Measurement of Trigonometrical Angles - Mathematical Problems and Solutions
We know,
∵ πc = 180º
or, 1c =
or, 1c = 57º 17′ 45ʺ
∵ D = 11º 15′ =
∴
or,
or,
or, [Substituting the value of D in this equation]
or,
∵ From , we get
∴
or,
or, [Substituting the value of R in this equation]
or, = 105º
We know,
∵
⇒
⇒
⇒
⇒
⇒ [∵ ]
∴ We can say that 60º is greater than one radian. So, one radian is not greater than 60º.
∵
∴
∴ The circular measure of the third angle
= [∵ Sum of the three angles of a triangle ]
=
=
We know,
∵
⇒
= [∵ The denominator is greater than the numerator, so the value of this fraction is less than one.]
=
🔹 07. Two angles of a triangle are 54º 11′ 48ʺ and 65º 48′ 12ʺ. Express the measure of the third angle of the triangle in radians.
🔹 08. The circular measures of two angles of a triangle are respectively and . Express the third angle in degrees and minutes.
🔹 09. One angle of a triangle is 60º and another angle is radians. Express the third angle in the sexagesimal system.
🔹 10. If the angles of a triangle are in the ratio 2 : 5 : 3, find the circular measure of the largest angle of the triangle.
🔹 11. The angles of a triangle are xº, (x + y)º, (x + 2y)º, and the ratio of the degree measure of the smallest angle to the radian measure of the largest angle is 60 : π. Express the angles of the triangle in degrees.
🔹 12. The sum of the two base angles of a triangle is 90º and their difference is 72º. Express the two angles in radians.
🔹 13. The sum of two angles is 135º and their difference is . Find the sexagesimal and circular measures of the two angles.
🔹 14. If the angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6, find their circular measures.
🔹 15. The radius of a circle is 10 cm. Find, in degrees and minutes, the angle subtended at the centre by an arc of length 6 cm.
🔹 16. Two circles of equal radius pass through each other's centres. Show that the length of the arc intercepted by one circle on the other circle is one-third of the circumference of that circle.
🔹 17. An arc of length s in a circle of radius r subtends an angle αº at the centre. Again, an arc of length S in another circle of radius R subtends an angle βº at its centre. Show that .
🔹 18. The side length of a regular hexagon is 7 cm. Find the circumference of its circumcircle.
🔹 19. In two circles, two arcs of equal length subtend angles of 60º and 30º respectively at their centres. Find the ratio of the radii of the two circles.
🔹 20. A horse runs along a circular path of radius 23 metres. In 5 seconds, the arc covered by the horse subtends an angle of 60º at the centre of the circle. Find the running speed of the horse.
🔹 21. If the length of the minute hand of a clock is 10 cm, find the distance travelled by the tip of the hand in 20 minutes.
🔹 22. An arc of length 40 metres of a circular railway track subtends an angle of 25º at the centre. Find the radius of the circular track.
🔹 23. A cow is tied to a post with a rope in such a way that, keeping the rope stretched, it can move along a circular path. If the cow moves 22 metres along this circular path and the angle subtended at the centre is 72º, find the length of the rope.