In this explainer, we will learn how to use the Pythagorean identities to find the values of trigonometric functions.
These values of trigonometric functions are often evaluated upon the application of one or more Pythagorean identity, which relate the different trigonometric and reciprocal trigonometric functions.
Trigonometric identities have several real-world applications in various fields such as physics, engineering, architecture, robotics, music theory, and navigation, to name a few. In physics, they can be used in projectile motion, modeling the mechanics of electromagnetic waves, analyzing alternating and direct currents, and finding the trajectory of a mass around a massive body under the force of gravity.
Let’s begin by recalling the trigonometric functions, whose Pythagorean identities we will examine in this explainer. Consider a right triangle:
The trigonometric functions can be expressed in terms of the ratio of the sides of the triangle as
These functions satisfy the following trigonometric identity:
We note that these trigonometric ratios are defined for acute angles , and the trigonometric functions for all values of are defined on the unit circle.
Suppose that a point moves along the unit circle in the counterclockwise direction. At a particular position on the unit circle with angle , the sine function is defined as and the cosine function as , as shown in the diagram above. In other words, the trigonometric functions are defined by using the coordinates of the point of intersection of the unit circle with the terminal side of in the standard position.
The reciprocal trigonometric equations are defined in terms of the standard trigonometric equations as follows.
Definition: Reciprocal Trigonometric Functions
The cosecant, secant, and cotangent functions are defined as
Let’s also state the Pythagorean identities, which we will explore in this explainer.
Definition: Pythagorean Identities for Trigonometric Functions
The Pythagorean identities for the trigonometric equations are given by
We note that we can manipulate the Pythagorean identity to derive the other identities for the reciprocal trigonometric functions. In particular, upon dividing by , we have
Similarly, dividing by , we can obtain the identity relating and :
This means we do not have to memorize all three Pythagorean identities, as we can derive all the other identities from the most basic one involving the sine and cosine functions.
Let’s find the value of a particular expression by applying an identity, with no other information. This means the values will always be the same, no matter what the angle is or the value of the trigonometric function at that angle.
Example 1: Applying the Pythagorean Identities to Evaluate Some Expressions
Find the value of .
Answer
In this example, we will find the value of a particular trigonometric expression by applying a Pythagorean identity.
Since this example involves sine and cosine functions, we recall the Pythagorean identity
If we distribute the parentheses in the given expression and apply this identity, we obtain
The sine function is equivalent to the cosine function by a translation to the left, which can be visualized by comparing the plots of both functions.
In particular, we have the following shift identities for angles and :
We can also illustrate these on the unit circle as follows:
Similarly, by replacing with , we obtain the following cofunction identities for the complementary angles and :
We can illustrate this as follows:
The figure depicts the right triangle with angle in the standard position, which intersects the unit circle at and its angle measure is acute, .
We can use these cofunction identities along with the Pythagorean identities to simplify trigonometric expressions. For example, if we have the expression
we can combine these identities and use them to determine identities for the other trigonometric functions that are defined in terms of the sine and cosine functions.
The trigonometric functions satisfy cofunction identities for all in their domains. In particular, we have
We can derive the cofunction identities for other trigonometric functions by relating them to the identities involving sine and cosine. For example, for the tangent function, we have
All of these identities also hold in radians, in particular, by replacing in degrees with in radians.
Now, let’s consider an example where we use this identity with the sine function to simplify a given trigonometric expression with a Pythagorean identity.
Example 2: Simplifying Trigonometric Expressions Using Cofunction and Pythagorean Identities
Simplify .
Answer
In this example, we will find the value of a particular trigonometric expression by applying a Pythagorean and cofunction identity.
Since this example involves sine function, we recall the Pythagorean identity
and the cofunction identity
Using this cofunction identity on the given expression and applying the Pythagorean identity, we obtain
We can determine the value of a trigonometric expression from the value of a trigonometric function. In the next example, we will determine the value of an expression using information about the lengths of a right triangle.
Example 3: Evaluating Pythagorean Identities for Angles in Right Triangles
Find , given that is a right triangle at , where and .
Answer
In this example, we will find the value of a particular trigonometric expression by using information about the lengths of a right triangle, to determine the value of a particular trigonometric function, and applying Pythagorean identities.
We can find the value of the desired expression using two different methods. Firstly, using Pythagorean identities, and secondly, using the Pythagorean theorem. Let’s begin with the method of finding this value using Pythagorean identities.
