MCQs On Introduction to trigonometry: Introduction & Explanation

Ans. b) 1

Explanation: The value of tan 60° = √3 and cot 30° = √3

Dividing , tan 60°/ cot 30° = √3/√3

= 1 (Ans)

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)

Ans. a) Sin2x

Explanation: From trignometric identities,

We know that cos2x + sin2x = 1

We get, 1 - cos2x = sin2x

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

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

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°

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

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

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

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

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

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

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

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

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

Ans a) 0

Explanation: given

sin2A = 2 sinA

When A = 0

sin2A = sin 0 = 0

And, 2sinA = 2* sin0 = 0

LHS = RHS

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