- Use and identify the sum and difference identities.
- Apply the sum and difference identities to solve trigonometric equations.
- Find the exact value of a trigonometric function for certain angles.
In this section we are going to explore , and . These identities have very useful expansions and can help to solve identities and equations.
Sum and Difference Formulas: Cosine
Is ? Upon appearance, yes, it is. This section explores how to find an expression that would equal . To simplify this, let the two given angles be and where .
Begin with the unit circle and place the angles and in standard position as shown in Figure A. Point Pt1 lies on the terminal side of , so its coordinates are and Point Pt2 lies on the terminal side of so its coordinates are . Place the in standard position, as shown in Figure B. The point A has coordinates and the Pt3 is on the terminal side of the angle , so its coordinates are .
Triangles in figure A and Triangle in figure B are congruent. (Two sides and the included angle, , are equal). Therefore the unknown side of each triangle must also be equal. That is:
Applying the distance formula to the triangles in Figures A and B and setting them equal to each other:
Square both sides to eliminate the square root.
FOIL all four squared expressions and simplify.
In , the difference formula for cosine, you can substitute to obtain: or . since and , then , which is the sum formula for cosine.
Using the Sum and Difference Identities of Cosine
The sum/difference formulas for cosine can be used to establish other identities:
Example 1: Find an equivalent form of using the cosine difference formula.
We know that is a true identity because of our understanding of the sine and cosine curves, which are a phase shift of off from each other.
The cosine formulas can also be used to find exact values of cosine that we weren’t able to find before, such as , among others.
Example 2: Find the exact value of
Solution: Use the difference formula where and .
Example 3: Find the exact value of .
Solution: There may be more than one pair of key angles that can add up (or subtract to) . Both pairs, and , will yield the correct answer.
You do not need to do the problem multiple ways, just the one that seems easiest to you.
Example 4: Find the exact value of , in radians.
Solution: , notice that and
Sum and Difference Identities: Sine
To find , use Example 1, from above:
In conclusion, , which is the sum formula for sine.
To obtain the identity for :
In conclusion, , so, this is the difference formula for sine.
Example 5: Find the exact value of
Solution: Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.
Example 6: Given , where is in Quadrant II, and , where is in Quadrant I, find the exact value of .
Solution: To find the exact value of , here we use . The values of and are known, however the values of and need to be found.
Use , to find the values of each of the missing cosine values.
For , substituting transforms to or , however, since is in Quadrant II, the cosine is negative, .
For use and substitute or and and since is in Quadrant I,
Now the sum formula for the sine of two angles can be found:
Sum and Difference Identities: Tangent
To find the sum formula for tangent:
In conclusion, . Substituting for in the above results in the difference formula for tangent:
Example 7: Find the exact value of .
Solution: Use the difference formula for tangent, with
To verify this on the calculator, and .
Using the Sum and Difference Identities to Verify Other Identities
Example 8: Verify the identity
Example 9: Show
Solution: First, expand left hand side using the sum and difference formulas:
Solving Equations with the Sum and Difference Formulas
Just like the section before, we can incorporate all of the sum and difference formulas into equations and solve for values of . In general, you will apply the formula before solving for the variable. Typically, the goal will be to isolate , or and then apply the inverse. Remember, that you may have to use the identities in addition to the formulas seen in this section to solve an equation.
Example 10: Solve in the interval .
Solution: First, get by itself, by dividing both sides by .
Now, expand the left side using the sine difference formula.
The when is .
Example 11: Find all the solutions for in the interval .
Solution: Get the by itself and then take the square root.
Now, use the cosine sum formula to expand and solve.
The is in Quadrants III and IV, so and .
Points to Consider
- What are the angles that have and as reference angles?
- Are the only angles that we can find the exact sine, cosine, or tangent values for, multiples of ? (Recall that would be , making it a multiple of )
- Find the exact value for:
- If , is in quad II, and , is in quad I find
- If , is in quad III, and , is in quad II find
- Prove the identity:
- Verify the identity:
- Verify that , using the sine sum formula.
- Reduce the following to a single term: .
- Find all solutions to , when is between .
- Solve for all values of between for .
- Find all solutions to , when is between .
- If and in Quadrant II, then by the Pythagorean Theorem . And, if and in Quadrant I, then by the Pythagorean Theorem . So, to find and
- If and in Quadrant III, then cosine is also negative. By the Pythagorean Theorem, the second leg is , so . If the and in Quadrant II, then the cosine is also negative. By the Pythagorean Theorem, the second leg is , so . To find , plug this information into the sine sum formula.
- This is the cosine difference formula, so:
- This is the expanded sine sum formula, so:
- Step 1: Expand using the cosine sum formula and change everything into sine and cosine Step 2: Find a common denominator for the right hand side. The two sides are the same, thus they are equal to each other and the identity is true.
- Step 1: Expand and using the sine sum and difference formulas. Step 2: FOIL and simplify. Step 3: Substitute for and for , distribute and simplify.
This could also be verified by using
- Step 1: Expand using the cosine and sine sum formulas. Step 2: Distribute and and simplify.
- Step 1: Expand left hand side using the sum and difference formulas Step 2: Divide each term on the left side by and simplify
- To find all the solutions, between , we need to expand using the sum formula and isolate the . This is true when , or
- First, solve for . Now, use the tangent sum formula to expand for when . This is true when or . If the tangent sum formula to expand for when , we get no solution as shown. Therefore, the tangent sum formula cannot be used in this case. However, since we know that when or , we can solve for as follows. Therefore, all of the solutions are
- To solve, expand each side: Set the two sides equal to each other: As a decimal, this is , so and .