Lesson Explainer: Related and Correlated Angle Identities Mathematics

In this explainer, we will learn how to write trigonometric functions, like sine, cosine, and tangent, and their reciprocals in terms of cofunctions and use their properties to compare two trigonometric functions.

We recall that if the angle πœƒ is in standard position on a set of axes with the unit circle centered at the origin, then we can use the coordinates of the intersection between the terminal side of the angle and the unit circle to define sinπœƒ and cosπœƒ.

This allows us to define the sine and cosine of any angle, and we can then use these values to evaluate the tangent of any angle or any reciprocal trigonometric function. This geometric interpretation of the trigonometric functions allows us to discover identities of the trigonometric functions.

For example, a full rotation of 360∘ (or 2πœ‹ radians) counterclockwise of the angle πœƒ will not change the position of its initial or terminal side. This will be true for any number of full rotations, or clockwise rotations; hence, sinsin(πœƒ)=(πœƒ+𝑛360)∘ and coscos(πœƒ)=(πœƒ+𝑛360)∘ for any integer 𝑛. This property is called periodicity of the sine and cosine function and can be extended to the other trigonometric functions as follows.

Definition: Periodicity of the Trigonometric Functions

For any integer 𝑛 and angle πœƒ measured in degrees,

  • sinsin(πœƒ)=(πœƒ+𝑛360)∘,
  • coscos(πœƒ)=(πœƒ+𝑛360)∘,
  • tantan(πœƒ)=(πœƒ+𝑛180)∘.

For any integer 𝑛 and angle πœƒ measured in radians,

  • sinsin(πœƒ)=(πœƒ+2πœ‹π‘›),
  • coscos(πœƒ)=(πœƒ+2πœ‹π‘›),
  • tantan(πœƒ)=(πœƒ+π‘›πœ‹).

By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.

We can determine other trigonometric identities by using geometric transformations on the unit circle. For example, if we reflect the angle πœƒ in the 𝑦-axis, we get the following:

Since this is a reflection in the vertical axis, the two triangles are congruent; hence, the height of the triangle is sin(πœƒ) and the width is cos(πœƒ). We can use this to determine the coordinates of the intersection between the hypotenuse and the unit circle as (βˆ’(πœƒ),(πœƒ))cossin. We can notice that the hypotenuse is the terminal side of the angle 180βˆ’πœƒβˆ˜ in standard position.

Then, by using the definition of the sine and cosine of an angle in standard position, we can show the coordinates of the intersection of its terminal side and the unit circle are ((180βˆ’πœƒ),(180βˆ’πœƒ))cossin∘∘. This gives us ((180βˆ’πœƒ),(180βˆ’πœƒ))=(βˆ’(πœƒ),(πœƒ)).cossincossin∘∘ Equating the coordinates gives coscosandsinsin(180βˆ’πœƒ)=βˆ’(πœƒ)(180βˆ’πœƒ)=(πœƒ).∘∘ This is true for any angle πœƒ measured in degrees, and we can find similar identities for angles measured in radians by using the same technique.

When the sum or difference of two angles is an integer multiple of right angles, we say the angles are related. A specific example of related angles is when two angles sum to give 180∘ or πœ‹ radians; we call these supplementary angles. We can summarize the identities for supplementary angles as follows.

Definition: Supplementary Angle Trigonometric Identities

For any angle πœƒ measured in degrees,

  • sinsin(180βˆ’πœƒ)=(πœƒ)∘,
  • coscos(180βˆ’πœƒ)=βˆ’(πœƒ)∘,
  • tantan(180βˆ’πœƒ)=βˆ’(πœƒ)∘.

For any angle πœƒ measured in radians,

  • sinsin(πœ‹βˆ’πœƒ)=(πœƒ),
  • coscos(πœ‹βˆ’πœƒ)=βˆ’(πœƒ),
  • tantan(πœ‹βˆ’πœƒ)=βˆ’(πœƒ).

By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.

