Review Questions
Find the exact value for:
If , is in quad II, and , is in quad I find
Find the exact value of
Verify the identity:
Verify
Show
Find all solutions to , when is between [0, ).
Review Answers
From the sum formula, we get:
If and in Quadrant II, then by the Pythagorean Theorem . And, if and in Quadrant I, then by the Pythagorean Theorem . So, to find
This is the cosine sum formula, so:
From the sum formula, we get:
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.
Expand using the cosine sum formula:
Step 1: Expand left hand side using the sum and difference formulas Step 2: Divide each term on the left side by cos cos and simplify
Step 1: Expand left hand side using the sum and difference formulas
Step 2: Substitute for and a for a and simplify
To find all the solutions, between [), we need to expand using the sum formula and isolate the cos x. This is true when or
Sum and Difference Identities for Sine and Tangent
Again, be careful to avoid confusing function notation with algebraic operations, as was seen previously.
Sum and Difference Identities for Sine
To find , use identity and identity as discussed previously.
In conclusion, , the sum formula for sines.
To obtain the identity for :
In conclusion, , the difference formula for sines
Example 1: Find the exact value of
or
or
In the following problem, the sum formula can be used, but the Pythagorean Trigonometric Identity is used first:
Example 2: Given: , where is in Quadrant II, and , where is in Quadrant I, find the exact value of .
To find the exact value of , here we use . The values of and . 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 and cosine is negative in Quadrant II,
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 for Tangent
To find the sum formula for tangent:
In conclusion, Substituting for in the above results in the difference formula for tangents:
Using the Sum and Difference Identities to Verify Other Identities
Example: Verify the identity
Lesson Summary
Trigonometry is a course that high schools (and thus in their admission process, colleges) require their students to know. In this light, Trigonometry is a liberal arts course. Think of this aspect of trigonometry when working through the continually growing list of identities and formulas that will need to be known. Think of this activity as a method of learning how to organize many thoughts efficiently, not unlike a set of folders in a file drawer- and the key organizing element is the Unit Circle. When asked to find the , first ask what quadrant will the point fall in, what will be the sign of the and values, what composition of angles can be a sum or difference that will equal the angle, etc. When substituting for in the difference formula for tangents visualize how this plays out on the unit circle. The successful trigonometry student will develop this visualizing as a habit.
Review Questions
Find the exact value:
If , is in quad III, and , is in quad II find
Verify the identity:
Simplify
Find the exact value of
Verify that , using the sine sum formula.
Reduce the following to a single term:
Solve for all values of between
Review Answers
Use the sine sum formula:
Use the sine sum formula:
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.
is the expanded sine sum formula, so it can be compressed to . The , thus
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.
Expand using the tangent sum formula.
To find the exact value of , expand it using the tangent difference formula.
Using the sine sum formula, we have:
Step 1: Expand using the cosine and sine sum formulas.
Step 2: Distribute and and simplify.
To find all the solutions, between [), we need to isolate , expand using the sum formula and then isolate the .
This is true when or .
Double-Angle Identities
There are ways for finding the value of a trigonometric function of a double angle if the value of the trigonometric function of the angle is known. For example: can be found in terms of trigonometric values of the angle “.”
Deriving the Double-Angle Identities
We can derive the double angle formulas by using the sum formulas with .
When we take if the formula becomes or
This is known as the Double Angle Formula for Sines the same procedure can be used in the sum formula for cosine:
Note: We can use the trigonometric identities to come up with alternate forms of these formulas. Since or , this can now be substituted into the above identity:
Similarly, in
Applying the Double-Angle Identities
If and a is in Quadrant II, both and can be found:
To use , the value of must be found first
or
, or . However since is in Quadrant II, is negative or
For , use
Finding Angle Values Given Double Angles
Example 1: Given and is a Quadrant II angle, find the value of
Simplify Expressions Using Double-Angle Identities
Verify:
Lesson Summary
The identities in this chapter widen the array of angles for which we can find trigonometric values, For example, suppose we know that for a first quadrant angle, . Now we can find the value of . By visualizing the unit circle and knowing that must be a bit larger than (because and is increasing in the first quadrant), must be an angle in the beginning of the second quadrant, and therefore must equal to a little less than (because is the maximum value of sine and sine is decreasing in the second quadrant).
If or , then or . Now using the double angle formula for sine:
Notice that the value for was a bit less than as predicted when visualizing the unit circle.
Example 2: Find the Notice that is in the quadrant, being between or and and in All Students Take Calculus mnemonic the (for the Quad) means that cosine is positive. Also notice that when visualizing the unit circle, being just a tad over , means that the cosine value is a little larger than or . Now use the sum formula for cosine: or
and
and
when found using a calculator.
Notice that this value corresponds to our prediction made at the beginning of the problem when visualizing the unit circle.
Review Questions
If and is in Quad II, find the exact values of and
Find the exact value of
Verify the identity:
Verify the identity:
If and is in Quad III, find the exact values of and
Find all solutions to if
Find all solutions to if
If you solve for you would get . This new formula is used to reduce powers of cosine by substituting in the right part of the equation
for . Try writing in terms of the first power of cosine.
If you solve for , you would get . Similar to the new formula above, this one is used to reduce powers of sine. Try writing in terms of the first power of cosine.
Rewrite in terms of the first power of cosine:
Review Answers
If and in Quadrant II, then cosine and tangent are negative. Also, by the Pythagorean Theorem, the third side is . So, and . Using this, we can find , , and .