Method 1
Since this example involves the tangent function, we recall the Pythagorean identity
By definition, the secant function is
Side is opposite to angle , and side is the hypotenuse of the right triangle. Hence, we can find the sine of this angle by taking the ratio of these lengths. We have
By the Pythagorean identity and the definition of the secant function,
Let’s first determine the denominator of the right-hand side of this expression, which depends on the value of , where we know the value of . Since this involves the sine and cosine functions, recall the Pythagorean identity relating these two values:
which we can rearrange as and where we substitute the given value for the sine function to obtain
Thus, for the given expression, we have
Method 2
We could also determine this value by finding the adjacent side of the right triangle from the Pythagorean theorem and using this to compute the value of , given as the ratio of the opposite and adjacent sides. By the Pythagorean theorem on the right triangle, we have
Thus, we have . Now we can determine the tangent of the angle by the ratio of the opposite side, , and the adjacent side, :
We note that this is the same as the ratio of the value given by the sine and cosine of the angle, since implies that , as is an acute angle, and hence
Now, we can square this value and add 1 to find the value of the required expression:
In the next example, we will determine the value of the square of a trigonometric function, , from the given value of the square of another trigonometric function, , by using the Pythagorean identities.
Example 4: Using the Pythagorean Identities to Evaluate a Trigonometric Function
Find the value of given .
Answer
In this example, we will find the value of the square of a trigonometric function from the value of the square of another trigonometric function.
Since this example involves cotangent and cosecant functions, we recall the Pythagorean identity
Thus, if we rearrange this expression to make the subject and substitute , we have
We can also determine the value of a trigonometric expression from the value of another expression, without the actual values of the functions, by applying the Pythagorean identity. Let’s consider an example of this.
Example 5: Using the Pythagorean Identities to Evaluate a Trigonometric Expression
Find the value of given .
Answer
In this example, we will find the value of a particular trigonometric expression from the value of another expression.
Since this example involves sine and cosine functions, we recall the Pythagorean identity
Since the Pythagorean identity involves squares of sine and cosine functions, it make sense to begin by squaring both sides of the equation. If we take the square of the given expression , distribute the parentheses, and apply this identity, we obtain
Now, let’s determine the value of another trigonometric expression, this time using multiple Pythagorean identities and the definition of the reciprocal trigonometric functions.
Example 6: Evaluating Trigonometric Expressions given Trigonometric Equations
Find the value of given .
Answer
In this example, we will find the value of a particular trigonometric expression from the value of another expression.
We can find the value of the given expression in two different ways: firstly, using Pythagorean identities and, secondly, by squaring the given equation and simplifying the resulting equation. Let’s begin with the method using Pythagorean identities.
Method 1
Since this example involves tangent and cotangent functions, we recall the Pythagorean identities
Note that the cosecant, secant, and cotangent functions are defined as
The left-hand side of the given trigonometric expression can be written as
Since the left-hand side of this expression involves the sine and cosine functions, we recall the Pythagorean identity
which can be used to simplify the numerator of the expression
Thus, the given expression is equivalent to
Now, for the expression we want to find the value of, we can apply the Pythagorean identities to obtain
Method 2
We note that we could have also found this value without applying the Pythagorean identities by squaring the given expression, , and using the definition of the cotangent function:
Thus, we obtain the value as
So far, the examples we have considered involved finding the value of a trigonometric expression by applying the Pythagorean identities. For some examples, we also used known values of a trigonometric function or another expression. But what if we want to determine the value of a trigonometric function from the value of another trigonometric expression? In some cases, we may need to know what quadrant the angle lies in because the signs of the trigonometric functions can be different for each quadrant.
The Pythagorean identities involve squares of values of trigonometric functions, so we may need to take the square root to obtain the value of another trigonometric function, which can be positive or negative. The sign of the trigonometric functions will depend on which quadrant we consider. The CAST diagram helps us to remember the signs of the trigonometric functions, for each quadrant.
Let’s recall the CAST diagram.
Rule: The Cast Diagram
In the first quadrant, all trigonometric functions are positive.
In the second quadrant, the sine function is positive.
In the third quadrant, the tangent function is positive.
In the fourth quadrant, the cosine function is positive.
The Pythagorean identities allow us to determine the value of a trigonometric function, say , from given the value of another trigonometric function, . From the first Pythagorean identity , we obtain
The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants. In other words, the value of the cosine function will depend on the value of and which quadrant angle lies.
Now, let’s look at an example where we determine the value of the cosine function from the given value of the sine value in a particular quadrant, using the Pythagorean identity.
Example 7: Using the Sine Ratio to Find the Cosine of an Angle
Find given , where .
Answer
In this example, we will find the value of a particular trigonometric function from the value of another function, in a particular quadrant.