For example, sinsinsin(135)=(180βˆ’45)=(45)=√22.∘∘∘∘

We can follow this same process to find trigonometric identities of other related angles. For example, consider the related angles πœƒ and πœƒ+180∘ as shown on the unit circle in standard position.

We can think of this as a rotation 180∘ counterclockwise about the origin or as a reflection in both the vertical and horizontal axes. By using the transformations and the congruency of the triangles, we see that coscosandsinsin(180+πœƒ)=βˆ’(πœƒ)(180+πœƒ)=βˆ’(πœƒ).∘∘

We can extend this to the tangent function and summarize these as follows.

Definition: Related Angle Trigonometric Identities

For any angle πœƒ measured in degrees,

  • sinsin(180+πœƒ)=βˆ’(πœƒ)∘,
  • coscos(180+πœƒ)=βˆ’(πœƒ)∘,
  • tantan(180+πœƒ)=(πœƒ)∘.

For any angle πœƒ measured in radians,

  • sinsin(πœ‹+πœƒ)=βˆ’(πœƒ),
  • coscos(πœ‹+πœƒ)=βˆ’(πœƒ),
  • tantan(πœ‹+πœƒ)=(πœƒ).

By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.

Another example of related angles is πœƒ and 360βˆ’πœƒβˆ˜; we can draw both of these angles in standard position as follows.

These angles are related by a reflection in the π‘₯-axis; this gives us the identities coscosandsinsin(360βˆ’πœƒ)=(πœƒ)(360βˆ’πœƒ)=βˆ’(πœƒ).∘∘ We can extend these identities to the tangent function and summarize them as follows.

Definition: More Related Angle Trigonometric Identities

For any angle πœƒ measured in degrees,

  • sinsin(360βˆ’πœƒ)=βˆ’(πœƒ)∘,
  • coscos(360βˆ’πœƒ)=(πœƒ)∘,
  • tantan(360βˆ’πœƒ)=βˆ’(πœƒ)∘.

For any angle πœƒ measured in radians,

  • sinsin(2πœ‹βˆ’πœƒ)=βˆ’(πœƒ),
  • coscos(2πœ‹βˆ’πœƒ)=(πœƒ),
  • tantan(2πœ‹βˆ’πœƒ)=βˆ’(πœƒ).

By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.

There is a special type of related angles called complementary angles, which are angles that sum to give a right angle. These are useful in trigonometry because if πœƒ is one angle in a right triangle, then the other angle is the complementary angle 90βˆ’πœƒβˆ˜. We can use this to find a set of identities called the cofunction identities. First, we draw πœƒ in standard position, as follows:

We can include the complementary angle in our diagram as shown. Since πœƒ+(90βˆ’πœƒ)=90∘∘, we can construct the following congruent triangle:

This is the angle 90βˆ’πœƒβˆ˜ in standard position, so the coordinates of the point of intersection give us the sine and cosine values of this angle. Since these triangles are congruent, we can equate the corresponding sides (where we must be careful of the signs) to get cossinandsincosπœƒ=(90βˆ’πœƒ)πœƒ=(90βˆ’πœƒ).∘∘

These are the cofunction identities, and they are true for any angle πœƒ measured in degrees. We can extend this to angles measured in radians and the tangent function as follows.

Definition: Cofunction Trigonometric Identities

For any angle πœƒ measured in degrees,

  • sincos(90βˆ’πœƒ)=(πœƒ)∘,
  • cossin(90βˆ’πœƒ)=(πœƒ)∘,
  • tancot(90βˆ’πœƒ)=(πœƒ)∘.

For any angle πœƒ measured in radians,

  • sincosο€»πœ‹2βˆ’πœƒο‡=(πœƒ),
  • cossinο€»πœ‹2βˆ’πœƒο‡=(πœƒ),
  • tancotο€»πœ‹2βˆ’πœƒο‡=(πœƒ).

By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.

These are not the only identities we can find from the congruencies of this triangle; we can also sketch the following congruent triangles on the same axes.

