Exercise (Extra Problems)
Trigonometric Ratios of Associated Angles
PROBLEMS.
1. Find the value of sin2120◦+ cos2150◦+ tan2120◦+ cos 180◦−tan 135◦.
2. Find the minimum positive values of xand y:sin (x−y) = 1
2;
cos (x+y) = 1
2.
3. For the minimum positive values of xand ysolve the given equations:
tan x+ tan y= 2 ;2 cos xcos y= 1.
4. Find the value of :
(i) sin21◦+ sin23◦+ sin25◦+· · · + sin285◦+ sin287◦+ sin289◦.
(ii) sin25◦+ sin210◦+ sin215◦+· · · + sin290◦.
(iii) sin26◦+ sin212◦+ sin218◦+· · · + sin290◦.
(iv) sin29◦+ sin218◦+ sin227◦+· · · + sin290◦.
(v) sin210◦+ sin220◦+ sin230◦+· · · + sin290◦.
(vi) cos (1◦)·cos (2◦)·cos (3◦)· · · cos (189◦).
(vii) tan (1◦)·tan (2◦)·tan (3◦)· · · tan (89◦).
(viii) tan (35◦)·tan (40◦)·tan (45◦)·tan (50◦)·tan (55◦).
(ix) sin (10◦) + sin (20◦) + sin(30◦) + sin (40◦) + · · · + sin (360◦).
(x) cos (10◦) + cos (20◦) + cos(30◦) + cos (40◦) + · · · + cos (360◦).
(xi) sin21◦+ sin23◦+ sin25◦+· · · + sin285◦+ sin287◦+ sin289◦.
(xii) sin2 π
18!+ sin2 π
9!+ sin2 4π
9!+ sin2 7π
18 !.
(xiii) tan (20◦)·tan (25◦)·tan (45◦)·tan (65◦)·tan (70◦).
(xiv) cos (18◦) + cos (234◦) + cos(162◦) + cos (306◦).
(xv) cos (20◦) + cos (40◦) + cos(60◦) + · · · + cos (180◦).
(xvi) sin (20◦) + sin (40◦) + sin(60◦) + · · · + sin (360◦).
5. Find the value of cos (θ1) + cos(θ2) + cos (θ3), if
sin (θ1) + sin (θ2) + sin(θ3)=3. (0≤θi≤π
2,∀i= 1,2,3).
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