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Trigonometric functions are often used to perform calculations in real-life fields, such as architecture, optics, and trajectories. In many cases, multiple trigonometric functions appear in one expression, potentially causing confusion. Fortunately, rules that relate different trigonometric functions exist, providing the ability to simplify calculations. This lesson will present some of them.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Firework's Height Given by Trigonometric Functions
Best friends Paulina and Maya graduated high school last Friday. At the graduation party, they, along with the rest of the graduates and guests, were admiring bright and colorful fireworks.
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Their physics teacher came up with a contest to play for a prize — three tickets to an Ali Styles concert! He said that the height of the firework and horizontal displacement were related by the following equation.
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Discussion
The Concept of a Trigonometric Identity and Some Examples
Consider the two given equalities. First, analyze if any of them can be simplified. Then, focus on how many values of make each of them true. What is the main difference between the equalities?
The first equality is an equation that can be solved to find the few, if any, values of that make the equation true. However, the second equality can be simplified to and, therefore, is an equation that is true for all values of for which the expressions in the equation are defined. Given that characteristic, the second equality is called an identity.
A trigonometric identity is an equation involving trigonometric functions that is true for all values for which every expression in the equation is defined.
Two of the most basic trigonometric identities are tangent and cotangent Identities. These identities relate tangent and cotangent to sine and cosine.
Rule
Tangent and Cotangent Identities
The tangent of an angle can be expressed as the ratio of the sine of to the cosine of
Similarly, the cotangent of can be expressed as the ratio of the cosine of to the sine of
Proof
Tangent IdentityTwo proofs will be written for this identity, one using a right triangle and the other using a unit circle.
Right Triangle
In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side
Unit Circle
Consider a unit circle and an angle in standard position.
It is known that the point of intersection of the terminal side of the angle and the unit circle has coordinates
Draw a right triangle using the origin and as two of its vertices. The length of the hypotenuse is and the lengths of the legs are and
Proof
Cotangent IdentityTwo more proofs will be written for this identity, one of them using just a right triangle and the other using a unit circle.
Right Triangle
In a right triangle, the cotangent of an angle is defined as the ratio of the length of the adjacent side to the length of the opposite side
Unit Circle
Using a unit circle, it has been already proven that the tangent of an angle is the ratio of the sine to the cosine of the angle
RearrangeEqn
Rearrange equation
\(\cot \theta =\dfrac{\cos \theta}{\sin \theta}\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
There are also trigonometric identities which show that some trigonometric functions are reciprocals of others.
Rule
Reciprocal Identities
The trigonometric ratios cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.
Proof
Consider a right triangle with the three sides labeled with respect to an acute angle
Example
Simplifying Expressions by Using Trigonometric Identities
Thinking of different ways to solve the firework challenge, Paulina found herself thinking about her time learning trigonometric identities earlier in the school year. She really enjoyed those lessons.
A few of her favorite exercises included the following where she was asked to simplify these expressions.
a
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b
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c
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Hint
a Recall the Tangent Identity.
b Use the Cotangent Identity and the Reciprocal Identity for secant.
Solution
a The first expression contains two different trigonometric functions — sine and tangent. This means the Tangent Identity could be useful here.
▼
Reduce by
b To simplify the second expression, recall the Cotangent Identity and the Reciprocal Identity for secant.
c The last expression is the sum of two terms.
MultFrac
Multiply fractions
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
Discussion
Suggestions for Verifying Identities
Some techniques, like the following, are helpful when verifying if trigonometric identities are true.
- Substitute one or more basic trigonometric identities to simplify the expression.
- Factor or multiply as necessary. Sometimes it is necessary to multiply both the numerator and denominator by the same trigonometric expression.
- Write each side of the identity in terms of sine and cosine only. Then simplify each side as much as possible.
Discussion
Presenting Pythagorean Identities
One of the most known trigonometric identities relates the square of sine and cosine of any angle This identity can be manipulated to obtain two more identities involving other trigonometric functions.
Rule
Pythagorean Identities
For any angle the following trigonometric identities hold true.
Proof
For Acute AnglesConsider a right triangle with a hypotenuse of
By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to can be expressed in terms of the angle.
