Lesson Explainer: Simplifying Trigonometric Expressions | Nagwa Lesson Explainer: Simplifying Trigonometric Expressions | Nagwa

Lesson Explainer: Simplifying Trigonometric Expressions Mathematics

In this explainer, we will learn how to simplify a trigonometric expression.

These expressions are often simplified upon the application of one or more trigonometric identity, which relate the different trigonometric and reciprocal trigonometric functions and their arguments. 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 us 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 sinOHcosAHtanOA𝜃=,𝜃=,𝜃=.

These functions satisfy the following trigonometric identity: tansincos𝜃=𝜃𝜃.

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

Suppose that a point moves along the unit circle in the counterclockwise direction. At a particular position (𝑥,𝑦) on the unit circle with angle 𝜃, the sine function is defined as 𝑦=𝜃sin and the cosine function as 𝑥=𝜃cos, as shown in the diagram above. In other words, the trigonometric functions are defined using the coordinates of the point of intersection of the unit circle with the terminal side of 𝜃 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 cscsinseccoscottancossin𝜃=1𝜃,𝜃=1𝜃,𝜃=1𝜃=𝜃𝜃.

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

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

Similarly, for the reciprocal trigonometric functions, we have csccscsecseccotcot(360+𝜃)=𝜃,(360+𝜃)=𝜃,(360+𝜃)=𝜃.

Similarly to the tangent function, the cotangent function is periodic by 𝜋, in radians, or 180 since we have cotcot(180+𝜃)=𝜃.

The trigonometric identities we will cover in this explainer hold for any angle 𝜃 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: if we have angle 𝜃degree, we can convert it to radians via 𝜃=𝜋180𝜃.radiansdegree

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 an example where we have to use reciprocal trigonometric functions to determine the value of a trigonometric expression.

Example 1: Using Reciprocal Identities to Evaluate Trigonometric Expressions

Find the value of 8𝜃×5𝜃sincsc.

Answer

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

One way of evaluating a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the cosecant function, which appears in the given expression: cscsin𝜃=1𝜃.

Therefore, the expression can be simplified to give 8𝜃×5𝜃=8𝜃×5=8𝜃×5𝜃=40×𝜃𝜃=40.sincscsinsinsinsinsinsin

Now, let’s consider an example where we simplify a particular trigonometric expression.

Example 2: Simplifying Trigonometric Expressions Using Trigonometric Identities

Simplify coscscsin𝜃𝜃𝜃.

Answer

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

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the cosecant function, which appears in the given expression: cscsin𝜃=1𝜃.

The given trigonometric expression becomes coscscsincossinsincos𝜃𝜃𝜃=𝜃×1𝜃×𝜃=𝜃.

In the next example, we will simplify a trigonometric expression by writing it out in terms of the sine and cosine functions.

Example 3: Simplifying Trigonometric Expressions Using Trigonometric Identities

Simplify tansinsec𝜃𝜃𝜃.

Answer

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

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definitions for the tangent and secant functions, which appear in the given expression: tansincosseccos𝜃=𝜃𝜃𝜃=1𝜃.

The given trigonometric expression becomes tansinsectansincossincossincossincoscossin𝜃𝜃𝜃=𝜃𝜃÷1𝜃=𝜃𝜃×𝜃𝜃=𝜃×𝜃𝜃=𝜃.

The next example involves a product of trigonometric and reciprocal trigonometric functions, which we can simply using the definition of the reciprocal functions and then rewriting the final answer in terms of another reciprocal function.

Example 4: Simplifying Trigonometric Expressions Using Reciprocal Identities

Simplify cosseccsc𝜃𝜃𝜃.

Answer

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

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definitions for the cosecant and secant functions, which appear in the given expression: cscsinseccos𝜃=1𝜃,𝜃=1𝜃.

Therefore, the expression can be simplified as cosseccsccoscossincossin𝜃𝜃𝜃=𝜃×1𝜃×1𝜃=𝜃𝜃.

Now, using the definition of the cotangent function, cotcossin𝜃=𝜃𝜃.

The given expression can be expressed in terms of the cotangent function as cosseccsccossincot𝜃𝜃𝜃=𝜃𝜃=𝜃.

The trigonometric and reciprocal trigonometric functions are even and odd functions as they satisfy the properties 𝑓(𝜃)=𝑓(𝜃), for even functions, and 𝑓(𝜃)=𝑓(𝜃), for odd functions. In particular, the sine function is odd, while the cosine function is even, since they satisfy sinsincoscos(𝜃)=𝜃,(𝜃)=𝜃, for any value of 𝜃 in either degrees or radians. From this, we can also determine the parity of the other trigonometric functions that are defined in terms of these. In particular, for the tangent function, we have tansincossincostan(𝜃)=(𝜃)(𝜃)=𝜃𝜃=𝜃.

