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 specific-angles. The angles 0⁰, 30⁰, 45⁰, 60⁰ and 90⁰ 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 90⁰, then any one angle is the complement of the other. Trigonometric ratios of complementary angles help in simplifying problems.
Trigonometric Ratios of 0⁰

sin 00 = 0 cos 00 = 1 tan 00 = 0

csc 00 = Not defined sec 00 = 1 cot 00 = undefined

Trigonometric Ratios of 30⁰

sin 300 = 1 / 2 cos 300 = √3 / 2 tan 300 = 1 / √3

csc 300 = 2 sec 300 = 2 / √3 cot 300 = √3

Trigonometric Ratios of 45⁰

sin 450 = 1 / √2 cos 450 = 1 / √2 tan 450 = 1

csc 450 = √2 sec 450 = √2 cot 450 = 1

Trigonometric Ratios of 60⁰

sin 600 = √3 / 2 cos 600 = 1 / 2 tan 600 = √3

csc 600 = 2 / √3 sec 600 = 2 cot 600 = 1 / √3

Some Solved Examples on trigonometry for specific angles:
1) Evaluate : ( sin² 45⁰ + cos² 45⁰) / tan² 60⁰
Solution :
( sin² 45⁰ + cos² 45⁰) / tan² 60⁰
= 1 / tan² 60⁰ [ since sin²θ + cos²θ = 1 ]
= 1/ (√3)²
= 1 / 3
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2) 2 sin²30⁰tan 60⁰ - 3 cos²60⁰sec²⁰
Solution :
2 sin²30⁰ - 3 cos²60⁰sec²⁰
= 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 0⁰ < A < 90⁰
Solution :
tan 5A =1
5A = 45
A = 45/5
A = 9⁰
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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 60⁰
A + 2B = 60 ------> (1)
cos (A + 4B ) = 0
cos (A + 4B ) = cos 90⁰
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

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Trigonometry

• SOHCAHTOA -Introduction to Trigonometry
• Trigonometric ratios and their Relation
• Trigonometry for specific angles
• Complementary angles in Trigonometry
• Trigonometric Equations

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