Key Formulas and Step-by-Step Solutions for Sum and Difference Angles
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 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.
FAQs on Sum and Difference of Angles in Trigonometry
1. What are the fundamental sum and difference formulas in trigonometry?
The fundamental sum and difference formulas, also known as trigonometric identities, are used to find the trigonometric values of angles that are expressed as the sum or difference of standard angles (like 30°, 45°, 60°, 90°). The primary formulas are:
- 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) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
- tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
2. What is the main practical application of sum and difference angle formulas?
The main application of these formulas is to calculate the exact trigonometric values for angles that are not standard, by breaking them down into a sum or difference of standard angles. For example, to find the value of sin(75°), you can express it as sin(45° + 30°) and apply the sum formula for sine, using the known values for 45° and 30°.
3. How are the double angle identities (like sin(2A)) related to the sum of angles formulas?
The double angle identities are a special case of the sum of angles formulas. You can derive them directly by setting the two angles to be equal (i.e., let B = A). For instance, in the formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B), if you substitute B with A, you get sin(A + A) = sin(A)cos(A) + cos(A)sin(A), which simplifies to sin(2A) = 2sin(A)cos(A). The same logic applies to derive cos(2A) and tan(2A).
4. Why is understanding the geometric derivation of these formulas important?
Understanding the geometric derivation, typically done using the unit circle, is crucial because it provides a conceptual foundation for why these formulas work. It demonstrates the relationship between the algebraic identities and the geometric properties of angles and coordinates. The derivation of cos(A - B) is often the starting point, using the distance formula between two points on the unit circle. This foundational proof helps in remembering the formulas and prevents common errors, reinforcing that they are not arbitrary rules but are derived from fundamental geometric principles.
5. How can you find the exact value of cos(15°) using the difference formula?
To find the exact value of cos(15°), you can express it as the difference between two standard angles, such as 45° and 30°. The steps are as follows:
- Rewrite the angle: cos(15°) = cos(45° - 30°).
- Apply the difference formula for cosine: cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
- Substitute A=45° and B=30°: cos(45°)cos(30°) + sin(45°)sin(30°).
- Insert the known values: (1/√2)(√3/2) + (1/√2)(1/2).
- Simplify the expression: (√3 + 1) / 2√2. This is the exact value of cos(15°).
6. Why is sin(A + B) not equal to sin(A) + sin(B)?
This is a common misconception. Trigonometric functions like sine are not linear operators, which means you cannot distribute them over addition or subtraction. A simple counter-example proves this: Let A = 30° and B = 60°. Then, sin(A + B) = sin(30° + 60°) = sin(90°) = 1. However, sin(A) + sin(B) = sin(30°) + sin(60°) = 0.5 + (√3/2), which is approximately 1.366. Since 1 ≠ 1.366, it is clear that sin(A + B) ≠ sin(A) + sin(B).
7. What is the importance of the sum and difference formulas for the tangent function?
The formulas for tan(A + B) and tan(A - B) are important because they allow for direct calculation without first finding the sine and cosine of the sum or difference. They are derived from the sine and cosine sum/difference formulas by using the identity tan(θ) = sin(θ)/cos(θ). These identities are particularly useful in solving trigonometric equations and simplifying complex expressions involving the tangent function, especially in topics like calculus and coordinate geometry.
8. Where are sum and difference angle formulas used in real-world scenarios?
Beyond textbook problems, these formulas have significant applications in various scientific fields. For example:
- In Physics, they are used to analyse the interference of waves (like sound or light) and to describe alternating currents (AC circuits) where phase differences are crucial.
- In Engineering, they are fundamental to signal processing, helping to analyse complex signals by breaking them into simpler sinusoidal components.
- In Navigation and Astronomy, they help in calculating positions and trajectories by relating different angular measurements in a complex system.