All of these would allow for more identities. The cofunction identities are one example of what are known as the correlated angle identities. The correlated angle identities are when the sum or difference of the angles is an integer multiple of a right angle; the other correlated angles are of the forms πœƒ and πœƒ+90∘. Consider an acute angle πœƒ in standard position, where we construct the flowing right triangle to determine the values of sinπœƒ and cosπœƒ.

By rotating the triangle 90∘ counterclockwise about the origin and by considering congruent triangles, we get the following:

This is the angle 90+πœƒβˆ˜ in standard position, so the coordinates of the point of intersection give us the sine and cosine values of this angle. Since these triangles are congruent, we can equate the corresponding sides (where we must be careful of the signs) to get βˆ’πœƒ=(90+πœƒ)πœƒ=(90+πœƒ).sincosandcossin∘∘ This was shown only by using rotation and congruency conditions; it will also hold for any angle πœƒ in standard position. We can also do the same for rotations of 180∘ or 270∘ counterclockwise to get the following:

Using the same argument above, we can construct more correlated angle identities. In fact, we can also use clockwise rotations instead. We summarize the following correlated angle identities.

Definition: Correlated Angle Trigonometric Identities

For any angle πœƒ measured in degrees,

  • sincos(90+πœƒ)=(πœƒ)∘,
  • cossin(90+πœƒ)=βˆ’(πœƒ)∘,
  • tancot(90+πœƒ)=βˆ’(πœƒ)∘.

For any angle πœƒ measured in radians,

  • sincosο€»πœ‹2+πœƒο‡=(πœƒ),
  • cossinο€»πœ‹2+πœƒο‡=βˆ’(πœƒ),
  • tancotο€»πœ‹2+πœƒο‡=βˆ’(πœƒ).

By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.

We can find many more identities by considering other triangle congruencies from transformations.

In our first example, we will use these identities to simplify a trigonometric expression.

Example 1: Simplifying Trigonometric Expressions Using Periodic Identities

Simplify coscosπœƒ+(180βˆ’πœƒ)∘.

Answer

There are a few different ways of simplifying this expression; for example, we could recall the following supplementary angle identity, which tells us that for any angle πœƒ measured in degrees, coscos(180βˆ’πœƒ)=βˆ’(πœƒ).∘ Substituting this into the expression then gives us coscoscoscosπœƒ+(180βˆ’πœƒ)=πœƒβˆ’(πœƒ)=0.∘

However, as we have seen, there are many trigonometric identities in similar forms, and committing all of these to memory is a difficult task. Instead, we should concentrate on understanding where these identities come fromβ€”symmetries of the unit circle. If we sketch an acute angle πœƒ in standard position, then the angle 180βˆ’πœƒβˆ˜ is the supplementary angle to πœƒ; together they form a straight line on the π‘₯-axis as shown.

The coordinates of the intersection between the unit circle and the terminal side of πœƒ give us the cosine and sine of angle πœƒ. If we reflect this in the vertical axis, we will then have the angle 180βˆ’πœƒβˆ˜ in standard position.

Since this is a reflection, the right triangles in both diagrams are congruent. In particular, the base of the triangle is length |πœƒ|cos, and equating this with the π‘₯-coordinate of the point of intersection (and being careful with the signs), we have coscos(180βˆ’πœƒ)=βˆ’(πœƒ).∘ This is true for any angle πœƒ measured in degrees; we then substitute this into the original expression to get coscoscoscosπœƒ+(180βˆ’πœƒ)=πœƒβˆ’(πœƒ)=0.∘

Hence, for any angle πœƒ measured in degrees, coscosπœƒ+(180βˆ’πœƒ)=0∘.

In our second example, we find an equivalent expression to a given trigonometric expression by using the unit circle.

Example 2: Finding Equivalent Expressions Using the Cofunction Identity for Sine and Cosine

Which of the following is equal to βˆ’πœƒsin?

