14.4: Sum and Difference Formulas
Objective
To use and derive the sum and difference formulas.
Review Queue
Using your calculator, find the value of each trig function below. Round your answer to 4 decimal places.
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Finding Exact Trig Values using Sum and Difference Formulas
Objective
Here you'll use the sum and difference formulas to find exact values of angles other than the critical angles.
Guidance
You know that etc... from the special right triangles. In this concept, we will learn how to find the exact values of the trig functions for angles other than these multiples of and . Using the Sum and Difference Formulas, we can find these exact trig values.
Sum and Difference Formulas
Example A
Find the exact value of .
Solution: This is an example of where we can use the sine sum formula from above, , where and .
In general, and similar statements can be made for the other sum and difference formulas.
Example B
Find the exact value of .
Solution: For this example, we could use either the sum or difference cosine formula, or . Let’s use the sum formula.
Example C
Find the exact value of .
Solution: This angle is the difference between and .
This angle is also the same as . You could have also used this value and done and arrived at the same answer.
Guided Practice
Find the exact values of:
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Answers
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Vocabulary
- Sum and Difference Formulas
Problem Set
Find the exact value of the following trig functions.
- Now, use from #1, and find . Do you arrive at the same answer? Why or why not?
- Using from #7, find . What is another way you could find ?
- Describe any patterns you see between the sine, cosine, and tangent of these “new” angles.
- Using your calculator, find the . Now, use the sum formula and your calculator to find the using and .
- Use the sine difference formula to find with any two angles you choose. Do you arrive at the same answer? Why or why not?
- Challenge Using and , show that .
Simplifying Trig Expressions using Sum and Difference Formulas
Objective
Here you'll use the sum and difference formulas to simplify expressions.
Guidance
We can also use the sum and difference formulas to simplify trigonometric expressions.
Example A
The and . is in the quadrant and is in the . Find .
Solution: First, we need to find and . Using the Pythagorean Theorem, missing lengths are 4 and 5, respectively. So, because it is in the quadrant and . Now, use the appropriate formulas.
Example B
Using the information from Example A, find .
Solution: From the cosine and sine of and , we know that and .
Example C
Simplify .
Solution: Expand this using the difference formula and then simplify.
Guided Practice
1. Using the information from Example A, find .
2. Simplify .
Answers
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Problem Set
and . Find the exact trig values of:
Simplify the following expressions.
Determine if the following trig statements are true or false.
Solving Trig Equations using Sum and Difference Formulas
Objective
Here you'll solve trig equations using the sum and difference formulas.
Guidance
Lastly, we can use the sum and difference formulas to solve trigonometric equations. For this concept, we will only find solutions in the interval .
Example A
Solve .
Solution: Use the formula to simplify the left-hand side and then solve for .
The cosine negative in the and quadrants. and .
Example B
Solve .
Solution:
In the interval, and .
Example C
Solve .
Solution:
At this point, we can factor the equation to be . , and 1, so . Be careful with these answers. When we check these solutions it turns out that does not work.
Therefore, is an extraneous solution.
Guided Practice
Solve the following equations in the interval .
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Answers
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Problem Set
Solve the following trig equations in the interval .
- Real Life Application The height, (in feet), of two people in different seats on a Ferris wheel can be modeled by and where is the time (in minutes). When are the two people at the same height?
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Date Created:
Apr 23, 2013Last Modified:
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