Since represents a side length, it is not Therefore, by diving both sides of the above equation by the second identity can be obtained.
The second identity was obtained.
Since represents a side length, it is not Therefore, by dividing both sides of by the third identity can be proven.
Finally, the third identity was obtained.
| Definition | Substitute | Simplify | |
|---|---|---|---|
It can be seen that if the hypotenuse of a right triangle is the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.
By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of and is equal to the square of
▼
Simplify
WriteSumFrac
Write as a sum of fractions
WritePow
Write as a power
CommutativePropAdd
Commutative Property of Addition
Proof
For Any AngleThe first identity can be shown using the unit circle and the Pythagorean Theorem. Consider a point on the unit circle in the first quadrant, corresponding to the angle A right triangle can be constructed with
Example
Trigonometric Functions of Angles on the Map
Enjoying the graduation party, Maya and Paulina shared some memories about some of the fun they had exploring their neighborhood as kids. After school they would buy junk food at the hour store, bird watch in the trees of the park, and play around the lake.
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The sine of the angle formed by the road connecting Paulina's and Maya's houses to the store is The cosine of the angle formed by the road connecting Paulina's and Maya's houses to the school is Interact with the map to view these angles.
a What is the cosine of
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b What is the tangent of
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c What is the sine of
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d What is the cotangent of
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Hint
a Apply the Pythagorean Identity.
b Recall the Tangent Identity.
c To determine the sign of the sine, identify in which quadrant is located.
d Use the Cotangent Identity.
Solution
a It is given that the sine of is To find the value of the cosine of the Pythagorean Identity can be used.
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The angle is in the first quadrant, where cosine is positive. Therefore, it can be concluded that the cosine of is
b To calculate the tangent of the Tangent Identity can be used.
▼
Simplify right-hand side
MultFrac
Multiply fractions
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
c This time the sine of the angle is known, while the cosine should be found. Again, the Pythagorean Theorem can be used. Substitute for and solve for
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The angle is in the second quadrant, where sine is positive. Therefore, the sine of is
d In order to find the cotangent of recall the Cotangent Identity.
▼
Simplify right-hand side
MultFrac
Multiply fractions
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
Pop Quiz
Practicing Applying Pythagorean Identities
Given the value of a trigonometric function and the quadrant of the angle, use one of the Pythagorean Identities to find the value of another trigonometric function. Round the answer to two decimal places.
Example
Verifying an Identity
Maya told Jordan, who was late for the graduation party, all about the contest and asked her to recall the trigonometric identities they learned on her way over. While thinking about the identities, Jordan remembered that, when they studied the topic, Maya sent her one interesting identity and decided to find that text.
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Hint
Use the known trigonometric identities to rewrite the left- and right-hand sides of the equation until they match.
Solution
To verify the given identity, its sides should be rewritten until they match. First, rewrite the left-hand side using the Reciprocal Identities for secant and cosecant.
Next, the right-hand side can be rewritten by applying the Tangent and Cotangent Identities.
Finally, by the Pythagorean Identity, the numerator of the fraction on the right-hand side is
A true statement was obtained, which means that the equation Maya sent is, indeed, a trigonometric identity.
Discussion
Cofunction and Negative Angles Identities
Studying angles and their trigonometric values in a right triangle more closely reveals further relationships between sine and cosine, and between tangent and cotangent.
Rule
Cofunction Identities
For any angle the following trigonometric identities hold true.
Proof
For Acute AnglesConsider a right triangle. The measure of its right angle is or radians. Let be the radian measure of one of the acute angles. Since the sum of two acute angles in a right triangle is the measure of the third acute angle must be
This identity is true for all angles, not just those that make it possible to construct a right triangle. Using similar reasoning, the corresponding identities for cosine and tangent can be proven.
When dealing with negative angles, identities that relate trigonometric values of negative and positive angles become very useful.
Rule
Negative Angle Identities
The function has odd symmetry, and has even symmetry, which can be seen from their graphs. As a result, the corresponding identities hold true.
Proof
andThe identities will be proven using a unit circle. Consider an arbitrary on a unit circle. Let be the point that the angle forms on the circle.