This is one of the formulas for .
Step 1: Use the cosine sum formula
Step 2: Use double angle formulas for and
Step 3: Distribute and simplify.
Step 1: Expand using the double angle formula.
Step 2: change and find a common denominator.
If and in Quadrant III, then and (Pythagorean Theorem, ). So,
Step 1: Expand
Step 2: Separate and solve each for .
OR
Expand and simplify
when
Using our new formula,
Now, our final answer needs to be in the first power of cosine, so we need to find a formula for . For this, we substitute everywhere there is an and the formula translates to .
Using our new formula,
Now, our final answer needs to be in the first power of cosine, so we need to find a formula for . For this, we substitute everywhere there is an and the formula translates to .
a) First, we use both of our new formulas, then simplify: b) For tangent, we using the identity and then substitute in our new formulas.
now, use the formulas we derived inn #8 and 9.
Half-Angle Identities
There are ways for finding the value of a trigonometric function of half of an angle if the value of the trigonometric function of the angle is known. For example: can be found in terms of trigonometric values of the angle “”.
Deriving the Half-Angle Formulas
The double angle formulas can be used to derive the half angle formulas, simply by solving for the inside term of the formula.
Note: Examining each of the half angle formulas, the answer appears to have two values- one positive, and the other negative (observe the “” in front of the radicals) When using any half angles formulas in a specific problem, there will be only one correct answer. Again, the unit circle can help determine which sign is correct. To obtain the appropriate sign, first identify which quadrant is in, and then assess the quadrant is in to determine whether the final answer is positive or negative.
Use Half-Angle Identities to Find Exact Values
Example 1: Use to find exact value of
Since , use the half angle formula for sine, where . In this example, the angle is a second quadrant angle, and the sin of a second quadrant angle is positive.
Find Half-Angle Values Given Angles
Example 2: Given that the , and that is a fourth quadrant angle, find
Using the Half- or Double-Angle Formulas to Verify Identities
Example 3:
Verify the following identity:
Lesson Summary
Remember that trigonometric identities and formulas usually do not follow algebraic patterns such as . The trigonometric formulas and identities are derived logically from basic principles of geometry and algebra.
Technology Notes
The graphing calculator can demonstrate that an apparently obvious pattern such as is incorrect. First graph: . Then graph: to observe that the two graphs are in not the same and therefore the obvious pattern does not have equivalent values.
Review Questions
Find the exact value of
If and is in Quad II, find
verify the identity:
Verify the identity:
If , find
Solve for
Solve for
Review Answers
Using the half angle formula, we get:
Using the half angle formula, we get:
Finally, we need to rationalize the denominator:
The tangent is negative because is in Quadrant II.
Finally, we need to rationalize the denominator:
If , then by the Pythagorean Theorem the third side is 24. Because is in the second quadrant, .
Step 1: Change right side into sine and cosine. Step 2: At the last step above, we have simplified the right side as much as possible, now we simplify the left side, using the half angle formula.
Step 1: change cotangent to cosine over sine, then cross-multiply.
First, we need to find the third side. Using the Pythagorean Theorem, we find that the final side is . Using this information, we find that . Plugging this into the half angle formula, we get:
To solve , first we need to isolate cosine, then use the half angle formula.
To solve , first isolate tangent, then use the half angle formula. Using your graphing calculator, when
Product-and-Sum, Sum-and-Product and Linear Combinations of Identities
Transformations of Sums, Differences of Sines and Cosines, and Products of Sines and Cosines
In some problems, the product of two trigonometric functions is more conveniently found by the sum of two trigonometric functions by use of identities such as this one:
This can be verified by using the sum and difference formulas:
The following variations can be derived similarly:
Transformations of Products of Sines and Cosines into Sums and Differences of Sines and Cosines
We present two formulas for transforming a product of sines or cosines into sums and differences of sines and cosines.
Triple-Angle Formulas and Beyond
By combining the sum formula and the double angle formula, formulas for triple angles can be found:
Example 1: Find the formula for
Example 2: Find the formula for
Linear Combinations
Finally, we present a formula which takes a linear combination of sines and cosines and converts it into a simpler cosine function.
where and
Example 3: Transform into the form
Therefore and The reference angle is or
Since cosine is positive and sine is negative, the angle must be a fourth quadrant angle. must therefore be or .
Lesson Summary
In this section, we discussed several trigonometric identities and formulas which when first observed do not seem correct. Trigonometric manipulations can produce patterns that may not seem correct, but are logically derived and are correct. Be sure to utilize a graphing calculator to confirm results that may appear surprising. And, as always, utilize the unit circle as a visual reference to help recall formulas and identities.
Review Questions
Express the sum as a product:
Express the difference as a product:
Verify the identity (using sum-to-product formula):
Express the product as a sum:
Transform to the form ,
Solve for all solutions .
Solve for all solutions .
Solve for all solutions .
In the study of electronics, the function is used to analyze frequency. Simplify this function using the sum-to-product formula.
Derive a formula for .
Review Answers
Using the sum-to-product formula:
Using the difference-to-product formula:
Using the difference-to-product formulas:
Using the product-to-sum formula:
If , then and . By the Pythagorean Theorem, and . So, because is negative, is in Quadrant IV, therefore . Our final answer is .
If , then and . By the Pythagorean Theorem, . Because and are both negative, is in Quadrant III, therefore Our final answer is .
Using the sum-to-product formula:
So, either or ,
Using the sum-to-product formula:
CK-12 Trigonometry Page 13