trigonometry

Q) If 1 + sin2 θ = 3 sin θ cos θ, then prove that tan θ = 1 or 1/2 Ans: Given that 1 + sin2 θ = 3 sin θ cos θ We know that sin2 θ + cos2 θ = 1 Hence, 1 + sin2 θ = 3 sin θ cos θ will become: […]

Q) Find the value of X if 2 cosec2 300 +  X sin2 600 – (3\4) tan2 300 = 10 Ans: Given that,

Q) If tan (A + B) = √3 and tan (A – B) = ; 0° < A + B < 90°; A > B, find A and B. Ans: Given that, tan (A + B) = √3 =  tan 60° Hence, A + B = 60° ………… (i) Next, its given that, tan (A –

Q) Prove that:  Ans: Let’s start from LHS = = = = We know that, a3−b3 formula is = (a−b)(a2 + b2 + ab) = = = = sec θ  cosec θ + 1 = 1 + sec θ  cosec θ = RHS Hence Proved 

Q) Prove that: = 2 cosec Ans: Let’s start from LHS LHS = Since sec A = LHS = = = = = = = = = = = = 2 cosec = RHS …………… Hence Proved 

Q) If a cos θ + b sin θ = m and  a sin θ – b cos θ = n, then prove that a2 + b2 = m2 + n2 Ans: Since a cos θ + b sin θ = m  By squaring on both sides, we get: (a cos θ + b sin θ)2

Q) If sin θ – cos θ = 0, then find the value of sin4 θ + cos4 θ. Ans: Since sin θ – cos θ = 0 sin θ = cos θ hence tan θ = 1 and θ = 450 and hence sin 450  = cos 450 = sin4 θ + cos4 θ  =

Q) Evaluate 2sec2 θ + 3 cosec2 θ – 2 sin θ cos θ if θ = 450. Ans: Since θ = 450, sec 450 = √2, cosec 450 = √2, sin 450 = ; cos 450 = 2sec2 θ + 3 cosec2 θ – 2 sin θ cos θ = 2 (√2)2  + 3 (√2)2

Q) If θ is an acute angle and sin θ = cos θ, find the value of tan2 θ + cot2 θ – 2. Ans: tan2 θ + cot2 θ – 2. = + – 2 By sin θ = cos θ (given), = + – 2 = 1 + 1 – 2 = 0

Q) Evaluate (5/cot2 30°) + (1/ sin2 60°) – cot2 45° + 2 sin2 90° Ans: Givent that, – cot2 45 + 2 sin2 90 = + – (1)2 + 2 (1)2 = + – 1 + 2 = 3 – 1 + 2 = 4