Why Do We Convert Degrees to Radians in Trigonometry?

In summary, when converting between degrees and radians, you can use the rule of multiplying by pi/180 to go from degrees to radians, and multiplying by 180/pi to go from radians to degrees. This is because a full circle is equivalent to 2*pi radians or 360 degrees. The fundamental reason for this conversion is that degrees and radians are different units for measuring the same thing, similar to miles and kilometers.
  • #1
freeofwork
44
0

Homework Statement



Well basically I'm trying to learn trigonometry from a textbook. It shows a rule,
to change from degrees to radians, multiply by pi/180.
to change from radians to degrees, multiply by 180/pi .


Homework Equations





The Attempt at a Solution


I'm the type of person that wants to know why what works. I cannot sleep when i cannot understand the background work of an rule. why do those rules work? ty
 
Physics news on Phys.org
  • #2
It's because a full circle is 2*pi radians and it's also 360 degrees. So (2*pi radians)=(360 degrees). So (dividing both sides by 2*pi), 1 radian=(360/(2*pi)*degree=(180/pi)*degree. That's your first conversion, you do the second.
 
  • #3
Dick's response is correct, but a more fundamental answer is that it is inherent in the definition of degrees and radians. They are just different units for the same thing, like miles and kilometers.
 
  • #4
The length of an arc s in a circle belonging to the central angle φ is s=(φ(degree)/360°) (2Rπ). Instead of degrees, we can measure the angle with the ratio of (arc length / radius): φ(radian)=s/R. Comparing with the previous equation φ(radian)=s/R=2π ( φ(degrees)/360°).

ehild
 
  • #5


Hi there! It's great that you're trying to understand the background behind the rules in trigonometry. The reason why these rules work is because of the relationship between degrees and radians. Degrees and radians are two different ways of measuring angles. Degrees are based on dividing a circle into 360 equal parts, while radians are based on dividing a circle into 2π (or 360°) equal parts. So when converting from degrees to radians, you are essentially converting the angle measurement from a fraction of 360° to a fraction of 2π. This is why you multiply by π/180, which simplifies to 1/180. Similarly, when converting from radians to degrees, you are converting the angle measurement from a fraction of 2π to a fraction of 360°. This is why you multiply by 180/π, which simplifies to 180/3.14, or approximately 57.3. I hope this explanation helps you in your understanding of trigonometry. Keep up the good work!
 

FAQ: Why Do We Convert Degrees to Radians in Trigonometry?

What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems involving angles, distances, and heights.

What are the three basic trigonometric functions?

The three basic trigonometric functions are sine, cosine, and tangent. These functions are used to find the ratios between the sides of a right triangle.

How do I find the sine, cosine, and tangent of an angle?

To find the sine, cosine, and tangent of an angle, you can use a calculator or reference a trigonometric table. These values represent the ratios between the sides of a right triangle and can also be found using the Pythagorean theorem.

What is the unit circle and how is it related to trigonometry?

The unit circle is a circle with a radius of 1 unit. It is used in trigonometry to help visualize the relationships between angles and the ratios of the sides of a right triangle. The coordinates of a point on the unit circle can also represent the sine and cosine of an angle.

How is trigonometry used in real life?

Trigonometry is used in many real-life applications, such as architecture, engineering, navigation, and astronomy. It is also used in various fields of science, such as physics and chemistry, to calculate and analyze data. Additionally, it can be used in everyday situations, such as determining the height of a tree or building.

Back
Top