Lesson Explainer: Simplifying Trigonometric Expressions Using Trigonometric Identities Mathematics

In this explainer, we will learn how to simplify trigonometric expressions by applying trigonometric identities.

These expressions are often simplified upon the application of one or more trigonometric identities, which relate the different trigonometric and reciprocal trigonometric functions. Their motivation is mathematical, but they also have applications in real-world problems.

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 the following right triangle:

The trigonometric functions can be expressed in terms of the ratio of the sides of the triangle as sinOHcosAHtanOAπœƒ=,πœƒ=,πœƒ=.

These functions satisfy the following trigonometric identity: tansincosπœƒ=πœƒπœƒ.

We note that these trigonometric ratios are defined for acute angles 0<πœƒ<90∘∘ and the trigonometric functions for all values of πœƒ are defined on the unit circle.

Suppose that a point moves along the unit circle in the counterclockwise direction. At a particular position (π‘₯,𝑦) on the unit circle with angle πœƒ, the sine function is defined as 𝑦=πœƒsin and the cosine function as π‘₯=πœƒcos, 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 πœƒ in the standard position.

The trigonometric functions are periodic, which means that if we add an integer multiple of 2πœ‹, in radians, or 360∘, in degrees, to the angle πœƒ, the value of the function stays the same: sinsincoscostantan(360+πœƒ)=πœƒ,(360+πœƒ)=πœƒ,(360+πœƒ)=πœƒ.∘∘∘

We can see these directly from the unit circle definition of the trigonometric functions. In fact, the tangent function is periodic by πœ‹, in radians, or 180∘, in degrees, since we have tantan(180+πœƒ)=πœƒ.∘

The domain and range of the trigonometric functions, as well the trigonometric identities we will cover in this explainer, hold for any angle πœƒ in the domain of the functions, in degrees or radians. In particular, we can convert an angle between degrees and radians using the following rule.

Rule: Converting between Degrees and Radians

If we have angle πœƒdegree in degrees, we can convert it to radians via πœƒ=πœ‹180πœƒ.radiansdegree

Example 1: Defining the Pythagorean Identity

The figure shows a unit circle and a radius with the lengths of its π‘₯- and 𝑦-components. Use the Pythagorean theorem to derive an identity connecting the lengths 1, cosπœƒ, and sinπœƒ.

Answer

In this example, we will use the Pythagorean theorem to derive an identity connecting the lengths 1, cosπœƒ, and sinπœƒ of the right triangle.

If we denote the adjacent side as π‘Ž, the opposite side as 𝑏, and the hypotenuse side as 𝑐 in a right triangle, the Pythagorean identity states that π‘Ž+𝑏=𝑐.

For the given figure, the adjacent side is π‘Ž=πœƒcos, the opposite side is 𝑏=πœƒsin, and the hypotenuse is the radius of the unit circle 𝑐=1. Thus, from the Pythagorean theorem, we have (πœƒ)+(πœƒ)=1πœƒ+πœƒ=1.cossinsincos

Let’s now summarize the Pythagorean identity for sine and cosine, which we will explore in this explainer.

Definition: Pythagorean Identities for Trigonometric Functions

The Pythagorean identity for the trigonometric functions sine and cosine is given by sincosοŠ¨οŠ¨πœƒ+πœƒ=1.

Example 2: Simplifying Trigonometric Expressions Using Pythagorean Identities

Simplify (πœƒ+πœƒ)βˆ’2πœƒπœƒsincossincos.

Answer

In this example, we want to simplify a particular expression involving trigonometric functions.

In order to simplify the expression, we will make use of the identity sincosοŠ¨οŠ¨πœƒ+πœƒ=1.

Therefore, the expression can be rewritten as (πœƒ+πœƒ)βˆ’2πœƒπœƒ=ο€Ίπœƒ+2πœƒπœƒ+πœƒο†βˆ’2πœƒπœƒ=ο€Ίπœƒ+πœƒο†+(2πœƒπœƒβˆ’2πœƒπœƒ)=πœƒ+πœƒ=1.sincossincossinsincoscossincossincossincossincossincos

The next example involves applying the Pythagorean identity to rewrite an expression involving a sine and a cosine, in terms of a sine function only.

Example 3: Using the Pythagorean Identities to Simplify a Trigonometric Expression

Simplify 31πœƒ+26πœƒsincos.

Answer

In this example, we want to simplify a particular expression involving trigonometric functions.

In order to simplify the expression, we will make use if the identity sincosοŠ¨οŠ¨πœƒ+πœƒ=1.

