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 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 and .
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 (or radians) counterclockwise of the angle 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, and for any integer . 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 and angle measured in degrees,
- ,
- ,
- .
For any integer and angle measured in radians,
- ,
- ,
- .
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 in the -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 and the width is . We can use this to determine the coordinates of the intersection between the hypotenuse and the unit circle as . We can notice that the hypotenuse is the terminal side of the angle 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 . This gives us Equating the coordinates gives This is true for any angle 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 or radians; we call these supplementary angles. We can summarize the identities for supplementary angles as follows.
Definition: Supplementary Angle Trigonometric Identities
For any angle measured in degrees,
- ,
- ,
- .
For any angle measured in radians,
- ,
- ,
- .
By taking the reciprocal of both sides of any of these identities, we can find similar identities for the reciprocal trigonometric functions.
For example,
We can follow this same process to find trigonometric identities of other related angles. For example, consider the related angles and as shown on the unit circle in standard position.
We can think of this as a rotation 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
We can extend this to the tangent function and summarize these as follows.
Definition: Related Angle Trigonometric Identities
For any angle measured in degrees,
- ,
- ,
- .
For any angle measured in radians,
- ,
- ,
- .
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 and ; we can draw both of these angles in standard position as follows.
These angles are related by a reflection in the -axis; this gives us the identities We can extend these identities to the tangent function and summarize them as follows.
Definition: More Related Angle Trigonometric Identities
For any angle measured in degrees,
- ,
- ,
- .
For any angle measured in radians,
- ,
- ,
- .
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 is one angle in a right triangle, then the other angle is the complementary angle . We can use this to find a set of identities called the cofunction identities. First, we draw in standard position, as follows:
We can include the complementary angle in our diagram as shown. Since , we can construct the following congruent triangle:
This is the angle 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
These are the cofunction identities, and they are true for any angle 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 measured in degrees,
- ,
- ,
- .
For any angle measured in radians,
- ,
- ,
- .
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 and . Consider an acute angle in standard position, where we construct the flowing right triangle to determine the values of and .
By rotating the triangle counterclockwise about the origin and by considering congruent triangles, we get the following:
This is the angle 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 This was shown only by using rotation and congruency conditions; it will also hold for any angle in standard position. We can also do the same for rotations of or 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 measured in degrees,
- ,
- ,
- .
For any angle measured in radians,
- ,
- ,
- .
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 .
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 measured in degrees, Substituting this into the expression then gives us
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 fromβsymmetries of the unit circle. If we sketch an acute angle in standard position, then the angle is the supplementary angle to ; together they form a straight line on the -axis as shown.
The coordinates of the intersection between the unit circle and the terminal side of give us the cosine and sine of angle . If we reflect this in the vertical axis, we will then have the angle 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 , and equating this with the -coordinate of the point of intersection (and being careful with the signs), we have This is true for any angle measured in degrees; we then substitute this into the original expression to get
Hence, for any angle measured in degrees, .
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 ?
Answer
There are several different ways of approaching this problem; one such way is to recall the cofunction identities. We know that so
Next, recall the following identity for supplementary angles: If we substitute , we see that
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 . For simplicity, we will assume 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 in standard position and using congruency.
We see that adding to the argument rotates the triangle radians counterclockwise about the origin and adding to the argument rotates the triangle 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 in standard position to find an expression for the - and -coordinates of each point of intersection; we get the following 4 identities: We see that the -coordinate of the point of intersection between the unit circle and the terminal side of in standard position is .
Hence, for any angle measured in radians, 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 , which of the following is equivalent to ?
Answer
Substituting into the identity gives us which is option C.
We can see why this is true by sketching the angle 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 point of intersection will be . We see that is the -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 . Therefore, if was drawn in standard position, it would be congruent to the triangle drawn for in standard position, as shown in the following.
In particular, the height of this triangle shows us that , 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 counterclockwise.
Example 4: Finding Equivalent Expressions Using the Cofunction Identity for the Tangent and Cotangent
Simplify .
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
Next, we recall the following two correlated trigonometric identities: Substituting the correlated identities into the expression for the tangent function gives us Finally, we recall that the cotangent function is the reciprocal of the tangent function to get
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 in standard position, then the coordinates of the point of intersection with the terminal side and the unit circle are .
By rotating this triangle counterclockwise about the origin, we get the following congruent triangle:
This is then angle in standard position, so the point of intersection is given by . By using the congruency of these triangles, and being careful to note that the -coordinate will change sign, we find two equivalent expressions by equating the coordinates of the point of intersection with the triangle lengths: Once again we get
It is important to reiterate that these arguments will work for any angle measured in degrees in standard position. Therefore, .
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 ?
Answer
We recall that the cofunction identity for the tangent function tells us that, for any angle measured in degrees, By substituting into this identity, we see that Multiplying through by 3, we have 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 . Which of the following must be true?
Answer
We note that the answers to this question all use βcosecβ, which is the abbreviated form of the cosecant function more typically denoted by csc.
We begin by rewriting the equation to make the subject: We now want to simplify this by using the following cofunction identity: To do this, we take the reciprocal of both sides of the identity and rewrite it in terms of the reciprocal trigonometric functions: Substituting into this identity and simplifying, we get Multiplying through by 2 then gives us 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 ?
Answer
To answer this question, we start by simplifying the expression by using the reciprocal trigonometric identity as follows: We can then rewrite this expression in terms of the cosine function by using the following cofunction identity: To make the right-hand side of this identity include our expression, we will first substitute into this cofunction identity and simplify to get Then, we multiply through by 7: which is option D.
Letβs 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 measured in degrees,- ,
- ,
- .
- ,
- ,
- .
- and ,
- and ,
- and .
- and ,
- and ,
- and .
- 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 top,
we have the related angle identities, and in the second diagram, we have
the correlated angle identities.
