In this explainer, we will learn how to derive the angle sum and difference identities, graphically or using the unitary circle, and use them to find trigonometric values.
The trigonometric angle sum and difference identities have been used in mathematics for centuries to solve real-world problems. The ancient Greeks used these formulae to solve astronomy problems such as finding the distance between Earth and the Sun.
Consider the expression . We will use the unit circle to demonstrate the identity that relates to this angle sum expression.
The following image shows a portion of the unit circle such that , , and are of unit length and is made up of two angles, and , where and . We observe that is in standard position and the initial side of is the terminal side of . Points and are the points of intersections of the terminal sides of and with the unit circle.
We will add some perpendiculars to this diagram to create a series of right triangles to which we can apply right triangle trigonometry; is perpendicular to , and are perpendicular to the -axis, and is perpendicular to as shown.
Next, we observe that and , since angles in a triangle sum to . Then , since vertically opposite angles are equal; hence, . Further, , as shown in the following diagram.
Using right triangle trigonometry on triangle allows us to find a relationship between and :
However, since ,
Then, using the sine ratio on triangle ,
Using the cosine ratio on triangle ,
Similarly, for triangle ,
And for triangle ,
Combining equations (1), (2), and (3), we obtain the angle sum identity for the sine function:
This is a demonstration of a proof of one of the three angle sum identities. While we made assumptions on and , and , our proof can be generalized for any angles and . We can take a similar approach to show that .
Definition: Angle Sum and Difference Identities
For any angles and measured in degrees or radians,
For instance, consider the expression . By writing the argument as , or anything similar, we can use the angle sum identity for cosine and the exact trigonometric values to evaluate the expression.
Letting and , we obtain
The table of exact trigonometric values allows us to evaluate this fairly easily. Recall, for an angle measured in degrees, we have the following.
We will now demonstrate how to apply these identities to solve more complicated problems which rely on us recognizing their general form.
Example 1: Using Angle Sum and Difference Identities to Simplify Trigonometric Expressions
First, we need to recognize that we are working with two angles, and , given in the form . Recall that for any two angles and ,
Letting and , we obtain
Hence, is simplified to .
In our previous example, we demonstrated how being able to recognize the form of an angle sum or difference identity can help us to simplify an expression. We will now repeat this process with the angle difference identity for sine.
Example 2: Using Trigonometric Sum and Difference of Angles Identities to Evaluate Trigonometric Expressions Involving Special Angles
Given that , find the value of , given that the angle is acute.
This equation contains two angles and given in the form . We can recognize that this is the same as the form for the angle difference identity for sine,
Letting and , we obtain
Equating the argument with the expression in the question,
In our previous examples, we demonstrated how to apply the identities for sine and cosine. Next, we will simplify an expression for the tangent of an angle.
Example 3: Using the Tangent Sum or Difference of Angles Identity to Simplify a Trigonometric Expression
In order to answer this question, we must recognize that this expression has two angles of and that are given in the form . This is in the form of the tangent angle sum identity which says that, for any angles and ,
Letting and , then
Hence, the expression can be simplified to , which is option A.
In our first few examples, we used the angle sum and difference identities to simplify expressions involving the trigonometric functions. In our next example, we will combine the use of these identities with right triangle trigonometry.
Example 4: Evaluating a Trigonometric Function of the Sum of Two Angles given Their Cosine Functions and Quadrants
Find given and , where and are acute angles.
Recall that the angle sum identity for cosine says that, for any angles and ,
Since we know the values of and , if we can find and , we will be able to calculate the value of . Given that is an acute angle and for a right triangle with included angle , we can construct a right triangle with included angle , adjacent side measuring 15 units, and hypotenuse measuring 17 units.
Using the sine ratio then, using the Pythagorean theorem, we can find the length of the opposite side:
Since we are dealing with length, we are only dealing with the positive square root, , which gives
Next, we repeat this process to find using a right triangle and the Pythagorean theorem. Since , the right triangle is constructed as shown.
So, we find
Finally, we substitute in the values of , , , and into the angle sum identity :
In our final example, we will use our knowledge of the trigonometric angle sum identities to find the tangent relationship of a non-right triangle.
Example 5: Using Right Triangle Trigonometry to Evaluate the Tangent Function of the Sum of Two Angles in a Triangle
The diagram shows triangle . Given that is perpendicular to , , , and , find the value of .
Triangle is not a right triangle. However, we are given that is perpendicular to , so triangle and triangle are both right triangles. Using right triangle trigonometry, we can find the values of and . Once we know these values, we can use the angle sum identity for tan to evaluate .
In right triangles, ; therefore,
Next, we substitute these values into the tangent angle sum identity:
In the previous examples, we have seen how the angle sum identities can help us to simplify algebraic expressions using exact values. It is important that we realize that we can also use these identities to help us simplify expressions whose value can be calculated using a calculator.
Example 6: Using Sum-to-Product Identities
Find the exact value of .
Recall that the angle sum and difference identity for cosine says that, for any angles and ,
We might notice that the expression can be factored as
Then, we look for a way to manipulate and to create an expression with terms with the same argument.
We write and
Then, using the angle sum identity,
We could use the exact values or type this into our calculator to demonstrate that
We will now recap some of the key concepts from this explainer.
- Angle sum and difference identities can be used to simplify expressions involving the sum of difference of two angles and to evaluate trigonometric expressions.
- The identities can be derived using the unit circle and right triangle trigonometry.
- For any two angles and ,