Since this example involves sine and cosine functions, we recall the Pythagorean identity
If we substitute and rearrange this, we find
Since , this corresponds to the fourth quadrant. We recall from the CAST diagram that the sign of the cosine function is positive. Thus, taking the positive sign,
In fact, we can write the values of the other trigonometric function in terms of and using the value for the cosine function, which follows from the other Pythagorean identities or the definition of the functions in terms of sine and cosine. For example, for the tangent function, we have
The sign of each trigonometric function will depend on the sign of and the quadrant in which angle lies.
In the next example, we will determine the value of the tangent function from the given value of the sine function in a particular quadrant using the Pythagorean identity.
Example 8: Using the Pythagorean Identity to Find the Value of a Trigonometric Function given Another One and the Quadrant of the Angle
Knowing that and , find .
Answer
In this example, we will find the value of a particular trigonometric function from the value of another function, in a particular quadrant.
Since this example involves sine and cosine functions, we recall the Pythagorean identity
Note that, by definition of the tangent function, we have
Let’s first determine the denominator of this expression. Rearranging the Pythagorean identity and taking the square root, we obtain
Since , which is the second quadrant, the cosine function is negative, and we take the negative root:
Substituting the given value , we obtain
Thus, for the tangent function, we obtain
Now, let’s consider an example where we determine the value of the secant function from a given trigonometric expression in the first quadrant, where all the trigonometric functions are positive, by applying a Pythagorean identity.
Example 9: Using the Pythagorean Identities to Evaluate a Trigonometric Function of an Angle
Find the value of given , where .
Answer
In this example, we will find the value of a particular trigonometric function from the value of a given trigonometric expression, in a particular quadrant.
Since this example involves tangent and secant functions, we recall the Pythagorean identity
We can also rewrite this identity as . Using the difference of squares formula, , we can rewrite this identity as
If we substitute the given expression, , we have
Now, we can substitute in order to eliminate from the expression and solve for :
Instead of finding the value of particular trigonometric functions, we can determine the value of a particular expression from a given value or expression.
Now, let’s consider an example where we determine the value of an expression from a given value in the third quadrant.
Example 10: Using the Pythagorean Identities to Evaluate a Trigonometric Expression given a Trigonometric Function and the Quadrant of an Angle
Find the value of given and .
Answer
In this example, we will find the value of a particular trigonometric expression from the value of a trigonometric function. We will first determine the value of the sine and cosine functions, separately, then take the product to find the value of the given expression.
Note that the cosecant and cotangent functions are defined as
For the given expression, we can substitute these and use the Pythagorean identity to obtain an expression in terms of as
Since the right-hand side involves a cosine function, and we are given the value of the sine function, we will make use of the Pythagorean identity:
hence, we obtain
Now, we can substitute the given value, , to obtain
Finally, let’s consider an example where we find the value of a trigonometric expression from a given value of the tangent function in the third and fourth quadrants by first determining the value of the sine and cosine functions and substituting this into the given expression.
Example 11: Using the Pythagorean Identities to Evaluate a Trigonometric Expression
Find the value of given , where .
Answer
In this example, we will find the value of a particular trigonometric expression from the value of a trigonometric function. We will first determine the value of the sine and cosine functions, separately, then take the product to find the value of the given expression.
We can write the given expression, , as a value for the tangent function:
We recall from the CAST diagram that the tangent function is positive in the third quadrant and negative in the fourth quadrant. Since the tangent function takes a negative value and we are given that the angle lies in either the third or the fourth quadrant, corresponding to the range , angle must lie in the fourth quadrant.
Since we know the value of the tangent function, we can determine the value of the cosine function from the Pythagorean identity involving the tangent and secant functions:
with the definition of the secant function in terms of the cosine function
We can rearrange this Pythagorean identity to make the subject as
Since angle lies in the fourth quadrant, we remember from the CAST diagram that the cosine function, hence the secant function, is positive in the fourth quadrant. Upon substituting the value of the tangent function from the given expression, we have
We can also obtain the value of the cosine function by taking the reciprocal of this to obtain
Now, since we know the value of the cosine function, we can determine the value of the sine function using the Pythagorean identity
We can rearrange this identity, involving the sine and cosine functions, to make the subject:
Since the sine function is negative in the fourth quadrant, we have
Now, we can determine the value of the trigonometric expression by substituting the values of sine and cosine:
Let’s finish by recapping a few important key points from this explainer.
Key Points
The Pythagorean identities are given by
The unit circle allows us to determine the correlated angle identities for sine and cosine.
For instance, the cofunction identities (in radians) are
The corresponding identities for the tangent and reciprocal trigonometric functions are found using their definitions in terms of the sine and cosine functions.
We may need to apply more than one Pythagorean identity, or type of identity, to simplify a trigonometric expression.
Pythagorean identities can only produce the absolute value of a trigonometric function. We need to use the CAST diagram to determine the sign of a trigonometric function within a quadrant.
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