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Trigonometry is all about the study of triangles. Pythagoras was a Greek philosopher who lived before 2500 years ago. He has contributed to many mathematical discoveries, one of which is known as the Pythagoras theorem. It is the building block of modern trigonometry. This theorem applies only to right-angled triangles (triangles having one angle 90 degrees), it says that the square of the hypotenuse of the triangle is equal to the sum of squares of the other two sides of the triangle.
Ques: What is the value of tan 60°/cot 30°
- 3
- 1
- 2
- 0
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Ans. b) 1
Explanation: The value of tan 60° = √3 and cot 30° = √3
Dividing , tan 60°/ cot 30° = √3/√3
= 1 (Ans)
Ques: (Sin 30° + cos 60°) - (sin 60° + cos 30°) equals
- 1 + √3
- 0
- 1 - √3
- 1 + 2√3
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Ans. c) 1 - √3
Explanation: The value of sin 30° = ½ , sin 60° =√3/2, cos 30° = √3/2, cos 60° = ½
Putting these values in the equation,
( ½ + ½ ) - ( √3/2 + √3/2 )
= 1 - (2 * √3/2)
1 - √3 (Ans)
Ques: What is the value of 1 - cos2x ?
- Sin2x
- Tan2x
- 1 - sin2x
- sec2x
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Ans. a) Sin2x
Explanation: From trignometric identities,
We know that cos2x + sin2x = 1
We get, 1 - cos2x = sin2x
Ques: If cosX = 2/3 . Find the value of tanX
- √5 /2
- 5/2
- 5 /√2
- √(5/2)
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Ans a) √5/2
Explanation: From trignometric identities we know,
1 + tan2x = sec2x
We also know that, cosx = 1 /secx
We get, 1 + tan2x = 1/cos2x
=> tan2x = (1/cos2x) - 1
=> tanx = √( (1/cos2x) -1 )
=> tanx = √ ( 9/4 - 1)
=> tanx = √(5/4)
=> tanx = √5 /4
Ques: If cosx = a/b , find the value of sinx
- √(b-a)/a
- (b2-a2)/b
- √(b2-a2)/b
- (b-a)/a
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Ans. c) √(b2-a2)/b
Explanation: given cosx = a/b
From trigonometric identities we know that,
sin2x + cos2x = 1
sin2x = 1 - cos2x
sinx = √(1 - cos2x)
sinx = √(1 - a2/b2)
sinx = √(b2 - a2)/b
Ques: 2 tan 30°/(1 + tan230°) is equal to
- sin 30°
- sin 60°
- cos 30°
- cos 60°
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Ans b) sin 60°
Explanation: We know the value, tan 30° = 1/√3
Putting this in the equation we get,
2 * 1/√3 / ( 1 + (1/√3) 2 )
= 2/√3 / (1 + 1/3)
= 2/√3 /((3+1)/3)
= 2/√3 / (4/3)
= 2/√3 * ¾
= √3 / 2
= sin 60°
Ques: Find the value of (sin 45° + cos 45°)
- 1/√2
- ½
- √3/2
- √2
Click here for the answer
Ans d) √2
Explanation: we know the value of sin45° and cos45°,
sin 45° + cos 45°
= 1/√2 + 1/√2
= ( 1 + 1 )/√2
= 2 / √2
= √2
Ques: If the value of sinA = ½ . Find the value of cotA
- 1/√3
- √3
- 1
- √3/2
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Ans b) √3
Explanation: Given sin A = ½
By trigonometric identities,
cos2A + sin2A = 1
cos2A = 1 - sin2A
cosA = √(1 - sin2A)
cosA = √(1 - (1/2)2)
cosA = √(1 - (¼))
cosA = √((4-1) / 4)
cosA = √(3/4)
cosA = √3/2
Now , cotA = cosA / sinA
cotA = (√3/2) / (½)
cotA = √3
Ques: if a triangle ABC is right-angled at C. What will be the value of cos(A+B)
- 1
- 0
- √3/2
- 1/2
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Ans b) 0
Explanation: Given to us is a right-angled triangle having angle C = 90°
We know that in a triangle, all the three angles sum upto 180°.
