We will learn how to find the values of trigonometric ratios of an angle. The questions are related to find the values of trigonometric functions of a real number x (i.e., sin x, cos x, tan x, etc.) at any values of x.
1. Find the values of cos (\(\frac{11\Pi}{3}\))
Solution:
cos (\(\frac{11\Pi}{3}\)) = cos (\(\frac{11\Pi}{3}\)), since cos ( θ) = cos θ
= cos (\(\frac{11 × 180°}{3}\))
= cos (\(\frac{1980°}{3}\))
= cos 660°
= cos (7 × 90° + 30°)
= sin 30°, [Since the angle 660° lies in the 4th quadrant and cos ratio is positive in this quadrant. Again, in the angle 660° = 7 × 90° + 30°, multiplier of 90° is 7, which is an odd integer ; for this reason cos ratio has changed to sin.]
= 1/2
2. Find the values of cot ( 855°)
Solution:
cot ( 855°) =  cot
855° [since, cot (θ) =  cot θ]
=  cot (9 × 90° + 45°)
=  ( tan 45°) [Since the angle 855° = 9 × 90° + 45° lies in the second quadrant and only sin and csc ratios are positive in the second quadrant, thus cot ratio has become negative. Again, in 855° = 9 x 90° + 45°, the number 9 i.e., an odd integer appears as a multiplier of 90°; for this reason cot ratio has changed to tan.]
= tan 45°
= 1.
3. Find the values of csc (1650°)
Solution:
csc (1650°) =  csc 1650°, [since, csc (θ) =  csc θ]
=  csc (18 × 90° + 30°)
=  ( csc 30°), [Since, the angle 1650° lies in the 3th quadrant and csc ratio is negative in this quadrant. Again, in 1650° = 18 × 90° + 30°, multiplier of 90° is 18, which is an even integer; for this reason csc ratio remains unaltered.]
= csc 30°
= 2
4. If sin 49° = 3/4, find the value of sin 581°.
Solution:
sin 581° = sin (7 × 90°  49°)
=  cos 49°, [Since the angle 581° = 7 × 90°  49° lies in the 3rd quadrant and only tan and cot ratios are positive in the 3rd quadrant, thus sin ratio has become negative. Again, in 581° = 7 × 90°  49°, the number 7 i.e., an odd integer appears as a multiplier of 90°; for this reason sin ratio has changed to cos.]
=  √(1 sin\(^{2}\) 49°)
=  \(\sqrt{1  (\frac{3}{4})^{2}}\)
= =  \(\sqrt{1  \frac{9}{16}}\)
=  \(\sqrt{\frac{16  9}{16}}\), [since, sin 49° = ¾]
= \(\frac{√7}{4}\)
11 and 12 Grade Math
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