Test: Trigonometric Identities - 2 - ACT MCQ

# Test: Trigonometric Identities - 2 - ACT MCQ

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## 15 Questions MCQ Test - Test: Trigonometric Identities - 2

Test: Trigonometric Identities - 2 for ACT 2024 is part of ACT preparation. The Test: Trigonometric Identities - 2 questions and answers have been prepared according to the ACT exam syllabus.The Test: Trigonometric Identities - 2 MCQs are made for ACT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Trigonometric Identities - 2 below.
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Test: Trigonometric Identities - 2 - Question 1

### If a sin 450 = b cosec 300, what is the value of a4/b4?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 1

Given a sin 450 = b cosec 300

So, a/b = cosec 300/ sin 450

a/b = 2/( 1/√2)

a/b = 2√2/1

a4/b4 = (2√2/1)4

a4/b4 = 64/1

or,

a4/b4 = 43

Test: Trigonometric Identities - 2 - Question 2

### If the value of α + β = 900, and α : β = 2 : 1, then what is the ratio of cos α to cos β ?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 2

Given α + β = 900, and α : β = 2 : 1

So, we can say that 2x + x = 900

3x = 900, which give

x = 300

So, α = 2x = 60

β = x = 30

cos α / cos β = cos 600 / cos 300

=> (1/2) / (√3/2)

or, 1/2 * 2/√3

= 1/√3

Or the ratio between cos α : cos β = 1 : √3

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Test: Trigonometric Identities - 2 - Question 3

### If tan θ - cot θ = 0, what will be the value of sin θ + cos θ?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 3

Given tan θ - cot θ = 0
Let's put θ = 450 in order to satisfy the above equation
tan 450 - cot 450 = 0
1 - 1 = 0 (equation satisfied with θ = 450)
Now, put θ = 450 in sin θ + cos θ, we will get
= sin 450 + cos 450
= 1/√2 + 1/√2
= √2

Test: Trigonometric Identities - 2 - Question 4

If the value of θ + φ = π/2, and sin θ = 1/2, what will be the value of sinφ?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 4

Given θ + φ = π/2
It can be written as, θ + φ = 900 (as π = 1800) …….(i)
sin θ = 1/2
or, θ = 300
On putting the value of θ = 300 in equation (i), we will get,
300 + φ = 900
So, φ = 600
Then, sin φ = sin 600 = √3/2

Test: Trigonometric Identities - 2 - Question 5

What will be the value of 1 - 2sin2 θ, if cos4 θ - sin4 θ = 2/3?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 5

Given cos4 θ - sin4 θ = 2/3
Now, here we can apply the formula -
a4 - b4 = (a2 - b2) (a2 + b2)
So, (cos2 θ - sin2 θ) (cos2 θ + sin2 θ) = 2/3
So, 1 x (cos2 θ - sin2 θ) = 2/3 (because cos2 θ + sin2 θ = 1)
⇒ (1 - sin2 θ) - sin2 θ = 2/3
So, 1 - 2sin2 θ = 2/3

Test: Trigonometric Identities - 2 - Question 6

What is the value of sin θ/(1 + cos θ) + sin θ/(1 - cos θ), where (00 < θ < 900)?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 6

Given, sin θ/(1 + cos θ) + sin θ/(1 - cos θ)
= [sin θ (1 - cos θ) + sin θ (1 + cos θ)] / [(1 - cos θ) (1 + cos θ)]
= [sin θ - sin θ cos θ + sin θ + sin θ cos θ] / [1 - cos2 θ]
= 2 sin θ / sin2 θ
= 2 cosec θ

Test: Trigonometric Identities - 2 - Question 7

What is the value of (tan2 θ - sec2 θ)?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 7

(tan2 θ - sec2 θ)
= sin2 θ/cos2 θ - 1/cos2 θ
= (sin2 θ - 1) / cos2 θ
= - cos2 θ/cos2 θ
= -1

Test: Trigonometric Identities - 2 - Question 8

What is the value of tan3θ, If tan7θ.tan2θ = 1?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 8