Therefore, the given expression can be rewritten, by rearranging the Pythagorean identity cossinοŠ¨οŠ¨πœƒ=1βˆ’πœƒ, as 31πœƒ+26πœƒ=31πœƒ+26ο€Ί1βˆ’πœƒο†=31πœƒβˆ’26πœƒ+26=26+5πœƒ.sincossinsinsinsinsin

The reciprocal trigonometric functions are defined in terms of the standard trigonometric functions as follows.

Definition: Reciprocal Trigonometric Functions

The cosecant, secant, and cotangent functions are defined as cscsinseccoscottancossinπœƒ=1πœƒ,πœƒ=1πœƒ,πœƒ=1πœƒ=πœƒπœƒ.

The reciprocal trigonometric functions are also periodic: csccscsecseccotcot(360+πœƒ)=πœƒ,(360+πœƒ)=πœƒ,(360+πœƒ)=πœƒ.∘∘∘

Similarly to the tangent function, the cotangent function is periodic by πœ‹, in radians, or 180∘, in degrees, since we have cotcot(180+πœƒ)=πœƒ.∘

When dealing with trigonometric expressions, it is useful to rewrite the reciprocal trigonometric identities in terms of sine and cosine in order to simplify. Let’s consider specific examples where we have to use reciprocal trigonometric functions to simplify trigonometric expressions.

The next example involves simplifying a trigonometric expression using the definition of a reciprocal trigonometric function and the Pythagorean identity.

Example 4: Simplifying Trigonometric Expressions Using Pythagorean and Reciprocal Identities

Simplify sincsccosπœƒπœƒβˆ’πœƒοŠ¨.

Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions.

In order to find the value of the expression, we note that cscsinπœƒ=1πœƒ.

We will also make use of the identity sincosοŠ¨οŠ¨πœƒ+πœƒ=1.

Therefore, the expression can be simplified as sincsccossinsincoscossinπœƒπœƒβˆ’πœƒ=πœƒΓ—1πœƒβˆ’πœƒ=1βˆ’πœƒ=πœƒ.

We note that we can manipulate the Pythagorean identity sincosοŠ¨οŠ¨πœƒ+πœƒ=1 to derive the other identities for the reciprocal trigonometric functions. In particular, upon dividing by sinοŠ¨πœƒ, we have sincossinsincossinsincotcscοŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨πœƒ+πœƒπœƒ=1πœƒ1+πœƒπœƒ=1πœƒ1+πœƒ=πœƒ.

Similarly, by dividing by cosοŠ¨πœƒ, we can obtain the identity relating tanπœƒ and secπœƒ. For trigonometric expressions, it can be useful to begin by looking at how to apply this identity in specific examples when simplifying expressions. The first example involves expanding a quadratic form, then applying the Pythagorean identity to evaluate the final expression.

Reciprocal Trigonometric Pythagorean Identities

Recall that 1+πœƒ=πœƒ,πœƒ+1=πœƒ.cotcsctansec

Now, let’s consider a few examples where we use reciprocal trigonometric Pythagorean identities.

Example 5: Simplifying Trigonometric Expressions Using Pythagorean and Reciprocal Identities

Simplify 1πœƒβˆ’1πœƒsintan.

Answer

In this example, we will simplify trigonometric expression using Pythagorean and reciprocal identities.

We will make use of the definitions cscsincottanπœƒ=1πœƒ,πœƒ=1πœƒ and the Pythagorean identity 1+πœƒ=πœƒ,cotcsc which we can rearrange as csccotοŠ¨οŠ¨πœƒβˆ’πœƒ=1.

From the given expression, upon using the definition of the reciprocal trigonometric function and applying the Pythagorean identity, we have 1πœƒβˆ’1πœƒ=(πœƒ)βˆ’(πœƒ)=πœƒβˆ’πœƒ=1.sintancsccotcsccot

Now, let’s look at some examples where we have to use these reciprocal trigonometric Pythagorean identities to simplify trigonometric expressions. The first example involves two such identities.

Example 6: Using Pythagorean Identities to Simplify a Trigonometric Expression

Simplify sincoscsccotοŠ¨οŠ¨οŠ¨οŠ¨πœƒ+πœƒπœƒβˆ’πœƒ.

Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions using trigonometric Pythagorean identities.

In particular, we will make use of the identities sincoscotcscοŠ¨οŠ¨οŠ¨οŠ¨πœƒ+πœƒ=1,1+πœƒ=πœƒ.

We can rearrange the second identity as csccotοŠ¨οŠ¨πœƒβˆ’πœƒ=1.

