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Test: Trigonometric Identities - 3 - SAT MCQ


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15 Questions MCQ Test Mathematics for SAT - Test: Trigonometric Identities - 3

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Test: Trigonometric Identities - 3 - Question 1

If the value of tan θ = 4/3, then which of the following is the correct value of (3 sin θ + 2 cos θ) / (3 sin θ - 2 cos θ) = ?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 1

It is given that, tan θ = 4/3
⇒ sin θ / cos θ = 4/3
So sin θ = 4, and cos θ = 3
Now, on putting the values of sin θ and cos θ in (3 sin θ + 2 cos θ) / (3 sin θ - 2 cos θ), we will get -
= 3x4 + 2x3/ 3x4 - 2x3
= 18/6
= 3

Test: Trigonometric Identities - 3 - Question 2

Which of the following is the correct relation between A and B, if A = tan 110 . tan 290, and B = 2 cot 610 . cot 790?

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

Given A = tan 110 . tan 290, and B = 2 cot 610 . cot 790
A / B = tan 110 . tan 290 / 2 cot 610 . cot 790
= [tan 110 . tan 290] / [2 cot (900 - 290) . cot (900 - 110)]
= tan 110 . tan 290 / 2 tan 110 . tan 290
= 1/2
So, A/B = 1/2
Or, 2A = B

Test: Trigonometric Identities - 3 - Question 3

If the value of sin(θ + 300) is 3/√12, then what is the value of cos2 θ?

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

Given sin (θ + 300) = 3/√12
It can be written as sin (θ + 300) = 3/2√3
Or, sin (θ + 300) = √3/2
⇒ sin (θ + 300) = sin 600
⇒ θ + 300 = 600
⇒ θ = 300
On putting θ = 300, in cos2 θ, we will get
cos2 300 = (√3/2)2
= 3/4

Test: Trigonometric Identities - 3 - Question 4

Which of the following is the correct value of cos2 550 + cos2 350 + sin2 650 + sin2 250?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 4

cos2 550 + cos2 350 + sin2 650 + sin2 250
⇒ cos2 (900 - 350) + cos2 350 + sin2 650 + sin2 (900 - 650)
⇒ (sin2 350 + cos2 350) + (sin2 650 + cos2 650)
⇒ 1 + 1
= 2

Test: Trigonometric Identities - 3 - Question 5

If cot 450.sec 600 = A tan 300.sin 600, then which of the following is the correct value of A?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 5

Given cot 450.sec 600 = A tan 300.sin 600
So, 1 x 2 = A 1/√3 x √3/2
⇒ 2 = A/2
So, A = 4

Test: Trigonometric Identities - 3 - Question 6

Which of the following is the correct value of (3 / 1+tan2 θ) + 2 sin2 θ + (1 / 1+cot2 θ)?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 6

(3 / 1+tan2 θ) + 2 sin2 θ + (1 / 1+cot2 θ) = ?
According to the trigonometric identities, the given equation can be written as -
= 3/sec2 θ + 2 sin2 θ + 1/cosec2 θ
= 3cos2 θ + 2 sin2 θ + sin2 θ
= 3cos2 θ + 3sin2 θ
= 3(cos2 θ + sin2 θ)
= 3

Test: Trigonometric Identities - 3 - Question 7

What will be the numerical value of (4 sec2 300 + cos2 600 - tan2 450) / (sin2 300 + cos2 300)?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 7

Given: (4 sec2 300 + cos2 600 - tan2 450) / (sin2 300 + cos2 300)
We have to put the numerical values,
= [4 (2/√3)2 + (½)2 - (1)2] / 1
⇒ sec2 A - 1 = 1 + 2 (sec2 B - 1)
⇒ sec2 A - 1 = 1 + 2 sec2 B - 2
⇒ sec2 A - 1 = 2 sec2 B - 1
⇒ 1/cos2 A = 2/cos2 B
⇒ cos2 B = 2cos2 A
⇒ or, cos B = √2 cos A
⇒ So, √2 cos A - cos B = 0

Test: Trigonometric Identities - 3 - Question 8

Which of the following is the correct value of (5/sec2 θ) + 3 sin2 θ + (2 / 1+cot2 θ)?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 8

