Example 1 :
Verify the following equality :
sin260°+ cos260° = 1
Solution :
sin260° + cos260° = (√3/2)2 + (1/2)2
sin260° + cos260° = (3/4) + (1/4)
sin260° + cos260° = (3 + 1)/4
sin260° + cos260° = 4/4
sin260° + cos260° = 1
Example 2 :
Verify the following equality :
1 + tan230° = sec230°
Solution :
1 + tan230° = 1 + (1/√3)2
1 + tan230° = 1 + (1/3)
1 + tan230° = 4/3 -----(1)
sec230° = (2/√3)2
sec230° = 4/3 -----(2)
From (1) and (2), we get
1 + tan230° = sec230°
Example 3 :
Verify the following equality :
cos90° = 1 - 2 sin245° = 2 cos245° - 1
Solution :
cos 90° = 0 -----(1)
1 - 2 sin245° = 1 - 2(1/√2)2
1 - 2 sin245° = 1 - 2(1/2)
1 - 2 sin245° = 0 -----(2)
2 cos2 45° - 1 = 2(1/√2)2 - 1
2 cos2 45° - 1 = 2(1/2) - 1
2 cos2 45° - 1 = 0 -------(3)
From (1), (2) and (3), we get
cos90° = 1 - 2 sin245° = 2 cos245° - 1
Example 4 :
Verify the following equality :
sin 30° cos 60° + cos 30° sin 60° = sin 90°
Solution :
sin 30° cos 60° + cos 30° sin 60° :
= (1/2) (1/2) + (√3/2) (√3/2)
= (1/4) + (3/4)
= (1 + 3)/4
= 4/4
= 1 -----(1)
sin 90° = 1 -----(2)
From (1) and (2), we get
sin 30° cos 60° + cos 30° sin 60° = sin 90°
Example 5 :
Find the value of the following :
(tan45°/cosec30°) + (sec60°/cot45°) - (5sin90°/2cos0°)
Solution :
(tan45°/cosec30°) + (sec60°/cot45°) - (5sin90°/2cos0°) :
= (1/2) + (2/1) - (5(1)/2(1))
= (1/2) + (4/2) - (5/2)
= (1 + 4 - 5)/2
= 0
Example 6 :
Find the value of the following :
(sin 90°+cos 60°+cos 45°) × (sin 30°+cos 0°-cos 45°)
Solution :
(sin 90°+cos 60°+cos 45°) × (sin 30°+cos 0°-cos 45°) :
= [1 + (1/2) + (1/√2)] × [(1/2) + 1 - (1/√2)]
= [(3/2) + (1/√2)] × [(3/2) - (1/√2)]
= [(3/2)2 - (1/√2)2]
= [(9/4) - (1/2)]
= (9 - 2)/4
= 7/4
Example 7 :
Find the value of the following :
sin230° - 2 cos360° + 3 tan445°
Solution :
sin230° - 2 cos360° + 3 tan445° :
= (1/2)2 - 2(1/2)3 + 3(1)4
= (1/4) - (1/4) + 3
= 3
Example 8 :
Verify :
cos 3A = 4 cos3A - 3 cos A , when A = 30°
Solution :
cos 3A = cos 3(30°)
cos 3A = cos 90°
cos 3A = 0 ---(1)
4 cos3A - 3 cosA = 4 cos330° - 3 cos 30°
4 cos3A - 3 cosA = 4 (√3/2)3 - 3 (√3/2)
4 cos3A - 3 cosA = (12√3/8) - (3√3/2)
4 cos3A - 3 cosA = (3√3/2) - (3√3/2)
4 cos3A - 3 cosA = 0 -----(2)
From (1) and (2), we get
cos 3A = 4 cos3A - 3 cos A
when A = 30°.
Example 9 :
Find the value of 8sin 2x cos 4x sin6x, when x = 15°.
Solution :
8 sin 2x cos 4x sin6x :
= 8 sin 2(15°) cos 4(15°) sin 6(15°)
= 8 sin 30° cos 60° sin 90°
= 8(1/2) (1/2) (1)
= 2
Example 10 :
Find the value of the following :
tan260° - 2 tan245° - cot230° + 2 sin230° + (3/4)csc245°
Solution :
tan260° - 2 tan245° - cot230° + 2 sin230° + (3/4)csc245°:
= (√3)2 - 2(1)2 - (√3)2 + 2(1/2)2 + (3/4)(√2)2
= 3 - 2(1) - 3 + 2(1/4) + (3/4)(2)
= 3 - 2 - 3 + 1/2 + 3/2
= 6/2 - 4/2 - 6/2 + 1/2 + 3/2
= (6 - 4 - 6 + 1 + 3) / 2
= 0
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