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Science > Mathematics > Trigonometry > You are Here 
Example – 01:
 If x^{c} = 405° and y° = – (π/12)^{c}. Find x and y
Given x^{c} = 405°
x containing term on R.H.S. is in radians. Hence we should convert L.H.S. into radians
∴ x^{c} = 405° = 405 x π/180 = (9π/4)^{c}
∴ x = 9π/4
Given y° = – (π/12)^{c}
y containing term on R.H.S. is in degrees. Hence we should convert L.H.S. into degrees
∴ y° = – (π/12)^{c }= – (π/12) x (180/π) = 15°
∴ y = 15
Ans: x = 9π/4 and y = 15
Example – 02:
 If θ° = – (5π/9)^{c} and Φ^{c} = 900°. Find θ and Φ
Given θ° = – (5π/9)^{c}
θ containing term on R.H.S. is in degrees. Hence we should convert L.H.S. into degrees
∴ θ° = – (5π/9)^{c}^{ }= – (5π/9) x (180/π) = – 100°
∴ θ = 100
Given Φ^{c} = 900°
Φ containing term on R.H.S. is in radians. Hence we should convert L.H.S. into radians
∴ Φ^{c} = 900° = 900 x π/180 = (5π)^{c}
∴ Φ = 5π
Ans: θ = 100 and Φ = 5π
Example – 03:
 Express following angles in radians

– 35°45’30”
– 35°45’30” = – [35° + (45/60)° + (30/3600)°]
– 35°45’30” = – [35° + 0.75° + 0.0083°] = – 35.7583°
– 35°45’30” = – 35.7583 x π/180 = 0.1987 π
– 35°45’30” = 0.1987 x 3.142 = 0.6242 radian

50°37’30”
50°37’30” = 50° + (37/60)° + (30/3600)°
50°37’30” = 50° + 0.6167° + 0.0083° = 50.625°
50°37’30” = 50.625 x π/180 = 0.2812 π
50°37’30” = 0.2812 x 3.142 = 0.8837 radian

– 10°40’30”
10°40’30” = 10° + (40/60)° + (30/3600)°
10°40’30” = 10° + 0.6667° + 0.0083° = 10.675°
10°40’30” = 10.675 x π/180 = 0.0593 π
10°40’30” = 0.0593 x 3.142 = 0.1863 radian
Interior Angle of Regular Polygon:
Steps to Find Interior Angle of Polygon:
 Find the measure of each exterior angle of regular polygon = 360°/No.of sides of polygon
 Find the measure of each interior angle of polygon = 180° – measure of exterior angle
Example – 04:
 Find Interior angles of following regular polygons in degrees and radians

Pentagon:
Pentagon has 5 sides
Each exterior angle = 360°/5 = 72°
Each interior angle = 180° – 72° = 108° = 108 x π/180 = (3π/5)^{c}
Ans: The interior angle of regular pentagon is 72° or (3π/5)^{c}

Hexagon:
Hexagon has 6 sides
Each exterior angle = 360°/6 = 60°
Each interior angle = 180° – 60° = 120° = 120 x π/180 = (2π/3)^{c}
Ans: The interior angle of regular hexagon is 120° or (2π/3)^{c}

Octagon:
Octagon has 8 sides
Each exterior angle = 360°/8 = 45°
Each interior angle = 180° – 45° = 135° = 135 x π/180 = (3π/4)^{c}
Ans: The interior angle of regular octagon is 135° or (3π/4)^{c}

Polygon with 20 sides:
Polygon has 20 sides
Each exterior angle = 360°/20 = 18°
Each interior angle = 180° – 18° = 162° = 162 x π/180 = (9π/10)^{c}
Ans: The interior angle of regular polygon with 20 sides is 162° or (9π/10)^{c}

Polygon with 15 sides:
Polygon has 15 sides
Each exterior angle = 360°/15 = 24°
Each interior angle = 180° – 24° = 156° = 156 x π/180 = (13π/15)^{c}
Ans: The interior angle of regular polygon with 15 sides is 156° or (13π/15)^{c}

Polygon with 12 sides:
Polygon has 12 sides
Each exterior angle = 360°/12 = 30°
Each interior angle = 180° – 30° = 150° = 150 x π/180 = (5π/6)^{c}
Ans: The interior angle of regular polygon with 12 sides is 150° or (5π/6)^{c}
Example – 05:
 Find the number of sides of polygon if each of its interior angle is (3π/4)^{c}.
Each interior angle = (3π/4)^{c }= (3π/4) x (180/π) = 135°
Hence each exterior angle = 180° – 135° = 45°
Number of sides of polygon = 360°/each exterior angle = 360°/45 = 8°
Ans: Thus the polygon has 8 sides
Angle Between Hour Hand and Minute Hand
Example – 06:
 Find the degree and radian measure of the angle between the hour hand and minute hand of a clock at following timings.

