12 Sum and Difference Identities
Learning Objectives
In this section, you will:
- Use sum and difference formulas for cosine.
- Use sum and difference formulas for sine.
- Use sum and difference formulas for tangent.
- Use sum and difference formulas for cofunctions.
- Use sum and difference formulas to verify identities.
How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.
The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.
In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.
Using the Sum and Difference Formulas for Cosine
Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in (Figure 2).
We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See (Figure).
Sum formula for cosine | |
Difference formula for cosine |
First, we will prove the difference formula for cosines. Letâ€™s consider two points on the unit circle. See (Figure). Pointis at an anglefrom the positive x-axis with coordinatesand pointis at an angle offrom the positive x-axis with coordinatesNote the measure of angleis
Label two more points:at an angle offrom the positive x-axis with coordinatesand pointwith coordinatesTriangleis a rotation of triangleand thus the distance fromtois the same as the distance fromto
We can find the distance fromtousing the distance formula.
Then we apply the Pythagorean identity and simplify.
Similarly, using the distance formula we can find the distance fromto
Applying the Pythagorean identity and simplifying we get:
Because the two distances are the same, we set them equal to each other and simplify.
Finally we subtractfrom both sides and divide both sides by
Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.
Sum and Difference Formulas for Cosine
These formulas can be used to calculate the cosine of sums and differences of angles.
How To
Given two angles, find the cosine of the difference between the angles.
- Write the difference formula for cosine.
- Substitute the values of the given angles into the formula.
- Simplify.
Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles
Using the formula for the cosine of the difference of two angles, find the exact value of
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Begin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.
Keep in mind that we can always check the answer using a graphing calculator in radian mode.
Try It
Find the exact value of Hint:
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Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine
Find the exact value of
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As we can evaluate as
Keep in mind that we can always check the answer using a graphing calculator in degree mode.
Analysis
Note that we could have also solved this problem using the fact that
Try It
Find the exact value of
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Using the Sum and Difference Formulas for Sine
The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.
Sum and Difference Formulas for Sine
These formulas can be used to calculate the sines of sums and differences of angles.
How To
Given two angles, find the sine of the difference between the angles.
- Write the difference formula for sine.
- Substitute the given angles into the formula.
- Simplify.
Using Sum and Difference Identities to Evaluate the Difference of Angles
To find , we can use the sum and difference identities to evaluate the difference of the angles. Note that we can find it in two ways, with the same results.
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- Letâ€™s begin by writing the formula and substitute the given angles.
Next, we need to find the values of the trigonometric expressions.
Now we can substitute these values into the equation and simplify.
- Again, we write the formula and substitute the given angles.
Next, we find the values of the trigonometric expressions.
Now we can substitute these values into the equation and simplify.
Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function
Find the exact value of Then check the answer with a graphing calculator.
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The pattern displayed in this problem is Let and Then we can write
We will use the Pythagorean identities to find and
Using the sum formula for sine,