Therefore, the expression can be simplified as sincoscsccotοŠ¨οŠ¨οŠ¨οŠ¨πœƒ+πœƒπœƒβˆ’πœƒ=1.

In the next example, we want to simplify an expression by expanding a quadratic form and applying the Pythagorean identity involving the tangent function.

Example 7: Simplifying Trigonometric Expressions Using Pythagorean Identities

Simplify (1βˆ’πœƒ)+(1+πœƒ)tantan.

Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions using a trigonometric Pythagorean identity.

In particular, we will make use of the identity tansecοŠ¨οŠ¨πœƒ+1=πœƒ.

Upon expanding the expression, we can rewrite it using the identity as (1βˆ’πœƒ)+(1+πœƒ)=ο€Ή1βˆ’2πœƒ+πœƒο…+ο€Ή1+2πœƒ+πœƒο…=2+2πœƒ=2ο€Ή1+πœƒο…=2πœƒ.tantantantantantantantansec

The sine function is equivalent to the cosine function by a translation 90∘ to the left, which can be visualized by comparing the plots of both functions.

In particular, we have the following shift identities for the angles πœƒ and 90+πœƒβˆ˜: sincoscossin(90+πœƒ)=πœƒ,(90+πœƒ)=βˆ’πœƒ.∘∘

We can also illustrate these on the unit circle as shown:

Similarly, by replacing πœƒ with βˆ’πœƒ, we obtain the following cofunction identities for the complementary angles πœƒ and 90βˆ’πœƒβˆ˜: sincoscossin(90βˆ’πœƒ)=πœƒ,(90βˆ’πœƒ)=πœƒ.∘∘

We can illustrate this as shown:

The figure depicts the right triangle with angle 𝐴𝑂𝐡 in the standard position, which intersects the unit circle at 𝐡(π‘₯,𝑦), and whose angle measure is acute: 0<πœƒ<90∘∘.

We can use these cofunction identities along with Pythagorean identities to simplify trigonometric expressions. For example, consider the expression sincoscossinsincosπœƒ(90βˆ’πœƒ)+πœƒ(90βˆ’πœƒ)=πœƒ+πœƒ=1.∘∘

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.

Definition: Trigonometric Corelated Angle Identities

The trigonometric functions satisfy cofunction identities for all πœƒ in their domains. In particular, we have sincoscossintancotcottancscsecseccsc(90Β±πœƒ)=πœƒ,(90Β±πœƒ)=βˆ“πœƒ,(90Β±πœƒ)=βˆ“πœƒ,(90Β±πœƒ)=βˆ“πœƒ,(90Β±πœƒ)=πœƒ,(90Β±πœƒ)=βˆ“πœƒ.∘∘∘∘∘∘

For example, for the tangent function, we have tansincoscossincossincot(90Β±πœƒ)=(90Β±πœƒ)(90Β±πœƒ)=πœƒβˆ“πœƒ=βˆ“πœƒπœƒ=βˆ“πœƒ.∘∘∘

All of these identities also hold in radiansβ€”in particular, by replacing 90∘ in degrees with πœ‹2 in radians.

Now, let’s consider an example where we use these identities to simplify a trigonometric expression.

Now, suppose we want to determine sin(180βˆ’πœƒ)∘. We can find this by repeatedly using the identities above. If we let πœƒ=90βˆ’π‘₯, then sinsincos(180βˆ’πœƒ)=(90+π‘₯)=π‘₯.∘∘

Now, substituting back π‘₯=90βˆ’πœƒ, we obtain sincossin(180βˆ’πœƒ)=(90βˆ’πœƒ)=πœƒ.∘∘

Similarly, we find coscos(180βˆ’πœƒ)=βˆ’πœƒ.∘

By repeatedly applying these identities or using the unit circle, we also have the identities for the angles πœƒ and πœƒΒ±180∘: sinsincoscos(180Β±πœƒ)=βˆ“πœƒ,(180Β±πœƒ)=βˆ’πœƒ.∘∘

For πœƒ and 180βˆ’πœƒβˆ˜, we have the picture shown:

And for πœƒ and 180+πœƒβˆ˜, we have the following:

Similarly, for the angles πœƒ and 270Β±πœƒβˆ˜, we have sincoscossin(270Β±πœƒ)=βˆ’πœƒ,(270Β±πœƒ)=Β±πœƒ.∘∘

Now, let’s consider an example where we use the cofunction and Pythagorean identities to simplify a trigonometric expression.

