Problems on Trigonometric Ratios of an Angle

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

11 and 12 Grade Math

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