How to Solve Problems Involving Compound Angles
Simply put, trigonometric functions—also called circular functions—are the functions of a triangle's angle. This means that these trigonometric functions provide the relationship between the angles and sides of a triangle. The fundamental trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.
We need to know trigonometric ratios to get a clear view of the compound angles.
Angles
An angle is created when two rays are united at a common point. The two rays are referred to as the arms of the angle, while the common point is the node or vertex. The symbol "$\pi$” stands for the angle. The Latin word "Angulus" is where the word "angle" originated.
Typically, an angle is measured with a protractor and expressed in degrees. Here, several angles are shown at 30 degrees, 45 degrees, 60 degrees, 90 degrees, and 180 degrees. The angles' degrees values determine the many types of angles.
Angles can be expressed in terms of pi ($\pi$) or radians.
Trigonometric Ratios
The ratios of the triangle's side lengths are known as trigonometric ratios. These ratios explain how the ratio of a right triangle's sides to each angle works in trigonometry. The sin, cos, and tan functions can obtain the other significant trigonometric ratios, cosec, sec, and cot.
Trigon, which means "triangle," and metrôn, which means "to measure," are the roots of the term "trigonometry." It is a field of mathematics that examines how a right-angled triangle's angles and sides relate to one another.
Trigonometric Functions
Trigonometric Identities
Trigonometric Identities come in handy whenever trigonometric functions are incorporated into an expression or equation. For every value of a variable appearing on both sides of an equation, a trigonometric identity is true. These trigonometric identities involve one or more angles' sine, cosine, and tangent.
What are Compound Angles?
An algebraic sum of two or more angles can define the compound angle. Compound angles represent different trigonometric identities.
Compound angles are denoted using trigonometric identities. Compound angles can be used to compute the fundamental operations of finding the sum and difference of functions.
Formula of Compound Angles
The formula for trigonometric ratios of compound angles is given below:
$\sin {\rm{ }}(A + B){\rm{ }} = \sin {\rm{ }}A{\rm{ }}\cos {\rm{ }}B{\rm{ }} + \cos {\rm{ }}A{\rm{ }}\sin {\rm{ }}B$
$\sin \left( {A - B} \right) = \sin A\cos B - \cos A\sin B$
$\cos {\rm{ }}(A + B){\rm{ }} = \cos {\rm{ }}A{\rm{ }}\cos {\rm{ }}B - \sin {\rm{ }}A{\rm{ }}\sin {\rm{ }}B{\rm{ }}$
$\cos {\rm{ }}(A - B){\rm{ }} = \cos {\rm{ }}A{\rm{ }}\cos {\rm{ }}B{\rm{ }} + \sin {\rm{ }}A{\rm{ }}\cos {\rm{ }}B{\rm{ }}$
$\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$
$\tan \left( {A - B} \right) = \dfrac{{\tan A - \tan B}}{{1 + \tan A\tan B}}$
$\sin (A + B)\sin \left( {A - B} \right) = {\sin ^2}A - {\sin ^2}B = {\cos ^2}B - {\cos ^2}A$
$\cos \left( {A + B} \right)\cos \left( {A - B} \right) = {\cos ^2}A - {\sin ^2}A - {\sin ^2}B = {\cos ^2}B - {\sin ^2}A$
Questions on Compound Angles
Question 1: If $\cos A = \dfrac{4}{5}$ and $\cos B = \dfrac{{12}}{{13}},\dfrac{{3\pi }}{2} < A,B < 2\pi$, find the value of $\cos (A+B) $.
Solution:
According to the given question, $\cos A = \dfrac{4}{5}$ and $\cos B = \dfrac{{12}}{{13}}$. Also, $\dfrac{{3\pi }}{2} < A,B < 2\pi$, both A and B lie in the fourth quadrant. Then we can say that both sin A and sin B are negative.
First we need to find out $\sin A$,
$\begin{array}{c}\sin A = - \sqrt {1 + {{\cos }^2}A} \\ = - \sqrt {1 - \dfrac{{16}}{{25}}} \\ = - \dfrac{3}{5}\end{array}$
We need to find out $\sin B$,
$\begin{array}{c}\sin B = - \sqrt {1 + {{\cos }^2}B} \\ = - \sqrt {1 - \dfrac{{144}}{{169}}} \\ = - \dfrac{5}{{13}}\end{array}$
Now, we can find out $\cos (A+B)$,
$\begin{array}{c}\cos (A + B) = \cos A\cos B - \sin A\sin B\\ = \dfrac{4}{5} \times \dfrac{{12}}{{13}} - \left( {\dfrac{{ - 3}}{5}} \right) \times \left( {\dfrac{{ - 5}}{{13}}} \right)\\ = \dfrac{{48}}{{65}} - \dfrac{{15}}{{65}}\\ = \dfrac{{33}}{{65}}\end{array}$
Question 2: Determine the value of $\sin {15^\circ }$.