Thus, the tangent function is odd, and we can deduce the parity of the other trigonometric functions in a similar manner. We can summarize these as follows.

Even and Odd Identities for Trigonometric Functions

The cosine and secant functions are even, which means for any value of 𝜃 in their respective domains, they satisfy the identities coscossecsec(𝜃)=𝜃,(𝜃)=𝜃.

And the sine, tangent, cosecant, and cotangent functions are odd, which means they satisfy the following identities for any value of 𝜃 in their respective domains: sinsintantancsccsccotcot(𝜃)=𝜃,(𝜃)=𝜃,(𝜃)=𝜃,(𝜃)=𝜃.

Now, let’s consider an example where we have to apply the parity of a trigonometric function in order to simply a particular trigonometric expression.

Example 5: Simplifying Trigonometric Expressions Involving Odd and Even Identities

Simplify tancsc(𝜃)𝜃.

Answer

In this example, we want to simplify a particular expression involving trigonometric and reciprocal trigonometric functions using an even/odd identity.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the cosecant function, which appears in the given expression: cscsin𝜃=1𝜃.

The tangent function is odd, hence the identity tantan(𝜃)=𝜃.

We can rewrite the tangent function using its definition in terms of the sine and cosine functions: tansincos𝜃=𝜃𝜃.

Therefore, the expression can be simplified as tancsctancscsincossincos(𝜃)𝜃=𝜃𝜃=𝜃𝜃×1𝜃=1𝜃.

Finally, we can rewrite this expression in terms of the secant function defined as seccos𝜃=1𝜃.

Thus, the expression becomes tancscsec(𝜃)𝜃=𝜃.

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

In particular, we have the following shift identities for the angles 𝜃 and 90+𝜃: sincoscossin(90+𝜃)=𝜃,(90+𝜃)=𝜃.

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

Similarly, by replacing 𝜃 with 𝜃, we obtain the following cofunction identities for the complementary angles 𝜃 and 90𝜃: sincoscossin(90𝜃)=𝜃,(90𝜃)=𝜃.

We can illustrate this as shown.

The figure depicts a right triangle with angle 𝐴𝑂𝐵 in standard position, which intersects the unit circle at 𝐵(𝑥,𝑦) and has an acute-angle measure 0<𝜃<90.

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 Correlated-Angle Identities

The trigonometric functions satisfy cofunction identities for all 𝜃 in their domains. In particular, we have sincoscossintancotcottancscsecseccsc(90±𝜃)=𝜃,(90±𝜃)=𝜃,(90±𝜃)=𝜃,(90±𝜃)=𝜃,(90±𝜃)=𝜃,(90±𝜃)=𝜃.

For example, for the tangent function, we have tansincoscossincossincot(90±𝜃)=(90±𝜃)(90±𝜃)=𝜃𝜃=𝜃𝜃=𝜃.

All of these identities also hold in radians, in particular, by replacing 90 with 𝜋2 in radians.

Now, let’s consider an example where we use this identity along with the parity of a trigonometric function in order to simplify an expression.

Example 6: Simplifying Trigonometric Expressions Using Correlated and Even Identities

Simplify sinsec𝜋2+𝜃(𝜃).

Answer

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

We will also make use of the correlated-angle identity sincos𝜋2+𝜃=𝜃 and the even identity secsec(𝜃)=𝜃.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definition for the secant function: seccos𝜃=1𝜃.

Using these, the expression becomes sinseccosseccoscos𝜋2+𝜃(𝜃)=𝜃𝜃=𝜃×1𝜃=1.

Now, suppose we want to determine sin(180𝜃). We can find this by repeatedly using the identities above. If we let 𝜃=90𝑥, then sinsinsincos(180𝜃)=(180[90𝑥])=(90+𝑥)=𝑥.

Now, substituting back 𝑥=90𝜃, we obtain sincossin(180𝜃)=(90𝜃)=𝜃.

Similarly, we find coscos(180𝜃)=𝜃.

By repeatedly applying these identities or using the unit circle, we also have the identities for the angles 𝜃 and 𝜃±180: sinsincoscos(180±𝜃)=𝜃,(180±𝜃)=𝜃.

For 𝜃 and 180𝜃, we have the following.

And for 𝜃 and 180+𝜃, we have the following.

