are identities.
i)
15) Solve the following equation
i) sin θ – cos θ = 0 ii) 2sin2 θ =
17) Find the value of the following expressions.
i) If 7 sin2 θ+3 cos2 θ =4, find the value of tanθ ii) If cot θ =
Prove that each side equal to 1.
19) Simplify:
21) Without using trigonometric tables, evaluate:
°
116 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
26) Without using the trigonometric tables, evaluate the following:
7
28) Without using trigonometric tables, evaluate the following:
°
29) Without using trigonometric tables, evaluate the following:
(cos2 25° + cos2 65°) + cosec θ sec(90° – θ) – cot θ. tan (90° – θ) [CBSE 08]
30) If 7 sin2θ + 3 cos2θ = 4, show that tan θ = 3
1 [CBSE 08]
31) Prove that: (1 + cot A + tan A) (sin A – cos A) = sin A tan A – cot A cos A. [CBSE 08]
32) Without using trigonometric tables, evaluate the following:
2 ⎟
MULTIPLE CHOICE QUESTIONS:
CW Exercise:
Chapter 08: Introduction of Trigonometry 117
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 117
a)
118 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume
Chapter 08: Introduction of Trigonometry 119
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 119
COLUMN MATCHING QUESTIONS:
1) Given in column I are some right angled triangles. For each item in Column I, choose correct option(s) in column II.
Column I Column II
i)
sin C = A)
AC AB
ii)
cos A = B)
AC BC
iii)
tan B = C)
AB CA
iv)
cot C = D)
AB BC
E) BC AB
2) From column II choose correct option(s) for each item in column I.
Column I Column II
i) 2sin 2θ = 3 then θ = A) 12°
ii) 2cos 3θ = 3 then θ = B) 30° iii) 3 tan 5θ – 3 = 0 then θ = C) 45°
D) 20°
3) It is given that θ < (A + B) ≤ 90° and A > B. For the value of A and B, choose correct option from column II for each item in column I.
Column I Column II
i) sin (A – B) = 2
1, cos (A + B) = 2
1 A) A = 45°, B = 30°
ii) tan (A + B) = 3, tan (A – B) = 3
1 B) A = 60°, B = 30°
iii) sin (A + B) = 1, cos (A – B) = 2
3 C) A = 45°, B = 15°
D) A = 30°, B = 15°
A B
C
C B
A
C A
B
A C
B
120 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume 4) Choose correct option in column II for each item in column I.
Column I Column II
i) sin x = cos 60° cos 30° – sin 60° sin 30°, then x = A) 45° ii) cot 3x = sin 45° cos 45° + sin 30°, then x = B) 60° iii) cos 2x = sin 60° cos 30° – cos 60° sin 30°, then x = C) 0°
iv) tan 2x =
°
−
°
°
°
60 cos 30 cos
30 cos 30 sin 2
2
2 , then x = D) 15°
E) 30° 5) For each item in column I choose correct option in column II.
Column I Column II
i) sin 3A = cos (A – 26°) where 3A is acute angle, then A = A) 22° ii) tan 2A = cot (A – 18°) where 2A is acute angle, then A = 0 B) 29° iii) sec 4A = cosec (A – 20°) where 4A is acute angle, then A = 0 C) 27°
iv) cot 4A = tan (A – 20°) where 4A is acute angle, then A = 0 D) 36°
6) For each item in column I choose correct option in column II.
Column I Column II
i) If sin A + sin B + sin C = 3, then A) m∠A = 90°
ii) In ΔABC, ∠B = 90°, then B) sin B > sin C iii) In ΔABC, ∠B > ∠C > ∠A C) sin2 A + sin2 C = 1
D) ∠B = ∠C
E) cosec B + cosec C > 2
7) For each item in column I choose correct option in column II.
Column I Column II
i) cos 3° × cos 6° × cos 9° × … × cos 90° = A) 45° ii) If sec2θ = cosec2θ, θ = B) 0
iii) cot 3° × cot 6° × … × cot 90° = C) 180° – 3θ
D) sin2 20° – cosec2 20° + sin2 70° + tan2 70 E) 90°
8) For each item in column I choose correct option in column II.
Column I Column II
i) Maximum value of sin A + cos A is A) 2 ii) Minimum value of cosec A + sec B is B) 2 iii) In a right triangle, sin2 A + sin2 B + sin2 C = C) 0
D) 3
E) 1
ANSWERS TO UNSOLVED EXERCISES:
CW Exercise 8.1:
1) a) 5 3,
4 3,
3 4,
4 5,
3 5 (b)
17 15,
17 8 ,
8 15,
8 17,
15 17 (c)
10 1 ,
10 3 ,
3 1, 3,
3 10
2) i) 1 (ii) 225
544 3) i) 12/13, 5/13 (ii) 5/13, 12/13 (iii) 12/5, 5/12
4) 5/12 5) 1 7) i) 5 (ii) –1/17 8) 12/13, 5/13, 12/5
Chapter 08: Introduction of Trigonometry 121
Volume Universal Tutorials – X CBSE (2012–13) – Mathematics 121 10) i) HW Exercise 8.1:
1) 7 CW Exercise 8.2:
1) i)
HW Exercise 8.2:
1) (i)
CW Exercise 8.3:
1) i) cot 22° + cosec 21° (ii) tan 5° + sin 15°
HW Exercise 8.3:
1) (i) 2 (ii)
122 Universal Tutorials – X CBSE (2012–13) – Mathematics Volume 2) i) 1 (ii) 1 3) c 7) (i, ii) No (iii) yes
HW Exercise 8.4:
1) 1 2) 1 6) (i, ii) Yes (iii) No Miscellaneous Exercise:
1) cot θ = 15
8 , tan θ = 8
15, sin θ = 17
15, cos θ = 17
8 , cosec θ = 15
17, sec θ = 8 17
2) (i) 2 (ii) 3 2 (iii)
3
−13 (iv) 6 23 (v)
8
−37 3) (i)
6 1 (ii)
3
1 (iii) 2 2
2
b a
b
+ 4) 4
7) (i) 3 (ii) 2 2
1
3+ 8) (i) 30° (ii) 60° (iii) 45°
9) (i) –1 (ii) 0 (iii) 1 (iv) 8
3 (v) 1 (vi) 2 (vii) 2 (viii) 2 (ix) 2 (x) 0 (xi) 1 (xii) –1 (xiii) 1 (xiv) ½ (xv) 3 1
(xvi) 1 (xvii) 2 (xviii) 2 10) (ii) 3 1 (iii)
2 17 (iv)
8 77 (v)
3 16 13) (a) (i) θ = 45° or θ = 30° (ii) θ = 60° 14) (i) Yes (ii, iii) No 15) (i) 45° (ii) 30° (iii) 45° (iv) 60° 17) (i)
3 1 (ii)
a b
a b
− +
26) 1 28) 2 29) 2 32) 1 33) 44°
34) , ,
Column Matching Question:
1) i–AE, ii–BAC, iii–C, iv–DE 2) i–B, ii–D, iii–A 3) i–C, ii–C, iii–B 4) i–C, ii–D, iii–E, iv–E 5) i–B, ii–D, iii–A, iv–A 6) i–AD, ii–CE, iii–BE 7) i–BD, ii–AC, iii–BD 8) i–B, ii–A, iii–A
A cot 1
1
+ 2 cotA
A cot 1+ 2
A cot
1