∠A + ∠B + ∠C = 180°
∠A + ∠B = 180° - ∠C
∠A + ∠B = 180° - 90°
We get ∠A + ∠B = 90°
Now taking cos both sides
cos(A+B) = cos 90°
cos(A+B) = 0
Ques: Find the value of (tan1° tan2° tan3°… tan89°) is
- ½
- 1
- 0
- 2
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Ans b) 1
Explanation:
tan1° tan2° tan3°… tan89°
= (tan1° tan2° tan3°..tan44°). (tan45). (tan46° tan 47°… tan89°)
= (tan1° tan2° tan3°..tan44°). (tan45).(tan(90° - 44°) tan(90°- 43°).....tan(90°-1°))
= [(tan1° * cot1°).(tan2° * cot2°)......(tan44° * cot(44°)] . 1
=1 * 1* 1 * 1 *1 * 1* 1 * 1* 1 ……* 1 (because cotx = 1/tanx )
=1
Ques: Find the value of expression [cosec(75° + θ) - sec(15° - θ) - tan(55° + θ) + cot(35° - θ)]
- 0
- 1
- -1
- 3/2
Click here for the answer
Ans a) 0
Explanation: [cosec(75° + θ) - sec(15° - θ) - tan(55° + θ) + cot(35° - θ)]
= [cosec(90° - (15° - θ)) - sec( 15° - θ) - tan(55° - θ) + cot(90° - (55 °+ θ))]
We know that cosec (90° - x ) = secx and cot(90° - x) = tanx
= [sec(15° - θ) - sec (15° - θ) - tan(55° + θ) + tan(55° + θ)]
= 0
Ques: If cos(α + β) = 0, then sin(α – β) can be reduced to
- Cos β
- Cos 2β
- Sin α
- Sin 2α
Click here for the answer
Ans b) cos 2α
Explanation: Given that cos(α + β) = 0
We get cos(α + β) = cos90 (cos 90 = 0)
α + β = 90
α = 90 - β
Now , sin(α - β) = sin(90 - β - β) (α = 90 - β)
= sin(90 - 2β)
= cos 2β
Ques: if sinA + sin2A = 1, then the value of expression (cos2A + cos4A)
- 1
- 2
- 3
- ½
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Ans. a) 1
Explanation: Given,
sinA + sin2A = 1
sinA = 1 - sin2A
sinA = cos2A (sin2A + cos2A = 1)
Squaring both sides
sin2A = cos4A
1 - cos2A = cos4A ( sin2A + cos2A = 1)
1 = cos2A + cos4A
cos2A + cos4A = 1
Ques: If cos 9α = sinα and 9α < 90°. Find the value of tan 5α
- 0
- √3
- 1/√3
- 1
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Ans d) 1
Explanation: Given,
cos 9α = sinα and 9α < 90°
It implies that 9α is acute angle
Cos 9α = cos (90° - α) (cos(90 - α) = sinα)
We get
9α = 90° - α
10α = 90°
α = 9°
Tan 5α = tan 5 * 9 = tan45° = 1
Ques: Find the value of the expression (sin6θ + cos6θ + 3 sin2θ cos2θ )
- 0
- 1
- 2
- 3
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Ans b) 1
Explanation: we know that sin2x + cos2x = 1
Cubing this equation,
(Sin2x + cos2x)3 = 1
(sin2x )3 + (cos2x)3 + 3sin2x cos2x (sin2x + cos2x ) = 1
Sin6x + cos6x + 3sin2x cos2x = 1
Ques: sin2A = 2sinA only when the value of A is
- 0
- 45
- 60
- 30
Click here for the answer
Ans a) 0
Explanation: given
sin2A = 2 sinA
When A = 0
sin2A = sin 0 = 0
And, 2sinA = 2* sin0 = 0
LHS = RHS
Ques: In a triangle ABC, right angled at B, AB = 24cm, BC = 7cm. Find the value of tanC
- 12/7
- 24/7
- 7/24
- 20/7
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Ans b) 24/7
Explanation: given AB = 24cm and BC = 7cm
The given triangle is right-angled at B
We get tanC = opposite side / adjacent side
tanC = 24 /7
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