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This lesson will focus on introducing and practicing the trigonometric identities that relate the trigonometric values of an angle to the trigonometric values of the double-angle and half-angle.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Rules of the Quest Game and a Bonus
Zosia and her classmates entered a mathematical quest game. In this quest, there is an old leather map showing the path to various sacred mathematical sites. At each site, it is either solve the problem and move forward, or suffer the consequences of the unknown of the math universe.
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At the very beginning of the quest, they were given a bonus task, which can be solved at the end of the quest. If solved successfully, they will get a huge amount of bonus points. What is the value of
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Discussion
Presenting Double-Angle Identities
To be able to solve the tasks of the quest, the class first stopped at the infopoint to recall the Double-Angle Identities. These identities relate the trigonometric values of an angle to the trigonometric values of twice that angle.
Rule
Double-Angle Identities
The double-angle identities materialize when two angles with the same measure are substituted into the angle sum identities.
Proof
Double-Angle IdentitiesStart by writing the Angle Sum Identity for sine and cosine.
Let and With this, becomes Then, these two formulas can be rewritten in terms of
Now, recall that, by the Pythagorean Identity, the sine square plus the cosine square of the same angle equals From this identity, two different equations can be set.
Next, substitute Equation (I) into the first identity for the cosine.
That way, the second identity for the cosine has been obtained. To obtain the third cosine identity, substitute Equation (II) into the first identity for the cosine.
Next, substitute the first sine identity in the numerator and the first cosine identity in the denominator.
Then, divide the numerator and denominator by
Finally, simplifying the right-hand side the tangent identity will be obtained.
Sine Identity
Approaching the first equation, the Commutative Property of Multiplication can be applied to the second term of its right-hand side. Then, by adding the terms on the right-hand side of this equation, the formula for is obtained.
Cosine Identities
Approaching the second equation, the Product of Powers Property can be used to rewrite its right-hand side. By doing this, the first identity for the cosine of the double of an angle is obtained.
▼
Substitute for and simplify
\(\cos 2\theta = 2\cos^2\theta - 1 \quad{\color{#009600}{\bm{\Large{\checkmark }}}}\)
▼
Substitute for and simplify
\(\cos 2\theta = 1-2\sin^2 \theta \quad{\color{#009600}{\bm{\Large{\checkmark }}}}\)
Tangent Identity
To prove the tangent identity, start by rewriting in terms of sine and cosine.▼
Simplify right-hand side
WriteDiffFrac
Write as a difference of fractions
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
\(\tan 2\theta = \dfrac{2\tan\theta}{1-\tan^2\theta} \quad{\color{#009600}{\bm{\Large{\checkmark }}}}\)
Extra
CalculatingTo calculate the exact value of these steps can be followed.
- To be able to use the double-angle identities, the angle needs to be rewritten as multiplied by another angle. Therefore, rewrite as
- Use the second formula for the cosine of twice an angle.
- Based on the trigonometric ratios of common angles, it is known that
Example
Searching for Clues
When the class got to the first mathematical sacred site, they were told to look for three clues. After eagerly searching the neighborhood, they found three parts to one task.
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The task says to calculate knowing that and What exact value should the class get to be able to move on to the next site? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=f051dab641182febb046c8ca59aafb9e&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 7.949ex;\" ><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{4\\sqrt{21}}{25}","\\dfrac{\\sqrt{21}4}{25}","4\\dfrac{\\sqrt{21}}{25}"]}}
Hint
Find the value of by using one of the Pythagorean Identities.
Solution
In order to find the values of recall the Double-Angle Identity for sine.
The value of is known, but the value of is not. To find it, one of the Pythagorean Identities can be used.
Substitute with and solve for
By the definition of absolute value, can have two possible values.
It is given that is between and which is the first quadrant. Sine has positive values in the first quadrant, so the negative value can be disregarded.
Now that both sine and cosine of are known, the value of can be calculated.
Therefore, the value of is
Example
Finding Trigonometric Values to Determine the Password
At the second site, a steel safe awaits them. It is locked! Inside is the paper with the information needed to get to the third site. To open the safe, they must figure out the password.
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The password is decoded sequentially by the integers that appear in the answers to the three given tasks. It is known that and
a Find the exact value of
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b Find the exact value of
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c Find the exact value of
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Hint
a Use the Pythagorean Identity to find the value of
b Recall the Double-Angle Identity for cosine.
c Apply the Tangent Identity.
Solution
a Start by recalling the Double-Angle Identity for sine.
b In order to find the value of rewrite it by using the Double-Angle Identity for cosine.
c Finally, to find the value of use the Tangent Identity.
MultFrac
Multiply fractions
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
Discussion
Presenting Half-Angle Identities
Before moving to the next station, the class stopped at another infopoint to learn about the Half-Angle Identities.
Rule
Half-Angle Identities
The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied.
The sign of each formula is determined by the quadrant where the angle lies.
These identities are useful when finding the exact value of the sine, cosine, or tangent at a given angle.
Proof
Half-Angle IdentitiesFirst, write two of the Double-Angle Identities for cosine.
Next, substitute for
Finally, substitute the half-identities for the sine and cosine into the equation above and simplify the right-hand side.
Sine Identity
Start by solving the first identity written above for Next, substitute for to obtain the half-angle identity for the sine.