(5 / sec2 θ) + 3 sin2 θ + (2 / 1+cot2 θ) = ?
According to the trigonometric identities, the given equation can be written as -
= 5cos2 θ + 3 sin2 θ + 2/cosec2 θ
= 5cos2 θ + 3 sin2 θ + 2sin2 θ
= 5cos2 θ + 5sin2 θ
= 5(cos2 θ + sin2 θ)
= 5

Test: Trigonometric Identities - 3 - Question 9

The value of cot 300/tan 600 is -

Detailed Solution for Test: Trigonometric Identities - 3 - Question 9

 tan 600 = √3, cot 300 = √3
So, cot 300/tan 600 = √3 / √3
= 1

Test: Trigonometric Identities - 3 - Question 10

Suppose cos θ + sin θ = √2 cos θ, then which of the following is the correct value of cos θ - sin θ?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 10

It is given that, cos θ + sin θ = √2 cos θ …..(i)
On squaring both sides, we will get,
(cos θ + sin θ)2 = (√2 cos θ)2
⇒ cos2 θ + sin2 θ + 2 sin θ cos θ = 2 cos2 θ
Or, 2cos2 θ - cos2 θ - sin2 θ = 2 sinθ cosθ
⇒ cos2 θ - sin2 θ = 2 sin θ cos θ
⇒ (cos θ + sin θ) (cos θ - sin θ) = 2 sin θ cos θ
⇒ (√2 cos θ) (cos θ - sin θ) = 2 sin θ cos θ [from equation (i)]
⇒ (cos θ - sin θ) = 2 sinθ cosθ / √2 cos θ
= √2 sin θ

Test: Trigonometric Identities - 3 - Question 11

Which of the following is the correct value of cot 100.cot 200.cot 600.cot 700.cot 800?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 11

Here, we can apply the formula -
cot A. cot B = 1 (when A + B = 900)
= (cot 200 . cot 700) x (cot 100 . cot 800) x cot 600
= 1 x 1 x 1/√3
= 1/√3
So, the correct value of cot 100.cot 200.cot 600.cot 700.cot 800 = 1/√3

Test: Trigonometric Identities - 3 - Question 12

If tan θ + cot θ = 2, then what is the value of tan100 θ + cot100 θ?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 12

Given tan θ + cot θ = 2
Put θ = 450, above equation will satisfy as,
1 + 1 = 2
So, θ = 450,
= tan100 450 + cot100 450
= 1100 + 1100
= 2

Test: Trigonometric Identities - 3 - Question 13

If θ is said to be an acute angle, and 7 sin2 θ + 3 cos2 θ = 4, then what is the value of tan θ?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 13

Given 7 sin2 θ + 3 cos2 θ = 4
=> 7 sin2 θ + 3 (1 - sin2 θ) = 4
=> 7 sin2 θ + 3 - 3sin2 θ = 4
Then, 4sin2 θ = 1
Or, sin θ = 1/2
So, θ = 300
Now, put θ = 300 in tan θ, we will get,
tan θ = 1/√3

Alternate
We can directly check the equation by putting values of θ. Let's put θ = 300
7 sin2 300 + 3 cos2 300= 4
Then, 7 * 1/4 + 3 * 3/4 = 4
So, 7/4 + 9/4 = 4
16/4 = 4
Or, 4 = 4 (so, it satisfy the condition)
Now, tan 300 = 1/√3

Test: Trigonometric Identities - 3 - Question 14

If θ is said to be an acute angle, and 4 cos2 θ - 1 = 0, then what is the value of tan (θ - 150)?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 14

Given 4 cos2 θ - 1 = 0
4 cos2 θ = 1
cos2 θ = 1/4
cos θ = 1/2
Or, θ = 600
So, tan (θ - 150) = ?
⇒ tan (600 - 150)
= tan 450
= 1

Test: Trigonometric Identities - 3 - Question 15

What will be the value of 2cos2 θ - 1, if cos4 θ - sin4 θ = 2/3?

Detailed Solution for Test: Trigonometric Identities - 3 - Question 15

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)
⇒ cos2 θ - (1 - cos2 θ) = 2/3 (because sin2 θ = 1 - cos2 θ)
So, 2cos2 θ - 1 = 2/3

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