Recall that the values on the axis are represented by cosine and the value on the axis are represented by sine. Therefore, the coordinates of are
Additionally, after the reflection, the angle between the origin, and the axis is This means that the lengths of the newly created horizontal and vertical segments are and Therefore, the coordinates of can also be written as
Proof
By expressing using sine and cosine, this identity can be shown.
Example
Solving an Online Quiz
Jordan, still on the bus ride to the graduation party, feels excited about getting to solve a trig problem to win tickets to see Ali Styles live. She still has a bit of time before arriving so she used her tablet to look up a minute online quiz as practice.
Here are the expressions that Jordan is asked to simplify. What answers should be submitted to score a perfect
a
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b
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c
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d
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Hint
a Use the Cofunction Identities for sine and cosine.
b Recall the Negative Angle Identities.
c Rewrite the expression by applying the Cofunction Identity for cosine and the Reciprocal Identity for cosecant.
d Begin by using the Cotangent Identity.
Solution
a The first expression consists of two similar terms. To rewrite it, the Cofunction Identities can be used.
b To simplify the second expression, first recall the Negative Angle Identities.
c First, analyze the given expression.
d Start by rewriting the cotangent by applying the Cotangent Identity.
Example
Building a Birdhouse
Back to remembering their fun times before graduation, Paulina and Maya laughed about a walk in the park they had. One day, they noticed a beautiful birdhouse hanging on a maple tree. Multiple birds were eating from it and others were singing! They just had to try and build one themselves.
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They measured the dimensions of the birdhouse and the angle of its roof incline as the applet can display. However, they decided to change one part of the birdhouse and agreed to calculate its length at home. Later, when they compared notes, they saw their results were different.
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Hint
Start by applying the Cofunction and Negative Angle Identities for cosine. Then find by which factor the fraction should be expanded so that the Pythagorean Identity could be used.
Solution
In order to determine whether the expressions are equivalent, try to rewrite Paulina's expression until it matches Maya's expression.
Since the arguments of the cosines are and use the Cofunction and Negative Angle Identities for cosine to change the arguments to
Next, to be able to apply another trigonometric identity to the denominator, expand the fraction by the factor of
Note that the denominator has the term This term is also in the Pythagorean Identity. Rewrite this identity to get a similar expression on one side of the identity as the one in the denominator.
The rewritten identity can now be used to simplify the denominator and reduce the fraction by the factor of Next, distribute in the numerator and split the fraction as a sum of fractions.
Finally, notice that the fractions can be rewritten by using the Cotangent Identity and the Reciprocal Identity for cosecant.
This expression is the same as Maya's result, which means that Paulina's and Maya's expressions are equivalent. Therefore, both of their calculations are correct — they just obtained different forms of the same answer. Eventually, they were able to build an awesome birdhouse. What a memory.
Closure
Trying to Win the Contest
Earlier, it was said that at the graduation party, Paulina and Maya, along with the rest of the graduates and guests, were admiring the bright sparkles of a firework.
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Their physics teacher came up with a contest to win a prize — three tickets to an Ali Styles concert! The teacher said that the height of the firework and horizontal displacement were related by the following equation.
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Hint
Use two of the Reciprocal Identities for tangent and secant. Then apply one of the Pythagorean Identities.
Solution
Paulina and Maya really want to win the prize, so much so that they even sent practice problems to Jordan on her bus ride to the party. Now that Jordan has finally joined, they started analyzing the given equation.
First, Maya noticed that one of the Reciprocal Identities can be used to rewrite the second fraction in terms of
They decided to substitute it into the equation.
Remember, the purpose is to rewrite the expression in terms of tangent. Paulina notices that cosine still remains in the equation. It hits her that there is another Reciprocal Identity that relates cosine and secant. Since the equation contains a square of cosine, the girls decided to square each side of the identity.
Then, they substituted the expression into the equation.
Finally, Jordan suggests the idea to use one of the Pythagorean Identities containing secant.
This way they arrived at the equation where the tangent is the only trigonometric function. Their physics teacher checked their work. The results are in, and they won the tickets to the concert!
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