Trigonometry for Specific Angles
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In this section we will discuss about trigonometry for specific angles.We have already learned that trigonometry is the study of relationships between the sides and angles of a triangle. The ratios of the sides in a right triangle with respect to some acute angles are called trigonometry for specificangles. The angles 0 ^{0} , 30 ^{0} , 45 ^{0} , 60 ^{0} and 90 ^{0} are useful angles in trigonometry, and their numerical values are easy to remember.These are trigonometry for specific angles.
When two angles add up to 90 ^{0} , then any one angle is the complement of the other. Trigonometric ratios of complementary angles help in simplifying problems.
Trigonometric Ratios of 0^{0}

sin 0^{0} = 0 cos 0^{0} = 1 tan 0^{0} = 0 
csc 0^{0} = Not defined sec 0^{0} = 1 cot 0^{0} = undefined 

sin 30^{0} = 1 / 2 cos 30^{0} = √3 / 2 tan 30^{0} = 1 / √3 
csc 30^{0} = 2 sec 30^{0} = 2 / √3 cot 30^{0} = √3 

sin 45^{0} = 1 / √2 cos 45^{0} = 1 / √2 tan 45^{0} = 1 
csc 45^{0} = √2 sec 45^{0} = √2 cot 45^{0} = 1 

sin 60^{0} = √3 / 2 cos 60^{0} = 1 / 2 tan 60^{0} = √3 
csc 60^{0} = 2 / √3 sec 60^{0} = 2 cot 60^{0} = 1 / √3 
Some Solved Examples on trigonometry for specific angles:
1) Evaluate : ( sin ^{2} 45 ^{0} + cos ^{2} 45 ^{0} ) / tan ^{2} 60 ^{0}
Solution :
( sin ^{2} 45 ^{0} + cos ^{2} 45 ^{0} ) / tan ^{2} 60 ^{0}
= 1 / tan ^{2} 60 ^{0} [ since sin ^{2} θ + cos ^{2} θ = 1 ]
= 1/ (√3) ^{2}
= 1 / 3
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2) 2 sin ^{2} 30 ^{0} tan 60 ^{0}  3 cos ^{2} 60 ^{0} sec ^{2}^{0}
Solution :
2 sin ^{2} 30 ^{0}  3 cos ^{2} 60 ^{0} sec ^{2}^{0}
= 2 ( 1/ 2) ^{2} x √3 – 3 ( ½) ^{2} x (2 / √3) ^{2}
= 2 x 1 / 4 x √ 3 – 3 x ¼ x 4 / 3
= √3 / 2 – 1
= ( √3 – 2 ) / 2
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3) Solve tan 5A = 1 for 0 ^{0} < A < 90 ^{0}
Solution :
tan 5A =1
5A = 45
A = 45/5
A = 9 ^{0}
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4) Find the acute angles A and B, if sin(A + 2B) = √3 / 2 and
cos (A + 4B ) = 0, A>B.
Solution :
sin(A + 2B) = √3 / 2
sin(A + 2B) = sin 60 ^{0}
A + 2B = 60 > (1)
cos (A + 4B ) = 0
cos (A + 4B ) = cos 90 ^{0}
A + 4B = 90 > (2)
Subtract equation (1) from (2) we get
2B = 30
B = 15
Equation (1)
A + 2(15) = 60
A + 30 = 60
A = 60 – 30
A = 30
Trigonometry
• SOHCAHTOA Introduction to Trigonometry
• Trigonometric ratios and their Relation
• Trigonometry for specific angles
• Complementary angles in Trigonometry
• Trigonometric Equations
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