Double Angle: Properties, Rules, Formula & Examples

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  • 0:04 What Is a Double Angle?
  • 1:53 The Double Angle Formula
  • 3:53 Some Examples of the…
  • 5:19 Lesson Summary
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Lesson Transcript
Instructor: Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

Understanding the double angle formula is important to trigonometry. This lesson will help you to see the connections between the trigonometric ratios as they relate to the double angle formula.

What Is a Double Angle?

The concept known as a double angle is associated with the three common trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios - sine, cosine, and tangent - are functions that show the relationship between the sides of a right triangle, with respect to certain angles in the triangle.

Double, as the word implies, means to increase the size of the angle to twice its size. We can accomplish this in two ways, by multiplying or by adding. If angle y is 100 degrees, when the angle is doubled, it becomes 200 degrees. In trigonometry, doubling the angle is similar in concept. However, caution needs to be exercised as to what exactly we're doubling.

Let's say that we have Cos 60 = 0.5. If we want to double the angle, then we may think to do one of the following:

A) 2 * Cos x would give 2 * 0.5 = 1

B) Cos 2x would give Cos 2* 60 = Cos 120 = - 0.5

In part A we are not doubling the angle, but doubling the cosine of the angle. In part B, however, we are doubling only the angle.

Therefore, doubling the angle refers to multiplying the angle by two. The other way to double a quantity is to add the same quantity to the original amount. For example, if you have 10 apples and we double your amount, we could add 10 more apples. By adding we also doubled your amount, just as when we multiply by 2.

Both of these concepts apply to doubling the angle of trigonometric ratios. Accordingly, doubling the angle indicates the following:

Sin (x + x) = Sin 2x

Cos (x + x) = Cos 2x

Tan (x + x) = Tan 2x

The Double Angle Formula

Now, remember that trigonometric ratios share mathematical relationships. For example, the trig function tangent can be expressed as a fraction of sine and cosine, Tanx = Sinx / Cosx.

The double angle formula, is the method of expressing Sin 2x, Cos 2x, and Tan 2x in congruent relationships with each other. In this lesson, we will seek to prove on a small scale, that these relationships are true. Understanding the basic proofs below will help with remembering each formula. Let's begin with sine:

Sin (x + x) = Sin x Cos x + Sin x Cos x

Sin (x + x) = 2 Sin x Cos x

Since we already established that Sin (x + x) = Sin 2x, then Sine 2x = 2 Sin x Cos x.

Next, we will look at Cos 2x:

Once again, we will look at Cos (x + x) = Cos x Cos x - Sin x Sin x.

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