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]

= \(\frac{√7}{4}\)

**●** **Trigonometric Functions**

**Basic Trigonometric Ratios and Their Names****Restrictions of Trigonometrical Ratios****Reciprocal Relations of Trigonometric Ratios****Quotient Relations of Trigonometric Ratios****Limit of Trigonometric Ratios****Trigonometrical Identity****Problems on Trigonometric Identities****Elimination of Trigonometric Ratios****Eliminate Theta between the equations****Problems on Eliminate Theta****Trig Ratio Problems****Proving Trigonometric Ratios****Trig Ratios Proving Problems****Verify Trigonometric Identities****Trigonometrical Ratios of 0°****Trigonometrical Ratios of 30°****Trigonometrical Ratios of 45°****Trigonometrical Ratios of 60°****Trigonometrical Ratios of 90°****Trigonometrical Ratios Table****Problems on Trigonometric Ratio of Standard Angle****Trigonometrical Ratios of Complementary Angles****Rules of Trigonometric Signs****Signs of Trigonometrical Ratios****All Sin Tan Cos Rule****Trigonometrical Ratios of (- θ)****Trigonometrical Ratios of (90° + θ)****Trigonometrical Ratios of (90° - θ)****Trigonometrical Ratios of (180° + θ)****Trigonometrical Ratios of (180° - θ)****Trigonometrical Ratios of (270° + θ)****Trigonometrical Ratios of (270° - θ)****Trigonometrical Ratios of (360° + θ)****Trigonometrical Ratios of (360° - θ)****Trigonometrical Ratios of any Angle****Trigonometrical Ratios of some Particular Angles****Trigonometric Ratios of an Angle****Trigonometric Functions of any Angles****Problems on Trigonometric Ratios of an Angle****Problems on Signs of Trigonometrical Ratios**

**11 and 12 Grade Math**

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