  1. cosο€Ό3πœ‹2+πœƒοˆ
  2. cosο€»πœ‹2+πœƒο‡
  3. sinο€Ό3πœ‹2+πœƒοˆ
  4. sinο€»πœ‹2+πœƒο‡

Answer

There are several different ways of approaching this problem; one such way is to recall the cofunction identities. We know that sincosπœƒ=ο€»πœ‹2βˆ’πœƒο‡, so βˆ’πœƒ=βˆ’ο€»πœ‹2βˆ’πœƒο‡.sincos

Next, recall the following identity for supplementary angles: βˆ’(π‘₯)=(πœ‹βˆ’π‘₯).coscos If we substitute π‘₯=πœ‹2βˆ’πœƒ, we see that βˆ’ο€»πœ‹2βˆ’πœƒο‡=ο€»πœ‹βˆ’ο€»πœ‹2βˆ’πœƒο‡ο‡=ο€»πœ‹2+πœƒο‡.coscoscos

However, memorizing all of the identities for the trigonometric functions is difficult, and it is much easier to understand where the identities come from. We will instead answer this question by sketching the arguments of the four options in standard position and considering their relation to βˆ’πœƒsin. For simplicity, we will assume πœƒ is acute. Sketching both arguments in standard position gives us the following:

We can determine the coordinates of the point of intersection with the unit circle by sketching πœƒ in standard position and using congruency.

We see that adding πœ‹2 to the argument rotates the triangle πœ‹2 radians counterclockwise about the origin and adding 3πœ‹2 to the argument rotates the triangle 3πœ‹2 radians counterclockwise about the origin. By using the fact that these triangles are congruent, we can use the side lengths of the triangle given by πœƒ in standard position to find an expression for the π‘₯- and 𝑦-coordinates of each point of intersection; we get the following 4 identities: cossinsincoscossinsincosο€Ό3πœ‹2+πœƒοˆ=πœƒ,ο€Ό3πœ‹2+πœƒοˆ=βˆ’πœƒ,ο€»πœ‹2+πœƒο‡=βˆ’πœƒ,ο€»πœ‹2+πœƒο‡=πœƒ. We see that the π‘₯-coordinate of the point of intersection between the unit circle and the terminal side of πœ‹2+πœƒ in standard position is βˆ’πœƒsin.

Hence, for any angle πœƒ measured in radians, βˆ’πœƒ=ο€»πœ‹2+πœƒο‡,sincos which is option B.

In our next example, we will apply a cofunction identity to find an equivalent expression for the cosine of an angle in terms of the sine function.

Example 3: Finding Equivalent Expressions Using the Cofunction Identity for Sine and Cosine

Using the fact that cossinπœƒ=(90βˆ’πœƒ)∘, which of the following is equivalent to cos35∘?

  1. 135sin∘
  2. sin35∘
  3. sin55∘
  4. sin145∘
  5. βˆ’35sin∘

Answer

Substituting πœƒ=35∘ into the identity gives us cossinsin35=(90βˆ’35)=(55),∘∘∘∘ which is option C.

We can see why this is true by sketching the angle 35∘ in standard position and considering the intersection between the terminal side of this angle and the unit circle centered at the origin.

The coordinates of the points of intersection will be (35,35)cossin∘∘. We see that cos35∘ is the π‘₯-coordinate of this point, so it is the width of this triangle. We can find the size of the non-right angle in this triangle as 90βˆ’35=55∘∘∘. Therefore, if 55∘ was drawn in standard position, it would be congruent to the triangle drawn for 35∘ in standard position, as shown in the following.

In particular, the height of this triangle shows us that sincos55=35∘∘, which is option C.

In our next example, we will use the cofunction identities to simplify an expression involving the tangent of an angle rotated 90∘ counterclockwise.

Example 4: Finding Equivalent Expressions Using the Cofunction Identity for the Tangent and Cotangent

Simplify tan(90+πœƒ)∘.

