Bank Math Master

Question: If y = sinx then for which value of x, y has the highest value?

Solution: 
y = sinx
y এর মান সর্বোচ্চ হবে যদি sinx এর মান সর্বোচ্চ হয়।
sinx এর সর্বোচ্চ মান 1
তাহলে, 
x = 90°

Question: What is the value of sinA if cosA = √5/3?

Solution:
cosA = ভূমি/অতিভুজ = √5/3

লম্ব = √{(অতিভুজ)২ - (ভূমি)২}
= √{(3)2 - (√5)2}
= 2

sinA = 2/3

Question: What is the value of cos150°?

Solution: 
cos150°
= cos(90° + 60°)
= - sin60°
= - √3/2

Question: If θ = 45°, what is the value of sec²θ - tan²θ?

Solution:
θ = 45°

given,
sec²θ - tan²θ
= (√2)² - (1)²
= 2 - 1
= 1

Question: cosec(90° - θ) = 3/2. What is the value of tanθ?

Solution: 
cosec(90° - θ) = 3/2
or, secθ = 3/2
or, sec²θ = 9/4
or, 1 + tan²θ = 9/4
or, tan²θ = 9/4 - 1
or, tan²θ = 5/4
∴ tanθ = √5/2

Question: If 3sec²θ - 2 = 2, find the value of θ.

Solution: 
3sec²θ - 2 = 2
3sec²θ = 4
sec²θ = 4/3
secθ = 2/√3
secθ = sec30°
∴ θ = 30°

Question: cosθ√(sec²θ - 1) = ?

Solution: 
cosθ√(sec²θ - 1)
= cosθ√tan²θ
= cosθ (sinθ/cosθ)
= sinθ

Question: If A = 30°, find the value of sin2A/cosA?

Solution: 
sin2A/cosA
= (2sinAcosA)/cosA
= 2sinA
= 2sin30°
= 1

Question: If A = π/2 and B = π/4, what is the value of sin(A + B)?

Solution:
A = π/2
B = π/4

sin(A + B) = sin(π/2 + π/4)
= sin(90° + 45°)
= cos 45°
= 1/√2

Question: If secA - tanA = 3/5, find the value of secA + tanA?

Solution:
We know that,
sec²A - tan²A = 1
or, (secA + tanA)(secA - tanA) = 1
or, (secA + tanA) = 1/(secA - tanA)
∴ secA + tanA = 5/3

Question: If cot3A = 1, what is the value of 1/cos2A?

Solution: 
cot3A = 1
cot3A = cot45°
3A = 45°
A = 15°

1/cos2A = sec2A
= sec30°
= 2/√3

Question: What is the maximum value of 3sec²A - 4tan²A?

Solution: 
3sec²A - 4tan²A
= 3sec²A - 3tan²A - tan²A
= 3(sec²A - tan²A) - tan²A
= 3 - tan²A
= 3 - 0 [the maximum value of the equation depends on the minimum value of tan²A]
= 3

Question: Find the value of A if tan(A + 28°) = √3.

Solution:
tan(A + 28°) = √3
or, tan(A + 28°) = tan60°
or, A + 28° = 60°
or, A = 60° - 28°
∴ A = 32°

Question: sin(17π/2 + θ) = ?

Solution: 
sin(17π/2 + θ)
= sin{17 × (π/2) + θ}
= sin(π/2 + θ)
= cosθ

Question: tan15° + tan75° + tan105° + tan165° = ?

Solution:
tan15° + tan75° + tan105° + tan165°
= tan15° + tan(90° - 15°) + tan(90° + 15°) + tan(2 × 90° - 15°)
= tan15° + cot15° - cot15° - tan15°
= 0

Question: If cosecθ + cotθ = 3, Find the value of cotθ.

Solution:
given,
cosecθ + cotθ = 3.......(i)

cosec²θ - cot²θ = 1
or, (cosecθ + cotθ)(cosecθ - cotθ) = 1
or, cosecθ - cotθ = 1/3.......(ii)

subtracting (i) from (ii) we get,
2cotθ = 8/3
cotθ = 4/3

Question: If tanθ = a/b, find the value of asinθ + bcosθ.

Solution: 
tanθ = লম্ব/ভূমি = a/b

অতিভুজ = √(a² + b²)

asinθ + bcosθ
= a{a/√(a² + b²)} + b{b/√(a² + b²)}
= a²/√(a² + b²) + b²/√(a² + b²)
= (a² + b²)/√(a² + b²)
= √(a² + b²)

Question: If A = 45°, What is the value of 3 - sec²A?

Solution: 
A = 30°

3 - sec²A
= 3 - (sec45°)²
= 3 - 2
= 1

Question: Find the value of sin⁴θ - cos⁴θ if sin²θ - cos²θ = 2.

Solution: 
given,
sin²θ - cos²θ = 2

sin⁴θ - cos⁴θ
= (sin²θ)² - (cos²θ)²
= (sin²θ + cos²θ)(sin²θ - cos²θ)
= 1 × 2
= 2