We also have identities for the other trigonometric functions, which follow from those for the sine and cosine functions, from their definitions: tantancotcotcsccscsecsec(180±𝜃)=±𝜃,(180±𝜃)=±𝜃,(180±𝜃)=𝜃,(180±𝜃)=𝜃.

The next example involves using the definitions of the reciprocal trigonometric functions along with the cofunction identities in radians in order to simplify an expression.

Example 7: Using Periodic and Cofunction Identities to Simplify a Trigonometric Expression

Simplify seccot𝜃(𝜋𝜃).

Answer

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

We will also make use of the cofunction and correlated identities: seccsccotcot𝜋2𝜃=𝜃,(𝜋𝜃)=𝜃.

One way of simplifying a trigonometric expression is to write it in terms of the sine and cosine functions using the following definitions for the cosecant and cotangent functions, which appear in the numerator and denominator: cscsincotcossin𝜃=1𝜃,𝜃=𝜃𝜃.

The numerator of the expression can be simplified as seccscsin𝜋2𝜃=𝜃=1𝜃.

And the denominator as cotcotcossin(𝜋𝜃)=𝜃=𝜃𝜃.

Therefore, the expression can be simplified as seccotsinsincoscos𝜃(𝜋𝜃)=1𝜃×𝜃𝜃=1𝜃.

Finally, we can rewrite this expression in terms of the secant function defined as seccos𝜃=1𝜃.

Thus, we obtain seccotsec𝜃(𝜋𝜃)=𝜃.

Similarly, for the angles 𝜃 and 270±𝜃, we have sincoscossintancotcottancscsecseccsc(270±𝜃)=𝜃,(270±𝜃)=±𝜃.(270±𝜃)=𝜃,(270±𝜃)=𝜃,(270±𝜃)=𝜃,(270±𝜃)=±𝜃.

This can be visualized as follows.

By using the periodicity of the trigonometric functions and the unit circle, we have sinsincoscostantancotcotcsccscsecsec(360±𝜃)=±𝜃,(360±𝜃)=𝜃,(360±𝜃)=±𝜃,(360±𝜃)=±𝜃,(360±𝜃)=±𝜃,(360±𝜃)=𝜃.

All of the identities also hold in radians, by replacing 360 with 2𝜋 in radians. They can also be visualized using the unit circle as shown.

All of the correlated-angle identities can be visualized using the following.

Now, let us consider a few examples where we have to apply the cofunction identities in order to simplify a trigonometric expression. In the next example, we will use this identity repeatedly on the cosine and sine functions, in degrees.

Example 8: Simplifying Trigonometric Expressions Using Cofunction Identities

Simplify sincos𝜃+(270+𝜃).

Answer

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

In order to simplify the given expression, we make use of the correlated-angle identity cossin(270+𝜃)=𝜃.

Therefore, we have sincossinsinsin𝜃+(270+𝜃)=𝜃+𝜃=2𝜃.

In the last example, we want to apply the cofunction identities to the tangent and cotangent functions repeatedly, in degrees, in order to simplify a trigonometric expression.

Example 9: Using Trigonometric Identities to Simplify a Trigonometric Expression

Simplify sectantan𝜃𝜃(270+𝜃).

Answer

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

We will also make use of the correlated-angle identity tancot(270+𝜃)=𝜃.

Since we have, by definition of the cotangent function, cottan𝜃=1𝜃, the correlated identity can be written in terms of the tangent function as tancottan(270+𝜃)=𝜃=1𝜃.

Therefore, the expression can be simplified as sectantansectantansectantansec𝜃𝜃(270+𝜃)=𝜃𝜃×1𝜃=𝜃×𝜃𝜃=𝜃.

Let us 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 tansincoscscsinseccoscottancossin𝜃=𝜃𝜃,𝜃=1𝜃,𝜃=1𝜃,𝜃=1𝜃=𝜃𝜃. and we can use these to simplify trigonometric expressions.
  • These trigonometric functions are all either even or odd. In particular, for the sine and cosine functions, we have coscossinsin(𝜃)=𝜃,(𝜃)=𝜃, and similarly for the other trigonometric functions that follow from the definitions. We can use the parity of the trigonometric functions to help us simplify trigonometric expressions.
  • The unit circle allows us to determine the correlated-angle identities for sine and cosine.
    For instance, the cofunction identities (in radians) are, sincoscossin𝜋2𝜃=𝜃,𝜋2𝜃=𝜃. 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.

Download the Nagwa Classes App

Attend sessions, chat with your teacher and class, and access class-specific questions. Download the Nagwa Classes app today!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy