Lesson Explainer: Angle Sum and Difference Identities Mathematics

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 sin(𝛼+𝛽). 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 𝑂𝑃, 𝑂𝐴, and 𝑂𝐵 are of unit length and 𝑂 is made up of two angles, 𝛼 and 𝛽, where 𝛼+𝛽<90 and 𝛼,𝛽>0. We observe that 𝛼 is in standard position and the initial side of 𝛽 is the terminal side of 𝛼. Points 𝐴 and 𝑃 are the points of intersections of the terminal sides of 𝛼 and 𝛽 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; 𝑃𝑅 is perpendicular to 𝐴𝑂, 𝑃𝑇 and 𝑅𝑆 are perpendicular to the 𝑥-axis, and 𝑄𝑅 is perpendicular to 𝑃𝑇 as shown.

Next, we observe that 𝑃𝑅𝐶=90 and 𝑂𝐶𝑇=90𝛼, since angles in a triangle sum to 180. Then 𝑃𝐶𝑅=90𝛼, since vertically opposite angles are equal; hence, 𝑇𝑃𝑅=180(90+90𝛼)=𝛼. Further, 𝑄𝑇=𝑅𝑆, as shown in the following diagram.

Using right triangle trigonometry on triangle 𝑂𝑃𝑇 allows us to find a relationship between sin(𝛼+𝛽) and 𝑃𝑇: sin(𝛼+𝛽)=𝑃𝑇1=𝑃𝑇.

However, since 𝑃𝑇=𝑃𝑄+𝑄𝑇=𝑃𝑄+𝑅𝑆,

sin(𝛼+𝛽)=𝑃𝑄+𝑅𝑆.(1)

Then, using the sine ratio on triangle 𝑂𝑅𝑃, sin𝛽=𝑃𝑅1=𝑃𝑅.

Using the cosine ratio on triangle 𝑃𝑄𝑅, coscos𝛼=𝑃𝑄𝑃𝑅𝑃𝑄=𝑃𝑅𝛼.

Hence,

𝑃𝑄=𝛽𝛼.sincos(2)

Similarly, for triangle 𝑂𝑆𝑅, sinsin𝛼=𝑅𝑆𝑅𝑂𝑅𝑆=𝑅𝑂𝛼.

And for triangle 𝑂𝑅𝑃, cos𝛽=𝑅𝑂1=𝑅𝑂.

Hence,

𝑅𝑆=𝛽𝛼.cossin(3)

Combining equations (1), (2), and (3), we obtain the angle sum identity for the sine function: sinsincossincos(𝛼+𝛽)=𝛼𝛽+𝛽𝛼.

This is a demonstration of a proof of one of the three angle sum identities. While we made assumptions on 𝛼 and 𝛽, 𝛼+𝛽<90 and 𝛼,𝛽>0, our proof can be generalized for any angles 𝛼 and 𝛽. We can take a similar approach to show that coscoscossinsin(𝛼+𝛽)=(𝛼)(𝛽)(𝛼)(𝛽).

Definition: Angle Sum and Difference Identities

For any angles 𝛼 and 𝛽 measured in degrees or radians, sinsincossincoscoscoscossinsintantantantantan(𝛼±𝛽)𝛼𝛽±𝛽𝛼,(𝛼±𝛽)(𝛼)(𝛽)(𝛼)(𝛽),(𝛼±𝛽)𝛼±𝛽1𝛼𝛽.

For instance, consider the expression cos120. By writing the argument 120 as 90+30, or anything similar, we can use the angle sum identity for cosine and the exact trigonometric values to evaluate the expression.

Letting 𝛼=90 and 𝛽=30, we obtain coscoscoscossinsin120=(90+30)=90309030.

The table of exact trigonometric values allows us to evaluate this fairly easily. Recall, for an angle 𝜃 measured in degrees, we have the following.

𝜃030456090
sin𝜃01222321
cos𝜃13222120

Hence, cos120=0×321×12=12.

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

Simplify coscossinsin2𝑋22𝑋2𝑋22𝑋.

Answer

First, we need to recognize that we are working with two angles, 2𝑋 and 22𝑋, given in the form coscossinsin(𝛼)(𝛽)(𝛼)(𝛽). Recall that for any two angles 𝛼 and 𝛽, coscoscossinsin(𝛼+𝛽)=𝛼𝛽𝛼𝛽.

Letting 𝛼=2𝑋 and 𝛽=22𝑋, we obtain coscossinsincoscos(2𝑋)(22𝑋)(2𝑋)(22𝑋)=(2𝑋+22𝑋)=(24𝑋).

Hence, coscossinsin2𝑋22𝑋2𝑋22𝑋 is simplified to cos(24𝑋).

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 sincoscossinsin60306030=𝜃, find the value of 𝜃, given that the angle is acute.

Answer

This equation contains two angles 30 and 60 given in the form sincoscossin𝛼𝛽𝛼𝛽. We can recognize that this is the same as the form for the angle difference identity for sine, sinsincoscossin(𝛼𝛽)=𝛼𝛽𝛼𝛽.