Given tan7θ.tan2θ = 1
As we know, if tanA . tanB = 1 then, A + B = 900
So, 7θ + 3θ = 900
⇒ 9θ = 900
Or, θ = 100
Now, we have to find tan3θ
So, put θ = 100 in tan3θ, we will get
tan 300 = 1/√3

Test: Trigonometric Identities - 2 - Question 9

If sin (θ + 180) = cos 600, then what is the value of cos5θ, where 00 < θ < 900?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 9

Given sin (θ + 180) = cos 600
sin (θ + 180) = cos (900 - 300)
So, sin (θ + 180) = sin300
Then, θ = 300 - 180
θ = 120
So, cos5θ = cos 5 x 120
= cos 600
= 1/2

Test: Trigonometric Identities - 2 - Question 10

What will be the simplified value of (sec A sec B + tan A tan B)2 - ( sec A tan B + tan A sec B)2?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 10

The question is in the form of (a + b)2
So, on applying the identity, and after expanding the given equation, we will get -
⇒ sec2 A sec2 B + tan2 A tan2 B + 2 sec A sec B tan A tan B - sec2 A tan2 B - tan2 A sec2 B - 2 sec A tan B tan A sec B
⇒ Then, sec2 A [sec2 B - tan2 B] - tan2 A [sec2 B - tan2 B]
So, it will be written as [sec2 A - tan2 A] [sec2 B - tan2 B]
= 1 x 1
= 1.

Test: Trigonometric Identities - 2 - Question 11

What will be the value of sec4 θ - tan4 θ, if sec2 θ + tan2 θ = 7/12?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 11

Given sec2 θ + tan2 θ = 7/12
Now, here we can apply the formula -
a4 - b4 = (a2 - b2) (a2 + b2)
sec4 θ - tan4 θ = (sec2 θ - tan2 θ) (sec2 θ + tan2 θ)
= 1 x (sec2 θ + tan2 θ) {because 1 + tan2 θ = sec2 θ}
= 1 x 7/12
= 7/12

Test: Trigonometric Identities - 2 - Question 12

If r cosθ = √3, and r sinθ = 1, what is the value of r2 tanθ?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 12

Given r cosθ = √3, and r sinθ = 1
r cosθ / r sinθ = 1/√3
tanθ = tan 300
Or, θ = 300
On putting, θ = 300, we will get,
r sin 300 = 1
r x ½ = 1
or r =2
Now, r2 tanθ = ?
= (2)2 tan 300
= 4 x 1/√3
= 4/√3

Test: Trigonometric Identities - 2 - Question 13

If the value of tan2 θ + tan4 θ = 1, what will be the value of cos2 θ + cos4 θ?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 13

Given, tan2 θ + tan4 θ = 1 …. (i)
From equation (i),
tan2 θ ( 1 + tan2 θ ) = 1
tan2 θ ( sec2 θ ) = 1 [As according to the trigonometric identity, sec2 θ - tan2 θ = 1]
tan2 θ = 1/ sec2 θ
tan2 θ = cos2 θ ….(ii)
Now, cos2 θ + cos4 θ = ?
⇒ cos2 θ + (cos2)2 θ
⇒ tan2 θ + (tan2)2 θ
⇒ tan2 θ + tan4 θ
= 1 {from equation (i)}

Test: Trigonometric Identities - 2 - Question 14

If the value of sin A + cosec A = 2, then what is the value of sin7 A + cosec7 A?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 14

It is given that sin A + cosec A = 2 ……(i)
On putting A = 900, then above condition will satisfy
sin 900 + cosec 900 = 2
or, 1 + 1 = 2 (as the equation satisfies, so, A = 900)
Now, sin7 A + cosec7 A = ?
⇒ sin7 900 + cosec7 900
⇒ 17 + 17
= 2

Test: Trigonometric Identities - 2 - Question 15

What is the value of (sin 300 + cos 600) - (sin 600 + cos 300)?

Detailed Solution for Test: Trigonometric Identities - 2 - Question 15

Let's see the values -
sin 300 = 1/2
cos 600 = 1/2
sin 600 = √3/2
cos 300 = √3/2
So, (1/2 + 1/2) - (√3/2 + √3/2)
= 1 - 2√3/2
Or, 1 - √3

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