Bank Math Master

Question: What is the sine of 45 degrees?

Solution: 
sin45 = 1/√2
= √2/2

Question: If cosθ = 0.8, what is sinθ?

Solution: 
sinθ = √(1 - cos²θ)
= √(1 - 0.8²)
= √0.36
= 0.6

প্রশ্ন: If cosA sinA = 1,then (cosA + sinA)² =?

সমাধান:
(cosA + sinA)²
= cos²A + 2 cosA sinA + sin²A
= 1 + 2.1 [sin²A + cos²A = 1]
= 1 + 2
= 3

Question:  If tanθ = 3/4, what is value of cotθ?

Solution: 
cotθ = 1/tanθ
= 1/3/4
= 4/3

Question: If tan45° = 2x, what is the value of x²? 

Solution: 
tan45 = 2x
⇒ 1 = 2x
⇒ x = 1/2
⇒ x² = 1/4

Question: If cos(2θ) = 0.5, what is the value of cosθ?

Solution: 
cos(2θ) = 0.5
⇒ cos(2θ) = cos60
⇒ 2θ = 60
θ = 30

cos30 =  √3/2 

প্রশ্ন: If (sinθ + cosθ)/(sinθ - cosθ) = 7 then, secθ =?

সমাধান:
দেয়া আছে,    
(Sinθ + Cosθ)/(Sinθ - Cosθ) = 7 
⇒ (Sinθ + Cosθ) + (Sinθ - Cosθ)/(Sinθ + Cosθ) - (Sinθ - Cosθ) = (7 + 1)/(7 - 1)
⇒ (Sinθ + Cosθ + Sinθ - Cosθ)/(Sinθ + Cosθ - Sinθ + Cosθ) = 8/6
⇒ 2Sinθ/2Cosθ = 4/3
⇒ Sinθ/Cosθ = 4/3
⇒ tanθ = 4/3
⇒ tan²θ = 16/9
⇒ Sec²θ - 1 = 16/9
⇒ Sec²θ = (16/9) + 1
⇒ Sec²θ  = (16 + 9)/9
⇒ Sec²θ  = 25/9
⇒ Sec²θ  = (16 + 9)/9
⇒ Sec²θ  = 25/9
     Secθ = ±5/3

প্রশ্ন: tanA + cotA =? 

সমাধান: 
 tanA + cotA 
= (sinA/cosA) + (cosA/sinA)
= (sin²A + cos²A)/cosA.sinA
= 1/cosA.sinA
= (1/cosA)(1/sinA)
= secA.cosecA

Question: If θ + ϕ = π/2, and sinθ = 1/2, then find out the value of sinϕ-

Solution:
দেওয়া আছে,
θ+ϕ = π/2
⇒ θ = π/2 - ϕ
⇒ sinθ = sin(π/2 - ϕ)
⇒ 1/2 = cosϕ
⇒ cos²ϕ = 1/4
⇒ 1 - sin²ϕ = 1/4
⇒ 1 - 1/4 = sin²ϕ
⇒ sin²ϕ = 3/4
∴ sinϕ = √3/2

প্রশ্ন: tanA + sinA = m and tanA - sinA = n, then (m² - n²)/4 = ?

সমাধান:
(m² - n²)/4
= {(tanA + sinA)² - (tanA - sinA)²}/4
= (4tanA . sinA)/4    [(a + b)² - (a - b)² = 4ab]
= √(tan²A . sin²A)
= √{tan²A (1- cos²A)}
= √(tan²A - tan²A . cos²A)
= √(tan²A - (sin²A/cos²A) . cos²A)
= √(tan²A - sin²A)
= √{(tanA + sinA)(tanA - sinA)}
= √mn

প্রশ্ন: secA - tanA = 3/5 , then secA + tanA - (5/3) =?

সমাধান:
আমরা জানি,
sec²A - tan²A = 1
⇒ (secA - tanA ) (secA + tanA) = 1
⇒ (3/5) (secA + tanA) = 1
⇒ secA + tanA = 5/3

∴ secA + tanA - (5/3)
= (5/3) - (5/3)
= 0

প্রশ্ন: Which one is undefined? 

সমাধান:
cosec0° এর মান অসংজ্ঞায়িত।
sin90° এর মান 1
cos90° এর মান 0
sec0° এর মান 1 

Question: The minimum value of 2sin²θ + 3cos²θ is?

Solution: 
2 sin²θ + 3cos²θ
= 2 sin²θ + 2cos²θ + cos²θ
= 2 + cos²θ
= 2 + 0 [minimum value of cos²θ is 0]
= 2

Question: {(1 - tan²60°)/(1 + tan²60°)} + sin²60° = ?

Solution: 
(1-tan²60°)/(1+tan²60°) + sin²60°
= (1-(√3)²)/(1+(√3)²) + (√3/2)²
= (1 - 3)/(1+3) + 3/4
= -2/4 + 3/4
= 1/4

Question: sec²A + tan² A = 7, find, sec⁴A - tan⁴A =?

Solution: 
Given, sec²A + tan²A = 7

sec⁴A - tan⁴A
= (sec²A)² - (tan²A)²
= (sec²A + tan²A) (sec²A - tan²A)
= 7 × 1
= 7

Question: 

Solution: 

Question: If A = π/3, B = A/2 then sin (A + B) =? 

Solution: 
B = A/2
= π/6 

A + B = ( π/3) + ( π/6)
=  π/2 

sin(π/2 )
= 1

প্রশ্ন: If sinθ = 1/√2 then, tanθ =? 

সমাধান:
sinθ = 1/√2
⇒ sinθ = sin45°
∴ θ = 45°

∴ tanθ
= tan45°
= 1