Answer

We can simplify this by applying two separate identities. First, we recall that the tangent function is the quotient of the sine and cosine functions, giving us tansincos(90+πœƒ)=(90+πœƒ)(90+πœƒ).∘∘∘

Next, we recall the following two correlated trigonometric identities: sincoscossin(90+πœƒ)=(πœƒ),(90+πœƒ)=βˆ’(πœƒ).∘∘ Substituting the correlated identities into the expression for the tangent function gives us tansincoscossincossin(90+πœƒ)=(90+πœƒ)(90+πœƒ)=(πœƒ)βˆ’(πœƒ)=βˆ’(πœƒ)(πœƒ).∘∘∘ Finally, we recall that the cotangent function is the reciprocal of the tangent function to get tancossincot(90+πœƒ)=βˆ’(πœƒ)(πœƒ)=βˆ’(πœƒ).∘

However, the above method relies on us committing the correlated angle identities to memory. We can also show this result by considering rotations and congruent triangles on the unit circle. Recall that if we sketch πœƒ in standard position, then the coordinates of the point of intersection with the terminal side and the unit circle are (πœƒ,πœƒ)cossin.

By rotating this triangle 90∘ counterclockwise about the origin, we get the following congruent triangle:

This is then angle 90+πœƒβˆ˜ in standard position, so the point of intersection is given by ((90+πœƒ),(90+πœƒ))cossin∘∘. By using the congruency of these triangles, and being careful to note that the π‘₯-coordinate will change sign, we find two equivalent expressions by equating the coordinates of the point of intersection with the triangle lengths: cossinsincos(90+πœƒ)=βˆ’πœƒ,(90+πœƒ)=πœƒ.∘∘ Once again we get tansincoscossincot(90+πœƒ)=(90+πœƒ)(90+πœƒ)=πœƒβˆ’πœƒ=βˆ’πœƒ.∘∘∘

It is important to reiterate that these arguments will work for any angle πœƒ measured in degrees in standard position. Therefore, tancot(90+πœƒ)=βˆ’(πœƒ)∘.

In our next example, we will apply a trigonometric identity to find an equivalent expression for the cotangent of an angle in terms of the tangent function.

Example 5: Finding Equivalent Expressions Using the Cofunction Identity for the Tangent and Cotangent

Which of the following is equivalent to 3(43)cot∘?

  1. 3(133)tan∘
  2. βˆ’3(47)tan∘
  3. 6(43)tan∘
  4. βˆ’6(133)tan∘
  5. 3(47)tan∘

Answer

We recall that the cofunction identity for the tangent function tells us that, for any angle πœƒ measured in degrees, tancot(90βˆ’πœƒ)=πœƒ.∘ By substituting πœƒ=43∘ into this identity, we see that cottancottan43=(90βˆ’43)43=(47).∘∘∘∘∘ Multiplying through by 3, we have 343=3(47),cottan∘∘ which is option E.

In our next example, we use the cofunction identities to simplify an equation.

Example 6: Finding Equivalent Expressions Using the Cofunction Identity for the Secant and Cosecant

Consider the equation 5(22βˆ’π‘₯)βˆ’π΄=3(22βˆ’π‘₯)secsec∘∘. Which of the following must be true?

  1. 𝐴=2(68+π‘₯)cosec∘
  2. 𝐴=2(2βˆ’π‘₯)cosec∘
  3. 𝐴=2(112+π‘₯)cosec∘
  4. 𝐴=2(68βˆ’π‘₯)cosec∘
  5. 𝐴=βˆ’2(22+π‘₯)cosec∘

Answer

We begin by rewriting the equation to make 𝐴 the subject: 5(22βˆ’π‘₯)βˆ’π΄=3(22βˆ’π‘₯)𝐴=2(22βˆ’π‘₯).secsecsec∘∘∘ We now want to simplify this by using the following cofunction identity: cossin(90βˆ’πœƒ)=πœƒ.∘ To do this, we take the reciprocal of both sides of the identity and rewrite it in terms of the reciprocal trigonometric functions: 1(90βˆ’πœƒ)=1πœƒ(90βˆ’πœƒ)=πœƒ.cossinseccsc∘∘ Substituting πœƒ=68+π‘₯∘ into this identity and simplifying, we get seccscseccsc(90βˆ’(68+π‘₯))=(68+π‘₯)(22βˆ’π‘₯)=(68+π‘₯).∘∘∘∘∘ Multiplying through by 2 then gives us 2(22βˆ’π‘₯)=2(68+π‘₯)𝐴=2(68+π‘₯),seccsccsc∘∘∘ which is option A.

