Question

Convert the following angles to radians, giving your answer to 4 d.p.: (a) \(40^{\circ}\) (b) \(100^{\circ}\) (c) \(527^{\circ}\) (d) \(-200^{\circ}\)

Short answer

Question: Convert the following angles from degrees to radians and round the result to 4 decimal places: (a) 40 degrees (b) 100 degrees (c) 527 degrees (d) -200 degrees Answer: (a) 0.6981 radians (b) 1.7453 radians (c) 9.1986 radians (d) -3.4907 radians

Step-by-step solution

(a) Converting 40 degrees to radians:

To convert 40 degrees to radians, use the formula: radians = degrees × (π/180). We have: radians = 40 × (π/180) radians ≈ 0.6981

(b) Converting 100 degrees to radians:

To convert 100 degrees to radians, use the formula: radians = degrees × (π/180). We have: radians = 100 × (π/180) radians ≈ 1.7453

(c) Converting 527 degrees to radians:

To convert 527 degrees to radians, use the formula: radians = degrees × (π/180). We have: radians = 527 × (π/180) radians ≈ 9.1986

(d) Converting -200 degrees to radians:

To convert -200 degrees to radians, use the formula: radians = degrees × (π/180). We have: radians = -200 × (π/180) radians ≈ -3.4907 So, the angles in radians are: (a) 0.6981 (b) 1.7453 (c) 9.1986 (d) -3.4907

Key concepts

Angle Conversion

Converting angles from degrees to radians is a fundamental concept in mathematics, especially in the field of trigonometry. An angle's measurement in degrees, which is based on dividing a circle into 360 equal parts, can be converted into radians, where a full circle is represented by the mathematical constant \( \pi \) multiplied by 2.

To convert degrees to radians, we use the formula:\[ \text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right) \]This formula derives from the fact that \( 180^\circ \) corresponds to \( \pi \) radians. For instance, a right angle, which is \( 90^\circ \) in degree measure, equals \( \frac{\pi}{2} \) radians because \( 90 \times \frac{\pi}{180} = \frac{\pi}{2} \).

It's important to remember that when converting to radians, the answer can be left in terms of \( \pi \) for exactness or it can be approximated using a numerical value for \( \pi \) (usually 3.1416 or 3.14159).

Radian Measure

Radian measure is another way of expressing angles, distinct from the more commonly known degree measure. One radian is defined as the angle formed at the center of a circle by an arc whose length is equal to the radius of the circle. This leads to a full circle having \( 2\pi \) radians, as the circumference of a circle is \( 2\pi \) times its radius.

Using radian measure provides a natural and direct connection to various mathematical concepts and formulas, such as those in calculus and trigonometry. This connection often makes calculations simpler and more intuitive. For example, the derivative and integral formulas for trigonometric functions are more straightforward when the functions' arguments are in radians.

Mathematical Constants

In the context of angle conversion, the most significant mathematical constant is \( \pi \) (pi), which is approximately 3.14159. \( \pi \) is a crucial number in mathematics, representing the ratio of a circle's circumference to its diameter, regardless of the circle's size.

When we talk about radians, \( \pi \) itself represents a half-circle or \( 180^\circ \) in degree measure. The constant's infinite and non-repeating decimal nature allows for precise calculations across various fields, including engineering, physics, and, of course, mathematics. Other mathematical constants may include 'e' for natural logarithms and the imaginary unit \( i \), but \( \pi \) is usually the star of the show in trigonometry and angle conversion.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right-angled triangles. The key functions in trigonometry are sine, cosine, and tangent, which have various applications in science and engineering.

These functions are often defined in a unit circle where the angle is measured in radians. Working with radians can simplify many trigonometric expressions and is particularly convenient when dealing with periodic phenomena, such as waves and oscillations, because it aligns with the unit circle's properties. The conversion from degrees to radians is thus not just a mathematical exercise but a fundamental skill in applying trigonometry to real-world problems.