7. Radians

Radians - an Alternative Measure for Angle

In science and engineering, radians are much more convenient (and common) than degrees. A radian is defined as the angle between 2 radii (radiuses) of a circle where the arc between them has length of one radius.

Another way of putting it is: "a radian is the angle subtended by an arc of length r (the radius)".

One radian is about 57.3°.

Radians are especially useful in calculus where we want to interchange angles and other quantities (e.g. length). For example, see how radians are required in Fourier Series. That stuff won't work if we try to use degrees.

Most computer programs use radians as the default.

Care with your calculator! Make sure your calculator is set to radians when you are making radian calculations.

Converting Degrees to Radians

Because the circumference of a circle is given by C = 2πr and one revolution of a circle is 360°, it follows that:

2π radians = 360°.

This gives us the important result:

π radians = 180°

From this we can convert:

radians → degrees and

degrees → radians.

Examples:

1. Convert the following to degrees:

a. 1 radian

b. 2 radians

Answer: a.

b.

2. Convert the following to radians:

a. 50°

b. 357°

Answer: a.

b.

Exercises

Convert the following:

1) 60° to radians

2) 3.1 radians to degrees

3) π/4 radians to degrees

4) 156.34° to radians

Your calculator can do these for you. However, you are encouraged to know what is happening under the hood.

Answers: 1)

2)

3) 45°

4)

Examples involving the Trig Ratios

Find the value of

1. cos(π/6)

2. sec 4.5

3. sec 4.5°

Answers: 1. cos(π/6) (= cos30°) = 0.8660254

2. sec 4.5 = -4.7439275

3. sec 4.5° = 1.0030922

Notice the difference between these last two. Without the degree sign, 4.5 means "4.5 radians". It is important to set your calculator properly before starting these problems.

Exercises:

1. Express in radian measure in terms of π:

a) 12° b) 225°

Answer: a) b)

2. Express the following angles in degrees:

a) b)

Answer: a) b)

3. Express in radian measure (use decimals): 168.7°

Answer

4. Express in terms of degrees: 1.703

Answer

5. Find math expression

Answer

6. Find sin 2.34

Answer: sin 2.34 = 0.7185

7. Find θ if cos θ = -0.9135 (0 ≤ θ < 2π)

Answer: cos θ = -0.9135 (0 ≤ θ < 2π )

α = 0.41899 radians

Since cos θ is negative, it means θ is in the second and third quadrants.

θ = π - 0.41899 = 2.723 or

θ = π + 0.41899 = 3.561

8. Find θ if csc θ = 3.940 (0 ≤ θ < 2π)

Answer: If csc θ = 3.940 , then

α = 0.2566.

Now θ is in the first and second quadrants because csc θ is positive.

So

θ = 0.2566 or

θ = π - 0.2566 = 2.8850

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