Example 8: Simplifying Trigonometric Expressions Using Corelated Angle Identities

Simplify sinsin∘∘(180βˆ’πœƒ)+(270βˆ’πœƒ).

Answer

In this example, we will simplify a trigonometric expression using a cofunction and a Pythagorean identity.

The Pythagorean identity we will make use of is sincosοŠ¨οŠ¨πœƒ+πœƒ=1, along with the corelated angle identities sinsinsincos(180βˆ’πœƒ)=πœƒ,(270βˆ’πœƒ)=βˆ’πœƒ.∘∘

Thus, the given expression becomes sinsinsincossincos∘∘(180βˆ’πœƒ)+(270βˆ’πœƒ)=(πœƒ)+(βˆ’πœƒ)=πœƒ+πœƒ=1.

We also have identities for the other trigonometric functions, which follow from those for the sine and cosine functions from their definitions: tantancotcotcsccscsecsec(180Β±πœƒ)=Β±πœƒ,(180Β±πœƒ)=Β±πœƒ,(180Β±πœƒ)=βˆ“πœƒ,(180Β±πœƒ)=βˆ’πœƒβˆ˜βˆ˜βˆ˜βˆ˜ and tancotcottancscsecseccsc(270Β±πœƒ)=βˆ“πœƒ,(270Β±πœƒ)=βˆ“πœƒ,(270Β±πœƒ)=βˆ’πœƒ,(270Β±πœƒ)=Β±πœƒ.∘∘∘∘

By using the periodicity of the trigonometric functions and the unit circle, we have sinsincoscostantancotcotcsccscsecsec(360Β±πœƒ)=Β±πœƒ,(360Β±πœƒ)=πœƒ,(360Β±πœƒ)=Β±πœƒ,(360Β±πœƒ)=Β±πœƒ,(360Β±πœƒ)=Β±πœƒ,(360Β±πœƒ)=πœƒ.∘∘∘∘∘∘

All of the identities also hold in radians by replacing 360∘ in degrees with 2πœ‹ in radians. They can also be visualized using the unit circle as follows:

All of the identities that we derived can be visualized using the unit circle as shown:

Let’s consider an example where we use these identities, in radians, to simplify a particular trigonometric expression.

Example 9: Simplifying Trigonometric Expressions Using Cofunction and Pythagorean Identities

Simplify 1+ο€»βˆ’πœƒο‡1+ο€»βˆ’πœƒο‡cottanοŠ¨οŠ©οŽ„οŠ¨οŠ¨οŽ„οŠ¨.

Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions using cofunction identities.

We will make use of the definitions of the reciprocal trigonometric functions tansincosseccoscscsinπœƒ=πœƒπœƒ,πœƒ=1πœƒ,πœƒ=1πœƒ; the Pythagorean identities 1+πœƒ=πœƒ,πœƒ+1=πœƒ;cotcsctansec and the corelated angle identities cottantancotο€Ό3πœ‹2βˆ’πœƒοˆ=πœƒ,ο€»πœ‹2βˆ’πœƒο‡=πœƒ.

Thus, the given expression becomes 1+ο€»βˆ’πœƒο‡1+ο€»βˆ’πœƒο‡=1+πœƒ1+πœƒ=πœƒπœƒ=1πœƒΓ—πœƒ=πœƒπœƒ=ο€½πœƒπœƒο‰=πœƒ.cottantancotseccsccossinsincossincostanοŠ¨οŠ©οŽ„οŠ¨οŠ¨οŽ„οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨

Finally, let’s look at an example where we use this identity, along with others, to simplify an expression.

Let’s finish by recapping a few important key points from this explainer.

Key Points

  • We can express the tangent and reciprocal trigonometric functions in terms of sine and cosine as tansincoscscsinseccoscottancossinπœƒ=πœƒπœƒ,πœƒ=1πœƒ,πœƒ=1πœƒ,πœƒ=1πœƒ=πœƒπœƒ.
  • The Pythagorean identities are given by sincoscotcsctansecοŠ¨οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨πœƒ+πœƒ=1,1+πœƒ=πœƒ,πœƒ+1=πœƒ.
  • The unit circle allows us to determine the related and corelated angle identities for sine and cosine:
    For instance, the cofunction identities (in radians) are sincoscossinο€»πœ‹2βˆ’πœƒο‡=πœƒ,ο€»πœ‹2βˆ’πœƒο‡=πœƒ. The corresponding identities for the tangent and reciprocal trigonometric functions are found using their definitions in terms of the sine and cosine functions.
  • We often need to apply more than one identity, or type of identity, to simplify a trigonometric expression.

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