What is a Radian?
The angle is made when the radius is wrapped around the circle. 1 radian is about 57.2958 degrees.
Play the animation button below and have a feel of how 1 radian is measured. Change the radius and feel is there a change in the radian quantity?
Some common Angles radian and degree mapping
Question: If you have a 10 m radius circle and if you move 1m along the circumference of that circle what will be the subtended arc angle?
Question: How many radians do you need to move around a complete circle? Check the next simulation for that.
How many Radians in One complete Revolution?
Imagine you cut pieces of the string exactly the length equal to the radius of the circle
… how many pieces do you need to go once around the circle?
Check the simulation below and try to get an answer. Click the animation button to start visualizing it.
Question: Approximately how many radians will be required to get 1/4th of a complete circle?
Here the radius of the smallest circle is 1cm and every other circle gets its radius increased by 1 unit. A person who starts from P moves 1cm along the first circle and then vertically moves outward to jump to the next circle then he moves 2cm along the circumference then jumps vertically to the next circle. a similar process is continued until he again crosses the x-axis. At what radius circle he will be when he will be crossing the x-axis? You can view his motion as seen by the red lines.
Degree to radian conversion
1 degree is equivalent to (π/180) radians. Since you know this, all you have to do is multiply the number of degrees you’re working with by π/180 to convert it to radian terms.
Play with the simulation to generate new problems and strengthen your own concepts
Angle between clock hands
Some commonly known properties of the clock are
- A clock is a complete circle having 360 degrees. It is divided into 12 equal parts i.e. each part is 360/12 = 30°.
- As the minute hand takes a complete round in one hour, it covers 360° in 60 minutes.
- In 1 minute it covers 360/60 = 6°/ minute.
- Also, as the hour hand covers just one part out of the given 12 parts in one hour. This implies it covers 30° in 60 minutes i.e. ½° per minute.
- This implies that the relative speed of the minute hand is 6 – ½ = 5 ½ degrees.Some facts about clocks:
- Every hour, both hands coincide once. In 12 hours, they will coincide 11 times.
- It happens due to only one such incident between 12 and 1’o clock.
- The hands are in the same straight line when they are coincident or opposite to each other.
- When the two hands are at a right angle, they are 15-minute spaces apart. In one hour, they will form two right angles and in 12 hours there are only 22 right angles. It happens due to right angles formed by the minute and hour hand at 3’o clock and 9’o clock.
- When the hands are in opposite directions, they are 30-minute spaces apart.
- If both the hour hand and minute hand move at their normal speeds, then both the hands meet after 65 minutes.Now, let’s try to visualize all the above points from the simulation below. Play with the simulation and confirm the above read properties.
Question What is the angle formed by the minute hand and the hour hand at 4:45?
Question What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?
Cow grazing problem
Check the simulation and move the slider to visualize the sectors formed when the cow is grazing. Try changing the length of the rope and also the dimensions of the rectangle by moving the big black points.
Question: What will the area the cow will be grazing when the length of the rope is 8m and dimension of rectangle is 8×4 ?
Question: What will the area the cow will be grazing when the length of the rope is 6m and dimension of rectangle is 4×2 ?
Play with the Simulation, Move the point R to change the angle or you can also use the bottom left corner buttons as well. One ccan change the length of the stick by dragging the point P. Answer below questions after playing with the simulation
Question: When angle is changed from 0 degree to 45 degree which qty. is greater sin or cos ?
Question: When angle is changed from 45 degree to 90 degree which qty. is greater sin or cos ?
Question: In which interval the change in the value of sin is greater? a) if you change the angle from 30 – 35 degree b) when you change the angle from 65-70 degree.
Question: At what interval is tan greater than 1?
Play with the slider to check how sine takes values at different angles
Play with the slider to check how Cos takes values at different angles
Play with the slider to check how Tan takes values at different angles