Letting 𝛼=60 and 𝛽=30, we obtain sincoscossinsinsin60306030=(6030)=(30).

Equating the argument with the expression in the question, 𝜃=30.

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

Simplify tantantantan33+299133299.

  1. tan332
  2. tan226
  3. tantan33299
  4. 233229tantan

Answer

In order to answer this question, we must recognize that this expression has two angles of 33 and 299 that are given in the form tantantantan(𝛼)+(𝛽)1(𝛼)(𝛽). This is in the form of the tangent angle sum identity which says that, for any angles 𝛼 and 𝛽, tantantantantan(𝛼+𝛽)=(𝛼)+(𝛽)1(𝛼)(𝛽).

Letting 𝛼=33 and 𝛽=299, then tantantantantantan33+299133299=(33+299)=332.

Hence, the expression can be simplified to tan332, 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 cos(𝐴+𝐵) given cos𝐴=1517 and cos𝐵=513, where 𝐴 and 𝐵 are acute angles.

Answer

Recall that the angle sum identity for cosine says that, for any angles 𝐴 and 𝐵, coscoscossinsin(𝐴+𝐵)=𝐴𝐵𝐴𝐵.

Since we know the values of cos𝐴 and cos𝐵, if we can find sin𝐴 and sin𝐵, we will be able to calculate the value of cos(𝐴+𝐵). Given that 𝐴 is an acute angle and cosadjacenthypotenuse𝜃= for a right triangle with included angle 𝜃, we can construct a right triangle with included angle 𝐴, adjacent side measuring 15 units, and hypotenuse measuring 17 units.

Using the sine ratio sin𝐴=𝑎17, then, using the Pythagorean theorem, we can find the length of the opposite side: 𝑎+15=17𝑎=1715𝑎=64.

Since we are dealing with length, we are only dealing with the positive square root, 𝑎=8, which gives sin𝐴=817.

Next, we repeat this process to find sin𝐵 using a right triangle and the Pythagorean theorem. Since cos𝐵=513, the right triangle is constructed as shown.

So, we find 5+𝑏=13𝑏=135𝑏=144𝑏=12.

Hence, sin𝐵=1213.

Finally, we substitute in the values of sin𝐴, sin𝐵, cos𝐴, and cos𝐵 into the angle sum identity coscoscossinsin(𝐴+𝐵)=𝐴𝐵𝐴𝐵: cos(𝐴+𝐵)=1517×513817×1213=7522196221=21221.

Hence, cos(𝐴+𝐵)=21221.

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 𝐴𝐵𝐶. Given that 𝐴𝐷 is perpendicular to 𝐵𝐶, 𝐴𝐷=15cm, 𝐵𝐷=10cm, and 𝐶𝐷=7cm, find the value of tan(𝑋+𝑌).

Answer

Triangle 𝐴𝐵𝐶 is not a right triangle. However, we are given that 𝐴𝐷 is perpendicular to 𝐵𝐶, so triangle 𝐴𝐵𝐷 and triangle 𝐴𝐷𝐶 are both right triangles. Using right triangle trigonometry, we can find the values of tan𝑋 and tan𝑌. Once we know these values, we can use the angle sum identity for tan to evaluate tan(𝑋+𝑌).

In right triangles, tanoppositeadjacent𝜃=; therefore, tanandtantanandtan𝑋=𝐵𝐷𝐴𝐷𝑌=𝐶𝐷𝐴𝐷𝑋=1015𝑌=715.

Next, we substitute these values into the tangent angle sum identity: tantantantantantan(𝐴+𝐵)=𝐴+𝐵1𝐴𝐵(𝑋+𝑌)=+1×=1715×4531.

Hence, tan(𝑋+𝑌)=5131.

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 3(75)3(15)coscos.

Answer

Recall that the angle sum and difference identity for cosine says that, for any angles 𝐴 and 𝐵, coscoscossinsin(𝐴±𝐵)=𝐴𝐵𝐴𝐵.

We might notice that the expression 3(75)3(15)coscos can be factored as 3(75)3(15)=3((75)(15)).coscoscoscos

Then, we look for a way to manipulate cos(75) and cos(15) to create an expression with terms with the same argument.

We write coscos(75)=(45+30) and coscos(15)=(4530).

Then, using the angle sum identity, coscoscoscossinsin(75)=(45+30)=(45)(30)(45)(30).

And, coscoscoscossinsin(15)=(4530)=(45)(30)+(45)(30).

Hence, 3((75)(15))=3(((45)(30)(45)(30))((45)(30)+(45)(30)))=3(2(45)(30)).coscoscoscossinsincoscossinsinsinsin

We could use the exact values or type this into our calculator to demonstrate that 3(75)3(15)=322.coscos

We will now recap some of the key concepts from this explainer.

Key Points

  • 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 𝛼 and 𝛽, sinsincossincoscoscoscossinsintantantantantan(𝛼±𝛽)𝛼𝛽±𝛽𝛼,(𝛼±𝛽)(𝛼)(𝛽)(𝛼)(𝛽),(𝛼±𝛽)𝛼±𝛽1𝛼𝛽.

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