Trigonometry is a captivating branch of maths that delves deep into the relationships between the angles and sides of triangles. One of the most intriguing and useful aspects of trigonometry is the compound and double angle formulas. These formulas play a pivotal role in simplifying trigonometric expressions, solving equations, and understanding the behaviour of trigonometric functions.
Introduction to Compound Angles
When we talk about compound angles, we're referring to the sum or difference of two angles. The trigonometric functions of these compound angles can be expressed in terms of trigonometric functions of the individual angles. This is immensely useful, especially when we're trying to simplify complex expressions or solve trigonometric equations. To deepen your understanding, explore how these principles apply to the graphs of sine and cosine.
Compound Angle Formulas
These formulas are the backbone of many trigonometric simplifications and proofs. They are:
Sine of Sum or Difference of Angles
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
Cosine of Sum or Difference of Angles
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Tangent of Sum or Difference of Angles
- tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
- tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
Additionally, examining the graphs of tangent and cotangent can provide further insight into the behaviour of these functions in relation to compound angles.
Delving into Double Angle Formulas
The double angle formulas are essentially a special case of the compound angle formulas. Here, the two angles being added or subtracted are identical. They provide a way to express trigonometric functions of double angles in terms of single angles. This concept is crucial when solving trigonometric equations, as it offers a methodical approach to find solutions.
Sine Double Angle
- sin(2A) = 2sin(A)cos(A)
Cosine Double Angle
- cos(2A) = cos2(A) - sin2(A)
- Alternatively, cos(2A) = 2cos2(A) - 1 or cos(2A) = 1 - 2sin2(A)
Tangent Double Angle
- tan(2A) = 2tan(A) / (1 - tan^2(A))
For those interested in the application of trigonometry in calculus, the study of parametric equations can further enhance your understanding of trigonometric functions in complex scenarios.
Practical Applications and Detailed Examples
Example 1: Determine the exact value of sin(75°).
Using the compound angle formula for sine: sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°) = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6) + sqrt(2)) / 4
Example 2: Express sin(2x) in terms of cos(x).
Using the double angle formula for sine: sin(2x) = 2sin(x)cos(x)
Given that sin2(x) = 1 - cos2(x), we can express sin(x) as sqrt(1 - cos2(x)): sin(2x) = 2sqrt(1 - cos2(x))cos(x)
Example 3: Solve the equation cos(2x) = 0.5 for x.
Using the double angle formula for cosine: cos(2x) = 2cos2(x) - 1
Setting the equation equal to 0.5 and solving for x, we get: 2cos2(x) - 1 = 0.5 cos2(x) = 0.75 cos(x) = sqrt(0.75) x = cos(-1)(sqrt(0.75))
Example 4: Determine the value of tan(90° - A).
Using the compound angle formula for tangent: tan(90° - A) = (tan(90°) - tan(A)) / (1 + tan(90°)tan(A)) Since tan(90°) is undefined, tan(90° - A) = cot(A). This is a prime example of graph transformations at work in trigonometry.
Example 5: Prove that sin2(A) + cos2(A) = 1.
Using the Pythagorean identity, which is a direct result of the compound and double angle formulas, we can easily prove this.
Proof using Compound and Double Angle Formulas:
From the compound angle formula for sine, we have:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
If we let B = A, then:
- sin(2A) = 2sin(A)cos(A)
From the compound angle formula for cosine, we have:
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
If we let B = A again, then:
- cos(2A) = cos2(A) - sin2(A)
Another formula for cos(2A) is:
cos(2A) = 2cos2(A) - 1
And also:
- cos(2A) = 1 - 2sin2(A)
If we equate the two expressions for cos(2A), we get:
2cos2(A) + 2sin2(A) = 2
Dividing everything by 2, we get:
- cos2(A) + sin2(A) = 1
And that's how we prove the identity sin2(A) + cos2(A) = 1 using two different methods.
FAQ
Yes, compound and double angle formulas can be used for angles greater than 90°. In trigonometry, angles are not restricted to acute values. The formulas remain valid for all angles, whether they are acute, obtuse, or even reflex. This is because the trigonometric functions are periodic in nature, and their values repeat after specific intervals. For instance, the sine and cosine functions have a period of 360°, so the formulas are applicable across all angles.
Yes, there are compound and double angle formulas for the tangent function as well. The compound angle formula for tangent is given by tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)). The double angle formula for tangent can be derived from this by setting B = A, which gives tan(2A) = 2tan(A) / (1 - tan2(A)). These formulas are useful for simplifying expressions involving the tangent of the sum or double of an angle.
The unit circle is a circle of radius 1 centred at the origin of a coordinate plane. The trigonometric functions can be defined in terms of the coordinates of points on the unit circle. The compound and double angle formulas provide a relationship between the trigonometric values of the sum or double of two angles and the trigonometric values of the individual angles. When we consider two angles on the unit circle, the compound angle formulas essentially give the trigonometric values of their sum or difference by relating them to the coordinates of the points corresponding to the individual angles on the unit circle.
The double angle formulas for sine and cosine can be derived from the compound angle formulas. For instance, to derive the double angle formula for sine, we can set both angles in the compound angle formula for sine to be the same. This means we consider sin(A + A) which is sin(2A). Using the compound angle formula for sine, we get sin(2A) = sin(A)cos(A) + cos(A)sin(A), which simplifies to 2sin(A)cos(A). Similarly, for cosine, using the compound angle formula and setting both angles to be the same, we can derive the double angle formula for cosine.
Compound and double angle formulas are fundamental in trigonometry because they provide a means to express the trigonometric functions of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. This is particularly useful in simplifying complex trigonometric expressions, solving trigonometric equations, and analysing the properties of trigonometric functions. Moreover, these formulas have applications in various fields such as physics, engineering, and computer science, especially in problems related to waveforms, oscillations, and rotations.
Practice Questions
To find cos(2A), we can use the double angle formula for cosine: cos(2A) = 2cos2(A) - 1. First, we need to determine cos(A). Using the Pythagorean identity: cos2(A) = 1 - sin2(A). cos2(A) = 1 - (3/5)2. cos2(A) = 1 - 9/25 = 16/25. So, cos(A) = 4/5. Substituting this value into the double angle formula: cos(2A) = 2(4/5)2 - 1 = 2(16/25) - 1 = 7/25.
To express tan(45° + x) in terms of tan(x), we can use the compound angle formula for tangent: tan(45° + x) = (tan(45°) + tan(x)) / (1 - tan(45°)tan(x)). Since tan(45°) = 1, the formula becomes: tan(45° + x) = (1 + tan(x)) / (1 - tan(x)). This is the expression for tan(45° + x) in terms of tan(x).
