Sum Difference Angles Trigonometry

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Sum and Difference of Angles Identities

Angle sum identities and angle difference identities can be used to find the function values of any angles however, the most feasible use of sum of angles trig identities is to identify the exact values of an angle that can be mathematically expressed as a sum or difference using the familiar values for the sine, cosine and tangent of the 30°, 45°, 60° and 90° angles as well their multiples.


The sum and difference formulas for sine and cosine are also helpful in dealing with the inverse trigonometric functions.


Sum and Difference Angle Formulas

The sum formula for cosines implies that the cosine of the sum of two angles is equivalent to the product of the cosines of the angles subtracting the product of the sines of the angles. On the other hand, the difference formula for cosines implies that the cosine of the difference of two angles is equivalent to the product of the cosines of the angles adding up the product of the sines of the angles.


Sum and Difference Angle Formulas

Sum of Angle Identities

Difference of Angle Identities

Sin [A + B] = sin A · cos B + cos A · sin B

Sin [A - B] = sin A · cos B - cos A · sin B

cos [A + B] = cos A · cos B - sin A · sin B

Cos [A - B] = cos A · cos B + sin A · sin B

tan [A + B] = tanA + tanB= 1- tanA· tan B

tan [A - B] = tanA - tanB= 1+ tanA· tan B


Sum Formula for Tangent

The sum formula for tangent trigonometry implies that the tangent of the sum of two    angles is equivalent to the sum of the tangents of the angles further divided by 1 minus    (-) the product of the tangents of the angles.


Difference Formula for Tangent

The Difference Formula for tangent implies that the tangent of the difference of two angles is equivalent to the difference of the tangents of the angles divided by 1 plus (+) the product of the tangents of the angles.


Use of Pythagorean Theorem With the Sum and Difference Formulas

The Pythagorean Theorem when used along with the sum and difference formulas helps to find multiple sums and differences of angles.


Solved Examples


Example 1:

Identify the exact value of sine 15° applying the difference formula.


Solution:

Seeing that, 15° is the Value of Difference between 45° and 30°. Thus, the Sine Difference formula can be applied

Now, using the sin difference formula i.e. sin(A - B) = sin A · cos B - cos A · sin B. We get,

Sin [45° - 30°] = sin 45° · cos 30° - cos 45° · sin 30°

= √2/2. √3/2 -√2/2. √1/2

= √6/4 - √2/4 =√6-√2/4

Hence, sin 15° =√6-√2/4


Example 2:

Identify the Exact value of cos 75° applying the sum formula.


Solution:

Seeing that, 75° is the sum of 45° and 30°. Thus, the cos sum formula can be applied

Now, using the cos sum formula i.e. cos (A + B) = cos A · cos B - sin A · sin B. We get,

cos (30° + 45°) = cos 30° · cos 30° · cos 45° - sin 30°· sin 45°

= √3/2. √2/2 -√1/2. √2/2

= √6/4 - √2/4 =√6-√2/4

Hence cos 75° =√6-√2/4


Key Takeaways

1. You can have more than one option. For Example If you want to Find the sine of 120°

Sin (90 + 30) applying the Sum Formula

or

sin (180 - 60) applying the Difference Formula

Both Formulas Will Produce Similar Results.

2. In order to identify a secant, cosecant, or cotangent function, you would need to change the expression to one of the 3 basic functions, and perform the requisite calculations.

3. Remember to identify in which quadrant the angle will lie.

FAQ (Frequently Asked Questions)

Q1. What is the Use of the Sum and Difference Formulas for Cosine?

Answer: The trigonometric sum and difference angle formulas are helpful in determining the exact value of the sine, cosine, or tangent of an angle. It also becomes much easier to find the value of an angle if we are able to rewrite the given angle in terms of two angles that have known trigonometric values. Sum and difference formulas are useful in cross checking the identities. Moreover, solving application problems becomes much simpler using sum and difference formulas.

 Q2. How Do We Find the Cosine of the Difference Between the Angles, When Given Angles?

Answer: Follow the given steps:

1. Put and use the difference formula for cosine.

2. Substitute the values of the given angles into the formula used

3. Simplify the expression

Q3. What Does the Sum Formula For Sines Signify?

Answer: The trigonometric sum formula for sines implies that the sine of the sum of two angles is equivalent to the product of the sine of the 1st angle and cosine of the 2nd angle plus (+) the product of the cosine of the 1st angle and the sine of the 2nd angle.

Q4. What Does the Difference Formula For Sines Signify?

Answer: The difference formula for sines implies that the sine of the difference of two angles is equivalent to the product of the sine of the 1st angle and cosine of the 2nd angle minus (-) the product of the cosine of the 1st angle and the sine of the 2nd angle.