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There are identities that relate the trigonometric values of two angles to the trigonometric values of the sum or difference of these two angles. In this lesson, they will be introduced and practiced.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Designing a City Park

Tiffaniqua, who works as a landscape designer, received a job to create a new design for an old city park. Since the park is quite huge, she divided its area into six rectangular sections. The first section contains a fountain and is crossed by a river at two points — south and north

The first section of the park with a river and a fountain

When she first came to analyze the park, she stood at the north-west corner of the first section, which she marked as point Then, she took notes of some measures of angles and distances.

A piece of paper with the following notes: angle MSF=30 degrees, angle MSN=45 degrees and MS=40 meters

Later when returning to her work space, Tiffaniqua used her notes to make additional calculations. What is the length of the river within the first section of the park? Round the answer to the first decimal place.

Discussion

Introducing Angle Sum and Difference Identities

There are identities that allow calculating the values of trigonometric functions of the sum or difference of two angles.

Rule

Angle Sum and Difference Identities

To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.

There are also similar identities for the difference of two angles.

Proof

Let be a right triangle with hypotenuse and an acute angle with measure

Right triangle with hypotenuse length of 1 and acute angle x+y
By definition, the sine of an angle is the ratio between the lengths of the opposite side and the hypotenuse.
The idea now is to rewrite in terms of and To do it, draw a ray so that is divided into two angles with measures and Let be a point on this ray such that and are right triangles.
Right triangles ACD and ABC with acute angles with measures x and y, respectively.
Consider By calculating the sine and cosine of the legs of this triangle can be rewritten.
Now consider Knowing that the sine of can be used to write in terms of and
Solve for
Let be the point of intersection between and Notice that by the Vertical Angles Theorem.
Point G is the point of intersection of AC and DF

By the Third Angle Theorem, it is known that Therefore,

Since the purpose is to rewrite plot a point on such that This way a rectangle is formed. The opposite sides of a rectangle have the same length, so and are equal. Also, makes a right triangle.

A right triangle DEC is highlighted
Consequently, and can be written in terms of and using the cosine ratio.
Finally, by the Segment Addition Postulate, is equal to the sum of and All these lengths have been rewritten in terms of the sine and cosine of and
This concludes the proof of the first identity. The other identities can be proven using similar reasoning.

Extra

Calculating

Consider the following process for calculating the exact value of

  1. To be able to use the angle sum identities, the angle needs to be rewritten as the sum of two angles for which the sine and cosine are known. For example, can be rewritten as
  2. Use the first formula for the angle sum.
  3. Based on the trigonometric ratios of common angles, it is known that and
Following these three steps, the value of can be found.

Simplify
Notice that could also be rewritten as because and are known values.
They can be used to calculate the exact value of the sine, cosine, or tangent of a given angle.
Example

Evaluating Trigonometric Expressions at Random Values

When Tiffaniqua came home from work, she saw that her son Davontay and his friend Zain came up with a game. Davontay assigned numbers through to the trigonometric functions of sine, cosine, and tangent, while Zain assigned numbers through to six angle measures.

Assigned numbers for the first die: 1-sin(x), 2-cos(x), 3-tan(x), 4-tan(x), 5-sin(x), 6-cos(x); assigned numbers for the second die: 1-75 degrees, 2-(-15)degrees, 3-195 degrees, 4-240 degrees, 5-105 degrees, 6-(-165)degrees
Next, they rolled the dice four times to identify which two trigonometric values each person should calculate. The die on the left determines the trigonometric function and the die on the right determines the angle measure.
Rolling two dice four times results in getting the following four expressions: sin(105), tan(195), cos(75), tan(-15)
Find the exact values of the expressions that Davontay and Zain obtained. Write the answers in such a way that there are no radicals in the denominators.
a
b
c
d

Hint

a Express as the sum or difference of two notable angles.
b Use the Angle Sum and Difference Identities for tangent.
c Note that can be expressed as the difference of and
d Try to simplify the fraction by multiplying both the numerator and denominator by such an expression that would remove a radical from the denominator.

Solution

a The exact value of should be found. Recall that the values of the trigonometric functions of some notable angles are known. Try to represent the angle of as the sum or difference of two notable angles.
Now, use the Angle Sum Identity for sine.
Substitute for and for and simplify.
Substitute values

b Similarly to Part A, start by expressing as the sum or difference of two notable angles.
Next, apply the Angle Difference Identity for tangent.
Substitute for and for and then find the value of by solving the equation.

Finally, to simplify the obtained expression, multiply both the numerator and denominator by
Simplify right-hand side
c First, note that the angle of can be represented as the difference of and Therefore, the Angle Difference Identity for cosine can be used to calculate the value of
Next, use the known values of the trigonometric functions of notable angles and
Substitute values

d To find the value of first express the angle of as the sum or difference of two notable angles.
Next, use the Angle Difference Identity for tangent. Substitute with and with and solve the equation for
Simplify right-hand side

The expression can be simplified by multiplying both the numerator and denominator of the fraction by
Simplify right-hand side
Example

Evaluating Trigonometric Functions

In the game that Davontay and Zain created and played, Davontay solved everything correctly. Zain, on the other hand, made one mistake. This was on Zain's mind as they came home, so they decided to practice by evaluating more trigonometric functions.

1. sin(23pi/12) 2. cos(-165) 3. tan(255)

Find the values of the given expressions along with Zain.

a
b
c

Hint

a The angle can be expressed as the sum of and
b Start by using the Negative Angle Identity for cosine.

Solution

a To calculate the value of start by expressing the angle of as the sum or difference of two notable angles.
Note that the angles and can be simplified to and which are the notable angles corresponding to