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NY Regents Exam - Geometry: Tutoring Solution10 chapters | 120 lessons

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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.

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 2*x* 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 2*x*

Cos *(x + x)* = Cos 2*x*

Tan *(x + x)* = Tan 2*x*

Now, remember that trigonometric ratios share mathematical relationships. For example, the trig function tangent can be expressed as a fraction of sine and cosine, Tan*x* = Sin*x* / Cos*x*.

The **double angle formula**, is the method of expressing Sin 2*x*, Cos 2*x*, and Tan 2*x* 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 2*x*, then Sine 2*x* = 2 Sin *x* Cos *x*.

Next, we will look at Cos 2*x*:

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

From this we will get the following:

Now, it is best to just remember the Tan *2x* formula. It's as follows:

However, if you're up for the challenge, you could try to show that Tan 2*x* = Sin 2*x* / Cos 2*x* by using substitution.

The use of the double angle formula for each trigonometric ratio may vary based on the desired result. The main uses are as follows:

1) To prove equivalent relationships. For example,

Simply put, all we did in this example is make equivalent substitutions to give the desired result.

2) To find the exact value of trigonometric ratios. For example:

If Cos *x* = 5/13 and Sin *x* = 12/13, find the value of Sin 2*x*

Since Sin 2*x* = 2 Sin *x* Cos *x* then, we can use substitution to give:

Sin 2*x* = 2 * 12/13 * 5/13

Sin 2*x* = 120/169. Therefore, Sin 2*x* would give approximately 0.71. Knowing this approximation can give us the approximate value of the angle *x*.

Let's take a couple moments to review what we've learned about the **double angle**, which is associated with the three common trigonometric ratios: sine (sin), cosine (cos), and tangent (tan).

The **double angle formula** is the method of expressing Sin *2x*, Cos *2x*, and Tan *2x* in congruent relationships with each other. Using the double angle formula of the trigonometric ratios can be tedious as well as simple to use. Becoming proficient in their use comes only with diligence and practice. They're used to prove other equivalent relationships and also to find exact measures of angles. They're summarized here, which you can see are as follows:

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NY Regents Exam - Geometry: Tutoring Solution10 chapters | 120 lessons

- Triangles: Definition and Properties 4:30
- Area of Triangles and Rectangles 5:43
- Classifying Triangles by Angles and Sides 5:44
- Perimeter of Triangles and Rectangles 8:54
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- How to Identify Similar Triangles 7:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Applications of Similar Triangles 6:23
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Converse of a Statement: Explanation and Example 5:09
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Properties of Concurrent Lines in a Triangle 6:17
- Angles and Triangles: Practice Problems 7:43
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Constructing Triangles: Types of Geometric Construction 5:59
- Constructing the Median of a Triangle 4:47
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
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