Cosine Identity
Start by solving the second identity written at the beginning for Next, substitute for to obtain the half-angle identity for the cosine.
Tangent Identity
To derive the tangent identity, start by recalling the definition of the tangent ratio.▼
Simplify right-hand side
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
MultFrac
Multiply fractions
\(\tan \dfrac{\theta}{2} = \pm\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}} \quad {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Extra
CalculatingConsider the calculation of the exact value of
- To be able to use the half-angle identities, the angle needs to be rewritten as a certain angle divided by Therefore, rewrite as
- Based on the trigonometric ratios of common angles, it is known that
- According to the diagram of the quadrants, an angle that measures is in the first quadrant. Therefore, the cosine ratio is positive.
Example
Solving the Tasks Found Inside the Balloons
Zosia and her classmates reached the third site, ready for any task. Three balloons floated steadily in mist. They caught the balloons one by one and noticed that there is something inside each one. Zosia found a needle and popped the balloons.
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Three notes fell to the ground. The classmates have been tasked with finding the values of the following trigonometric expressions by using the Half-Angle Identities. The answers must be written without any radicals in the denominators.
a
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b
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c
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Solution
a First, review the Half-Angle Identity for sine.
b To calculate the cosine of use the Half-Angle Identity for cosine.
c The value of can be calculated by applying the Half-Angle Identity for tangent.
Example
Solving a Task from a Fortune Cookie
The class has successfully made it to the fourth mathematical sacred site which appears to be in a kitchen. Something seems strange. They door immediately close behind them. They are not in a kitchen at all, but an escape room! In order to get out, the class needed to find the task and solve it.
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After searching for a while, someone found a fortune cookie. Inside of it, the task challenges them to find the exact value of two expressions given that and
a
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b
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Hint
a Start by calculating by using the Pythagorean Identity.
b Use the Half-Angle Identity for cosine.
Solution
a In order to find the value of the Half-Angle Identity for sine can be used.
b To find the value of recall the Half-Angle Identity for cosine.
Example
Choosing the Right Direction
The class successfully solved the task from the fortune cookie and arrived at the fifth site. There they saw two potential roads to the next site. To determine which road to choose, they need to solve the task written on the placard.
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Which way should the class turn to end up at the sixth site? {"type":"choice","form":{"alts":["Left","Right"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
Use the Pythagorean Identity and the Double-Angle Identity for sine.
Solution
To verify the identity, rewrite its sides until they match. First, add and subtract from both sides of the equation. Then, factor out
Recall the Pythagorean Identity and rewrite it to match the expression in the parentheses.
Now, substitute for into the equation and simplify it. Next, use the Double-Angle Identity for sine.
As can be seen, the left-hand side is the same as the right-hand side. Therefore, the identity is verified. This means that the class should turn right to end up at the sixth site.
Example
Matching the Sides of Identities
The class made the correct turn to the right and arrived at the sixth site. They notice a rustic table with torn pieces of paper, on which the left-hand and right-hand sides of different identities were written. The task at hand is to match at least one of the identities.
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Zosia placed two random pieces in the middle of the table and started checking whether they form an identity. {"type":"choice","form":{"alts":["It is an identitiy.","It is not an identity."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
Rewrite one or both sides of the equation by using the Half-Angle Identity or the Double-Angle Identity.
Solution
In order to check whether the equation Zosia formed is an identity, rewrite the sides until they match. There are two ways to do this — using the Half-Angle Identity or using the Double-Angle Identity. Each way will be shown one at a time.
Using the Half-Angle Identity
Start by recalling the Half-Angle Identities for sine and cosine.
CalcRoot
Calculate root
\(\pm \dfrac{\sin\theta}{2}=\dfrac{\sin\theta}{2}\quad {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Using the Double-Angle Identity
Start by reviewing the Double-Angle Identity.Example
Simplifying Trigonometric Expressions
The class has arrived at the seventh mathematical sacred sit — the finale. A huge board with plenty of colorful stickers lies ahead. They are told to choose two random cards.
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The classmates go through and and turn the cards. Lo and behold, two trigonometric expressions appear that need to be simplified. Once this is done, they will have completed the math quest!
a
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b
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":["THETA"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\cos\\theta-\\sin\\theta"]}}
Hint
a Start by factoring out
b Use the Double-Angle Identity.
Solution
a Start by analyzing the first given expression.
b Again, to simplify the expression, use the Double-Angle Identity for cosine.
AssociativePropAdd
Associative Property of Addition
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
Closure
Solving the Task to Get the Bonus
When the math quest game for Zosia and her classmates began, they were going to face some tough tasks. After passing all the sites, the rules stated that they could try for a bonus task and get major points. Naturally, they decided try to solve it.
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What is the value of {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":["THETA"]},"rendered":false},"text":""},"formTextBefore":"<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=e587275dfa57291f677414f5ba8a0a04&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -1.838ex;height: 5.176ex; width: 5.337ex;\" ><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{1}{4}\\tan \\theta"]}}
Hint
Use the Double-Angle Identity for sine and the Tangent Identity.
Solution
To find the value of substitute the corresponding expressions to and and then simplify.
Now, the Double-Angle Identity for sine can be used.
Therefore, the quotient of and simplifies to
MultFrac
Multiply fractions
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
Multiply
Multiply
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
WriteProdFrac
Write as a product of fractions
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