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Exercise - 3.1

Trigonometric Functions

**Question-1 :-** Find the radian measures corresponding to the following degree measures:

(i) 25° (ii) – 47°30′ (iii) 240° (iv) 520°.

(i) 25° We know that 180° = π radian. Hence, 25° = π/180 x 25 radian = 5π/36 radian Therefore, 25° = 5π/36 radian. (ii) – 47°30′ We know that 180° = π radian. Hence, -47° 30′ = -47 + 1/2 degree = π/180 x (-95/2) radian = -93π/360 radian = -31π/120 Therefore, -47° 30′ = -31π/120 radian (iii) 240° We know that 180° = π radian. Hence, 240° = π/180 x 240 radian = 4π/3 radian Therefore, 240° = 4π/3 radian. (iv) 520° We know that 180° = π radian. Hence, 520° = π/180 x 520 radian = 26π/9 radian Therefore, 540° = 26π/9 radian.

**Question-2 :-** Find the degree measures corresponding to the following radian measures (Use π = 22/7)

(i) 11/16 (ii) -4 (iii) 5π/3 (iv) 7π/6.

(i) 11/16 We know that π radian = 180°. Hence, 11/16 radians = 180/π x 11/16 degree = (45 x 11 x 7)/ (22 x 4) degree = 315/8 degree = 39° + 3/8 degree = 39° + (3 x 60)/8 minute [1° = 60'] = 39° + 180/8 minute = 39° + 22' + 1/2 minute [1' = 60"] = 39° + 22' + 30" 11/16 radians = 39° 22' 30". (ii) -4 -4 radians = 180/π x (-4) degree = [180 x 7 x (-4)]/22 degree = -2520/11 degree = -229° + 1/11 degree = -229° + (1 x 60)/11 minute [1° = 60'] = -229° + 60/11 minute = -229° + 5' + 5/11 minute [1' = 60"] = -229° + 5' + 27" -4 radians = -229° 5' 27" approximately. (iii) 5π/3 5π/3 radians = 180/π x 5π/3 degree = 300° 5π/3 radians = 300°. (iv) 7π/6 7π/6 radians = 180/π x 7π/6 degree = 210° 7π/6 radians = 210°.

**Question-3 :-** A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

No. of revolution made by wheel in 1 minute = 360° No. of revolution made by wheel in 1 second = 360/60 = 6 In one complete revolution, the wheels turns an angle of 2π radian. Hence, in 6 complete revolutions, it will turn an angle of 6 x 2π radian. i.e., 12π radian Thus, in one second, the wheels turns an angle of 12π radian.

**Question-4 :-** Find the degree measure of the angle subtended at the centre of a circle of
radius 100 cm by an arc of length 22 cm (Use π = 22/7).

We know that in a circle of radius r unit, length of arc l and angle θ. Then, l = r.θ Given : radius of circle (r) = 100 cm length of arc (l) = 22 cm angle (θ) = ? θ = l/r = 22/100 radian = 180/π x 22/100 degree = (180 x 22 x 7)/ (22 x 100) degree = 126/10 degree = 12° + 3/5 degree = 12° + (3 x 60)/5 minute [1° = 60'] = 39° + 36' = 39° 36' The required angle is 39° 36'.

**Question-5 :-** In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

Given: Diameter of circle = 40 cm radius of circle = 40/2 = 20 cm length of chord = 20 cm length of minor arc of chord = ? Let AB be a chord(length = 20 cm) of the circle. In ∆AOB, OA = OB = radius of circle = 20 cm Also, AB = 20 cm Thus, ∆AOB is a an equilateral triangle. So, θ = 60° = π/3 radian We know that in a circle of radius r unit, length of arc l and angle θ. Then, l = r.θ length of arc (AB) = 20 x π/3 = 20π/3 cm Length of minor arc of chord is 20π/3 cm.

**Question-6 :-** If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Let the radii of two circles are r₁ and r₂. Let an arc of length l subtend an angle of 60° at the centre of the circle of radius r₁ Let an arc of length l subtend an angle of 75° at the centre of the circle of radius r₂ Now, 60° = π/3 radian and 75° = 5π/12 radian We know that in a circle of radius r unit, length of arc l and angle θ. Then, l = r.θ l = r₁.θ₁ and l = r₂.θ₂ l = π/3 . r₁ and l = 5π/12 . r₂ π/3 . r₁ = 5π/12 . r₂ r₁/r₂ = 5π/12 x 3/π r₁/r₂ = 5/4 r₁ : r₂ = 5 : 4 Ratio of radii is 5 : 4

**Question-7 :-**
Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length

(i) 10 cm (ii) 15 cm (iii) 21 cm

We know that in a circle of radius r unit, length of arc l and angle θ. Then, l = r.θ Given: (i) 10 cm length of arc (l) = 10 cm radius of circle (r) i.e.,Pendulum = 75 cm angle (θ) = ? θ = l/r = 10/75 = 2/15 radian (ii) 15 cm length of arc (l) = 15 cm radius of circle (r) i.e.,Pendulum = 75 cm angle (θ) = ? θ = l/r = 15/75 = 1/5 radian (iii) 21 cm length of arc (l) = 21 cm radius of circle (r) i.e.,Pendulum = 75 cm angle (θ) = ? θ = l/r = 21/75 = 7/25 radian

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