Trigonometry for Specific Angles
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