Twenty minutes past seven:
At twenty minutes past seven, the minute hand is at 4 and hour hand crossed 7
Angle traced by hour hand in 1 minute = 0.5°
Angle traced by hour hand in 20 minutes = 0.5° x 20 = 10°
Thus the hour hand is 10° ahead of 7 th Mark
The angle between 4 and 7 is 90°
Thus angle between hour hand and minute hand = 90° + 10° = 100°
100° = 100 x π/180 = (5π/9)^{c}
Ans: The angle between hour hand and minute hand is 100° or (5π/9)^{c}

Twenty minutes past two:
At twenty minutes past two, the minute hand is at 4 and hour hand crossed 2
Angle traced by hour hand in 1 minute = 0.5°
Angle traced by hour hand in 20 minutes = 0.5° x 20 = 10°
Thus the hour hand is 10° ahead of 2 nd Mark
The angle between 2 and 4 is 60°
Thus angle between hour hand and minute hand = 60° – 10° = 50°
50° = 50 x π/180 = (5π/18)^{c}
Ans: The angle between hour hand and minute hand is 50° or (5π/18)^{c}

Quarter past six:
At quarter past six, the minute hand is at 3 and hour hand crossed 6
Angle traced by hour hand in 1 minute = 0.5°
Angle traced by hour hand in 15 minutes = 0.5° x 17 = 7.5°
Thus the hour hand is 7.5° ahead of 6th Mark
The angle between 3 and 6 is 90°
Thus angle between hour hand and minute hand = 90° + 7.5° = 97.5°
97.5° = 97.5 x π/180 = (13π/24)^{c}
Ans: The angle between hour hand and minute hand is 97.5° or (13π/24)^{c}

Ten past eleven:
At ten past eleven, the minute hand is at 2 and hour hand crossed 11
Angle traced by hour hand in 1 minute = 0.5°
Angle traced by hour hand in 10 minutes = 0.5° x 10 = 5°
Thus the hour hand is 5° ahead of 11th Mark
The angle between 11 and 2 is 90°
Thus angle between hour hand and minute hand = 90° – 5° = 85°
85° = 85 x π/180 = (17π/36)^{c}
Ans: The angle between hour hand and minute hand is 85° or (17π/36)^{c}
Example – 07:
 Show that the minute hand of a clock gains 5°30′ on hour hand in one minute.
Angle traced by hour hand in 1 minute = 0.5°
Angle traced by minute hand in 1 minute = 6°
Thus angle between hour hand and minute hand = 6° – 0.5° = 5.5° = 5°30′
Ans: Thus the minute hand of a clock gains 5°30′ on hour hand in one minute.
Example – 08:
 Determine which of the following pairs of angles are coterminal.

210° and – 150°
– 150° = – 150° + 360° = 210°
Thus the two angles have the same initial arm and terminal arm.
Hence the angles 210° and – 150° are coterminal angles.

330° and – 60°
– 60° = – 60° + 360° = 300°
Thus the two angles do not have the same initial arm and terminal arm.
Hence the angles 330° and – 60° are not coterminal angles.

405° and – 675°
405° = 405° – 360° = 45°
– 675° + 360° x 2 = 45°
Thus the two angles have the same initial arm and terminal arm.
Hence the angles 405° and – 675° are coterminal angles.

1230° and – 930°
1230° = 1230° – 360° x 3 = 150°
– 930° + 360° x 3 = 150°
Thus the two angles have the same initial arm and terminal arm.
Hence the angles 1230° and – 930° are coterminal angles.
Example – 09:
 A wheel makes 360 revolutions in one minute. Trough how many radians does it turn in 1 second?
 Given: No. of revolutions = 360 per minute
 To Find: Radians per second = ?
 Solution:
No. of revolutions per second = 360/60 = 6
In one revolution the wheel turns through 2π radians
Radians per second = 2π x 6 = 12π^{c}
Ans: The wheel will turn through 12π^{c} in 1 second
Science > Mathematics > Trigonometry > You are Here 
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