We will learn how to solve different types of problems on trigonometric ratios of an angle.

**1.** Which of the six trigonometric function are positive for x = -10π/3?

**Solution: **

Given, x = -10π/3

We know that terminal position of x + 2nπ, where n ∈ Z, is the same as that of x.

Here, -10π/3 + 2 × 2π = 2π/3, which lies in the second quadrant.

**Note:** This process of finding a co-terminal angle or reference number results in an angle or number α, 0 ≤ α < 2π, so that we can determine in which quadrant the given angle or number lies.

Therefore, x = -10π/3 lies in the second quadrant.

Hence, sin x and csc x are positive while the other four trigonometric functions i.e. cos x, tan x, cot x and sec x are negative.

**2.** Express cos (- 1555°) in terms of the ratio of a positive
angle less than 30°.

**Solution: **

cos(- 1555°) = cos 1555°, since we know cos (- θ) = cos θ]

= cos (17 × 90° + 25°)

= - sin 25°; since the angle 1555° lies in the second
d quadrant and cos ratio is negative in this quadrant. Again, in the angle 1555° = 17 × 90° + 25°, multiplier
of 90° is 17, which is an odd integer ; for this reason cos ratio has changed
to sin.

**Note:**
The trigonometrical ratio of an angle of any magnitude can always be expressed in terms of ratio
of a positive angle less than 30°.

**3.** If θ = 170° find the sign of
(sin θ + cos θ)

**Solution: **

sin θ = sin 170° = sin (2 × 90° - 10°) = sin 10°

and cos θ = cos 170° =
cos (1 × 90° + 80°)= - sin 80°

Therefore, sin θ + cos θ = sin 10° - sin 80°

Since sin 10° > 0, sin 80° > 0 and sin 80°
> sin 10°, thus sin 10° - sin 80° < 0 (i.e. negative) so, the value of (sin θ +
cos θ) is negative.

**4.** Find the value of cos
200° sin 160° + sin (- 340°) cos (- 380°).

**Solution: **

Given, cos 200° sin 160° + sin
(- 340°) cos (- 380°)

= cos (2 × 90° + 20°) sin (1 × 90° + 70°) + (- sin 340°) cos 380°

= - cos 20° cos 70° - sin (3 × 90° + 70°) cos (4 × 90° + 20°)

= - cos 20° cos 700 - (- cos 70°) cos 20°

= - cos 200 cos 70° + cos 70° cos 20°

= 0

**●** **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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