Lesson Explainer: Simplifying Trigonometric Expressions Using Trigonometric Identities Mathematics • First Year of Secondary School

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

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 trigonometric functions are periodic, which means that if we add an integer multiple of , in radians, or , in degrees, to the angle , the value of the function stays the same:

We can see these directly from the unit circle definition of the trigonometric functions. In fact, the tangent function is periodic by , in radians, or , in degrees, since we have

The domain and range of the trigonometric functions, as well the trigonometric identities we will cover in this explainer, hold for any angle 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.

Example 1: Defining the Pythagorean Identity

The figure shows a unit circle and a radius with the lengths of its - and -components. Use the Pythagorean theorem to derive an identity connecting the lengths 1, , and .

Answer

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

If we denote the adjacent side as , the opposite side as , and the hypotenuse side as in a right triangle, the Pythagorean identity states that

For the given figure, the adjacent side is , the opposite side is , and the hypotenuse is the radius of the unit circle . Thus, from the Pythagorean theorem, we have

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

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.

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

The reciprocal trigonometric functions are also periodic:

Similarly to the tangent function, the cotangent function is periodic by , in radians, or , in degrees, since we have

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.

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, by dividing by , we can obtain the identity relating and . 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.

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

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.

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

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 the angles and :

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

Similarly, by replacing with , we obtain the following cofunction identities for the complementary angles and :

We can illustrate this as shown:

The figure depicts the right triangle with angle in the standard position, which intersects the unit circle at , and whose angle measure is acute: .

We can use these cofunction identities along with Pythagorean identities to simplify trigonometric expressions. For example, consider 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.

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 these identities to simplify a trigonometric expression.

Now, suppose we want to determine . We can find this by repeatedly using the identities above. If we let , then

Now, substituting back , we obtain

Similarly, we find

By repeatedly applying these identities or using the unit circle, we also have the identities for the angles and :

For and , we have the picture shown:

And for and , we have the following:

Similarly, for the angles and , we have

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

We also have identities for the other trigonometric functions, which follow from those for the sine and cosine functions from their definitions: and

By using the periodicity of the trigonometric functions and the unit circle, we have

All of the identities also hold in radians by replacing in degrees with 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.

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.