Short Answer Type & Objective Questions::
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1) Evaluate:
A) sin 15. (√3-1)/2√2
B) cos 15. (√3+1)/2√2
C) tan 15. 2-√3
D) sun 75. (√3+1)/2√2
E) cos 75. (√3-1)/2√2
F) tan 75. 2 + √3
2) If tan(A+B)= 1/2, tan(A-B)=1/3, then find
A) tan 2A. 1
B) tan 2B. 1/7
3) evaluate: Cos 20+ cos100 + cos140. 0
4) A positive acute angle is divided into two parts whose tangents are 1/2 and 1/3. Find the angle. 45°
5) If A+B+C= π and cosA= cosB cosC, Prove that, cotB cotC= 1/2.
6) If x,y,z are in AP, show that, cot y= (sinx - sin z)/(cos z - cos x).
7) If tanA= 1/2 and A+B= π/4, find tan B. 1/3
8) If sinA= 3/5 and If tanB= 5/12, given K being obtuse and B acute, find sin(A+B). 16/65
9) If A+ B= π/4,Prove that, (1+ tanA) (1+ tanB)= 2.
10) Find the maximum and minimum values of 3 sinx+ 4 cosx.
5, -5
11) Choose the correct option:
a) The value of tan 75 - Cot 75 is
A) 2√3. B) 2 + √3 C) 2 - √3. D) n
b) If (1+ tanA)(1+ tan B)= 2, then the value of A+ B is
A) π/4. B) π/2 C) 3π/4 D) none
c) The value of sin 50 - sin 70+ sin 10 is
A) 1 B) 0. C) 1/2 D) 2
d) value of (cos 36- sin 36)/(cos 36 + sin 36) is
A) tan. 9 B) tan 54 C) tan 81 D) tan 36
e) Value of cos 15+√3cos15 is
A) -1 B) √2. C) -√2 D) 1
f) If A= π/13, the value of (sin 11A - sin 3A)/(sin 3A +sin 15A) is
A) -1. B) 0 C) 1 D) none
g) Value of tan 20+ tan 50+ tan 110 - tan 20 tan 50 tan 110 is
A) 1 B) 0. C) -1 D) undefined
h) If A+ B+ C=π and tanA= 1, tanB= 2, then tan C is
A) -1 B)-2 C) -3 D) 3.
I) If A+ B+ C=π the value of sin(A+B) sinC - cos(A+B) cosC is
A) -1 B) 0 C) 1 D) none
ESSAY TYPE
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1) Prove :
a) tan 70= 2tan 50 tan 20.
b) tan 70=tan 20+ 2tan 40+ 4tan 10.
c) sin²(45+A) - sin(30- A) - sin 15 cos(15+ 2A) = sin 2A.
d) tan(30-A) tan 2A + tan 2A tan(60 - A) + tan(60-A) tan(30 -A) = 1
e) sin 16 + cos 16 = 1/√2 (sin 1+ √3 cos 1)
f) cot 16 cot 44 + cot 44 cot 76 - cot 76 cot 16 = 3.
2) If cos(A+B) = cos C, show that 1- cos²A - cos²B - cos²C + 2 cosA cosB cosC= 0.
3) An angle a is divided into two parts, such that the ratio of their tangents is K and their difference is x, show that sinx={(K-1)/(K+1)}. Sin a.
4) If cotA, cotB, cotC are in AP, show that cot(B-A), cotB, cot(B-C) are also in AP.
5) If a cosx= b cos(x+120)= c cos(x+240), prove that ab+ bc+ ca= 0.
6) Given A+ B-C=π, prove that, sin²A +sin² B - sin² C= 2 sinA sinB cosC.
7) If sinA= K sin(A+B), prove that tan(A+ B)= sinB/(cos B - K).
8) If the angles A, B, C of a ∆ ABC are in AP, show that √3 tanA tanC - tanA - tan C= √3.
9) If m tan(A-30) = n tan(A+ 120), prove that cos 2A = (m+n)/{2(m-n)}.
10) If {tan(A-B)}/tan A + sin² C/sin²A = 1, show that tan A tan B= tan² C.
11) If x/a = cos(x - A) and y/b = cos(x - B), show that x²/a² + y²/b² - 2xy/ab cos(A- B)= sin²(A-B).
12) If A+ B+ C= 180, show that (cotA + cotB)/(tanA + tanB) + (cotA + cotC)/(tanB + tanC) + (cotC + cotA)/(tanC + tanA) = 1
13) If cos(A +B) cos(C+ D) + cos(A -B) cos(C- D)= 0, then show that tanA tanB tan C tanD= -1.
14) 2 cos x= a+ 1/a, 2 cos y= b+1/b, show that, 2 cos(x-y)= a/b + b/a.
15) If a/b = cosx/cosy, then show that a tan x + b tan y= (a+b) tan{(x+y)/2}.
16) If A+ B+ C=π, Show that, cotA cot B cot C + cosec A cosec B cosec C = cot A + cot B + cot C.
17) If (a+b) tan(A -B) = (a-b) tan(A+ B) , and a cos 2B + bcos 2A= c, prove that a² - b² + c²= 2ac cos 2B.
18) If A+ B+ C=π, Show that tanA/(tan B tan C) + tan B/(tanC tan A) + tan C/(tanA tan B) = (tanA + tanB + tan C) - 2(cotA + cotB + cot C).
19) If tanA, tanB are the roots of x² + px + q= 0, Show that the value of sin²(A+B) + p sin(A+B) cos(A+B) + q cos²(A+ B) is q.
20) If √2 cos A = cos B= cos³B and √2 sinA = sin B - sin³B, then show that sin(A- B)= ±1/3.
21) If cos(x-y)+ cos(y-z)+ cos(z-x)= -3/2 show that
A) cosx + cos y+ cos z = 0 and sin x + sin y + sin z= 0
B) cos(x-y)= cos(y-z)= cos(z-x)= -1/2 .
22) If the angle C of a triangle ABC be obtuse, then show that, tanA tanB < 1.
23) A, B, C are the angles of an acute angled triangle, show that, cos A cos B cos C ≤ 1/8.
24) Evaluate:
a) sin 10 sin 50 + sin 50 sin 250 +
sin 250 sin 10. -3/4
b) If 9x= π, then cosx cos 2x cos 3x cos 4x. 1/16
c) If A, B, C are the angles of a triangle, find the maximum value
A) SinA+ Sin B + Sin C. 3√3/2
B) cosA + cos B + cos C. 3/2
C) SinA. Sin B. Sin C. 3√3/8
D) cosA cos B cos C. 1/8
d) If tan(A -B)= sin 2B/(5- cos 2B), Find tanA/tanB. 3/2
e) 1/2 sin 10 - 2 sin 70. 1
25) Eliminate A and B from,
A) tanA + tanB= a, cotA + cotB= b and A+B= K.
B) sinA + sinB= a, cosA + cosB= b and A-B= π/3.
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