Double angle practice pdf

Double angle and half angle identities are very important in simplification of trigonometric functions and assist in performing complex calculations with ease. Practice finding the exact value of trig expressions, evaluate trig equations using the double and half angle formula, verify and prove the identities with this assemblage of printable worksheets, ideal for high school students.

Use the double angle identities and half angle identities charts as a precursor to the exercises. Commence your practice with our free worksheets! Double angle identities. Employ these worksheets with answer keys to find the exact value of the trigonometric expression using the double angle identities sin 2x, cos 2x and tan 2x. Half angle identities. Utilize these half angle identities PDFs to find the exact value of a trigonometric expression given as degrees and radians.

Mixed Review Double and half angle identities. Recapitulate the application of double and half angle formulas with these printable high school worksheets. Observe the angle measure, check if it can be expressed in double or half angle and then apply the appropriate formula to simplify.

Express as a single trigonometric function. Simplifying complex trigonometric expressions becomes easy with double and half angle identities. Express each trigonometric expression as a known angle measure by doubling or halving to simplify it and express as a single trigonometric function.

Value of a trig expression.

double angle practice pdf

Apply double angle or half angle identities to determine the value of trig expression based on the trig ratio given and the angle specified in the given interval. Verify using double and half angle formulae. Rattle your brains to verify the double and half angle formulas using logical steps to show that one side of the equation is equivalent to the other with this set of pdf worksheets. Members have exclusive facilities to download an individual worksheet, or an entire level.

double angle practice pdf

Login Become a Member. Double angle identities Employ these worksheets with answer keys to find the exact value of the trigonometric expression using the double angle identities sin 2x, cos 2x and tan 2x. Download the set 3 Worksheets.By Mary Jane Sterling. Identities for angles that are twice as large as one of the common angles double angles are used frequently in trig. These identities allow you to deal with a larger angle in the terms of a smaller and more-manageable one.

Because tangent is equal to the ratio of sine and cosine, its identity comes from their double-angle identities. Note that the cosine function has three different versions of its double-angle identity. Finding the cosine of twice an angle is easier than finding the other function values, because you have three versions to choose from. You make your choice depending on what information is available and what looks easiest to compute.

To show you where the first of the double-angle identities for cosine comes from, this example uses the angle-sum identity for cosine. Putting this result back into the double-angle identity for cosine and simplifying, you get. Then substitute this result into the first angle-sum identity for cosine:.

The biggest advantage to having three different identities for the cosine of a double angle is that you can solve for the cosine with just one other function value. The sum and difference identities for sine and cosine, on the other hand, as well as the double-angle identity for sine, all involve both the sine and cosine of the angles. The resulting cosine is positive. The cosine is positive in the first and fourth quadrants, so how do you know which of those two quadrants the terminal side of this double angle lies in?

Go back to the beginning of the problem — you know that the original angle is in the fourth quadrant. An angle in the fourth quadrant measures between degrees and degrees. The angles between those two values lie in the third and fourth quadrants. The cosine is positive in the fourth quadrant, so this double angle lies in the fourth quadrant. She has been teaching mathematics at Bradley University in Peoria, Illinois, for more than 30 years and has loved working with future business executives, physical therapists, teachers, and many others.

Using the Double-Angle Identity for Cosine.One interesting element about the trigonometric functions is that there is a way to compute the value of the trigonometric function of the double of a given angle, by using relatively simple formulas, by using the so-called double angle formulas. Then, the following formulas are used for the double angle.

So, say that you know the trigonometric values for 30 othen you can use the formulas above to compute the trigonometric values for 60 o. We said that the double angle could be very useful for calculation purposes, but actually, it is more of a theoretical use for them. I mean, trigonometric tables are not computed using the double angle starting from some notable angles, but using Taylor approximation instead.

Double angle formulas are extremely useful in identities used to make certain calculation of trigonometric integrals possible. Forgot password? This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept Read More.

Necessary Always Enabled.Bicycle ramps made for competition see [link] must vary in height depending on the skill level of the competitors.

double angle practice pdf

The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one. In the previous section, we used addition and subtraction formulas for trigonometric functions.

Now, we take another look at those same formulas. Deriving the double-angle for cosine gives us three options. Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more interpretations. The first one is:. Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.

If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. Use the Pythagorean Theorem to find the length of the hypotenuse:. This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side. We will work on the right side of the equal sign and rewrite the expression until it matches the left side. Part of being successful in mathematics is the ability to recognize patterns.

double angle practice pdf

While the terms or symbols may change, the algebra remains consistent. In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.

Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side.

Using the Double-Angle Identity for Cosine

However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show. When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result.

There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier. The double-angle formulas can be used to derive the reduction formulaswhich are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine.

They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine.The first question is already posted on the board. I encourage my students to work on this first problem in groups. As the students work I listen to the conversations.

For groups that are stuck, I ask, "How could you rewrite 2A so that you can use the formulas mentioned in the directions? Some students have more perseverance than others. I provide further assistance, if necessary. I pick a student that has made some progress to give me a hint on what might work. I ask, "How did you start? Why did you do that? To conclude today's bell work, I choose a student to go to the board and demonstrate how to verify the identity. I try to pick a student who has been working quietly thus far.

Having begun the class with a demonstration, I now ask the students to continue to work on the identities on the Double Angle worksheet.

During the Bell Work, they saw an approach to these problems. I expect my students to appreciate the structure of the problem and work through the remaining examples, beginning with the identity for cos 2A. In problems 4 and 5 of the worksheet students see how cos 2A can be written in 3 different forms.

Using the trig angle addition identities

This can sometimes challenge students who do not see the structure of the Pythagorean Identities and how those identities can be rearragned. S tudents share their reasoning with the class after working a few minutes. On page 3 of the examples the students determine the identity for tan 2A.

After working with sin 2A and cos 2A the students are seeing how to replace u and v in the sum formula. The students find the tan 2A without many problems. Once the students verify all the identities, they put these identities on their reference sheets. The last page of the examples gives the students a problem to work. The students find the sin 2A after finding the value we discussed whether the result seems reasonable.

I want students to think about where the terminal side of angle 2A is on the coordinate plane. It is important for students to analyze the accuracy of answers.

If a student determines the angle should be in quadrant III but their answer for sin 2A is positive then they know they have an error.

On standardized test using this kind of strategy will help student eliminate answers which is one of the test taking strategies taught to students. After doing sin 2A I discuss how you can choose which formula to use for cos 2A. Students need to recognize that using one formula may be easier depending on the information given in the problem.

In the above problems students are given information like the practice problems. Problem 14 gives some student trouble. The problem asks students to find sec 2 theta. I ask the students if they can rewrite sec x as another function. Questions requires students to find sin 2ucos 2u and tan 2u.The angle is divided in half for novices.

What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one. In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. Deriving the double-angle formula for sine begins with the sum formula. Deriving the double-angle for cosine gives us three options.

Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The first variation is:. How to: Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value. If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. Use the Pythagorean Theorem to find the length of the hypotenuse:.

Substitute these values into the equation, and simplify. Substitute this value into the equation, and simplify. This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function. Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas.

Choose the more complicated side of the equation and rewrite it until it matches the other side. We will work on the right side of the equal sign and rewrite the expression until it matches the left side. Part of being successful in mathematics is the ability to recognize patterns.

While the terms or symbols may change, the algebra remains consistent. In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.

Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show. When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result.

There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier. The double-angle formulas can be used to derive the reduction formulas, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine.The following questions are meant to guide our study of the material in this section.

After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. The sum and difference identities can be used to derive the double and half angle identities as well as other identities, and we will see how in this section.

Again, these identities allow us to determine exact values for the trigonometric functions at more points and also provide tools for solving trigonometric equations as we will see later. This leads to solving the equation. These formulas are called double angle identities. In our Beginning Activity we found that.

A similar identity for the. So we use the Pythagorean identity. There is also a double angle identity for the tangent. We leave the verification of that identity for the exercises. To summarize:. Approximate the smallest positive solution in degrees, to two decimal places, to the range equation. The fact that the two trigonometric functions have different periods makes this equation a little more difficult.

We can use the Double Angle Identity for the sine to rewrite the equation as. Now we have a product that is equal to 0, so at least one of the factors must be 0. This yields the two equations. We solve each equation in turn. The Cosine Half Angle Identity shows us that. We can also rewrite the expression under the square root sign to obtain.

Focus Questions The following questions are meant to guide our study of the material in this section.

Double Angle Identities & Formulas - Exact Value of Sin(2x), Cos(2x), Tan(2x)

What are the Double Angle Identities for the sine, cosine, and tangent? What are the Half Angle Identities for the sine, cosine, and tangent? What are the Product-to-Sum Identities for the sine and cosine? What are the Sum-to-Product Identities for the sine and cosine? Why are these identities useful? Verify your answer. Answer 1.


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