In our final example, we will use the reciprocal trigonometric identities and cofunction identities to simplify a trigonometric expression.

Example 7: Finding Equivalent Expressions Using Reciprocal Trigonometric Identities and Cofunction Identities

Which of the following is an equivalent expression to 7(π‘₯+12)csc∘?

  1. 7(78βˆ’π‘₯)sin∘
  2. 7(π‘₯+12)cos∘
  3. 7(π‘₯βˆ’12)cos∘
  4. 7(78βˆ’π‘₯)cos∘
  5. sin(78βˆ’π‘₯)7∘

Answer

To answer this question, we start by simplifying the expression by using the reciprocal trigonometric identity as follows: 7(π‘₯+12)=7ο€Όοˆ=7(π‘₯+12).cscsin∘(ο—οŠ°οŠ§οŠ¨)∘sin∘ We can then rewrite this expression in terms of the cosine function by using the following cofunction identity: cossin(90βˆ’πœƒ)=πœƒ.∘ To make the right-hand side of this identity include our expression, we will first substitute πœƒ=π‘₯+12∘ into this cofunction identity and simplify to get cossincossin(90βˆ’(π‘₯+12))=(π‘₯+12)(78βˆ’π‘₯)=(π‘₯+12).∘∘∘∘∘ Then, we multiply through by 7: 7(78βˆ’π‘₯)=7(π‘₯+12)=7(π‘₯+12),cossincsc∘∘∘ which is option D.

Let’s finish by recapping some of the important points of this explainer.

Key Points

  • We can show the properties of trigonometric functions by considering the congruency of triangles formed by angles in standard position on the unit circle. In particular, we have the related angle trigonometric identities.
    For any angle πœƒ measured in degrees,
    • sinsin(180+πœƒ)=βˆ’(πœƒ)∘,
    • coscos(180+πœƒ)=βˆ’(πœƒ)∘,
    • tantan(180+πœƒ)=(πœƒ)∘.
    For any angle πœƒ measured in radians,
    • sinsin(πœ‹+πœƒ)=βˆ’(πœƒ),
    • coscos(πœ‹+πœƒ)=βˆ’(πœƒ),
    • tantan(πœ‹+πœƒ)=(πœƒ).
    We also have the following correlated angle identities:
    • sincos(90βˆ’πœƒ)=(πœƒ)∘ and sincos(90+πœƒ)=(πœƒ)∘,
    • cossin(90βˆ’πœƒ)=(πœƒ)∘ and cossin(90+πœƒ)=βˆ’(πœƒ)∘,
    • tancot(90βˆ’πœƒ)=(πœƒ)∘ and tancot(90+πœƒ)=βˆ’(πœƒ)∘.
    For any angle πœƒ measured in radians,
    • sincosο€»πœ‹2βˆ’πœƒο‡=(πœƒ) and sincosο€»πœ‹2+πœƒο‡=(πœƒ),
    • cossinο€»πœ‹2βˆ’πœƒο‡=(πœƒ) and cossinο€»πœ‹2+πœƒο‡=βˆ’(πœƒ),
    • tancotο€»πœ‹2βˆ’πœƒο‡=(πœƒ) and tancotο€»πœ‹2+πœƒο‡=βˆ’(πœƒ).
  • All of these identities and more can be directly shown from the symmetries of the unit circle. For the above identities, we can see these by considering congruent triangles in the following diagram. In the diagram on the left, we have the related angle identities, and in the diagram on the right, we have the correlated angle identities.

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