Solution:
According to the given question, we need to find the value of $\sin {15^\circ }$.
$\sin \left( {A - B} \right) = \sin A\cos B - \cos A\sin B$
$\sin \left( {45 - 30} \right) = \sin 45\cos 30 - \cos 45\sin 30$
$\begin{array}{c}\sin {15^ \circ } = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2} - \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{2}\\ = \dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}\end{array}$
Question 3: If A, B, and C are angles of a triangle, then prove that $\dfrac{A}{2} = \cot \left( {B + C} \right)/2$.
Solution:
According to the given question, we know A, B, and C are the angles of a triangle,
$\begin{array}{l}A + B + C = n\\B + C = n - A\\\dfrac{{\left( {B + C} \right)}}{2} = \dfrac{n}{2} - \dfrac{A}{2}\end{array}$
Then,
$\begin{array}{l}\cot \dfrac{{\left( {B + C} \right)}}{2} = \cot \left( {\dfrac{n}{2} - \dfrac{A}{2}} \right)\\ = \tan \dfrac{A}{2}\end{array}$
Hence, proved.
Conclusion:
Trigonometry was developed to solve problems and take measurements involving triangles. Trigonometry is a tool used by meteorologists, engineers, scientists, engineers, engineers, engineers, and navigators. It is referred to as sum and difference formulas in American mathematics, but it is known as compound angles in British Mathematics. Trigonometric functions are utilised to obtain consistent results, including compound angles, trigonometric ratios, and multiple angles.
FAQs on Compound Angles: Definition, Properties & Examples
1. What is a compound angle in trigonometry?
A compound angle is an angle that is formed by the algebraic sum or difference of two or more angles. For example, if A and B are two angles, then (A + B) and (A - B) are compound angles. We use specific trigonometric identities, known as compound angle formulas, to find the trigonometric ratios of these angles.
2. What are the primary formulas for compound angles involving sine and cosine?
The fundamental compound angle formulas for sine and cosine are essential for solving trigonometric problems. They 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)
3. How do compound angle formulas help in finding the exact value of trigonometric functions like cos(15°)?
Compound angle formulas are incredibly useful for finding exact trigonometric values for angles that are not standard (like 0°, 30°, 45°, 60°, 90°). For example, to find the exact value of cos(15°), you can express 15° as a difference of two standard angles, such as (45° - 30°). You can then apply the formula for cos(A - B):
cos(15°) = cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°).
By substituting the known values, you can calculate the precise answer without a calculator.
4. Why is sin(A + B) not equal to sin(A) + sin(B)?
This is a common misconception. The reason sin(A + B) is not sin(A) + sin(B) is that 'sin' is a trigonometric function, not a number or variable that can be distributed over the terms in the parenthesis. A simple 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 clearly not equal to 1. The correct expansion is given by the compound angle formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
5. How is the formula for tan(A + B) derived from the sine and cosine formulas?
The formula for tan(A + B) is derived directly from the fundamental definitions of tan, sin, and cos. The derivation follows these steps:
1. Start with the identity tan(θ) = sin(θ) / cos(θ). So, tan(A + B) = sin(A + B) / cos(A + B).
2. Substitute the compound angle formulas for sin(A + B) and cos(A + B) in the numerator and denominator.
3. Divide both the numerator and the denominator by the term cos(A)cos(B).
4. Simplifying the resulting expression gives the final formula: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)).
6. What is the difference between compound angles and multiple angles in trigonometry?
The key difference lies in how the angles are combined:
- A compound angle is the algebraic sum or difference of two or more distinct angles, such as (A + B) or (A - B).
- A multiple angle is an integer multiple of a single angle, such as 2A, 3A, etc.
Importantly, the formulas for multiple angles (like sin(2A)) are actually special cases derived from compound angle formulas. For instance, you can find the formula for sin(2A) by taking the sin(A + B) formula and substituting B with A.
7. What are some real-world applications where understanding compound angles is important?
Compound angle concepts are applied in various scientific and technical fields. For example:
- In Physics, they are used to analyse the superposition of waves (like light or sound waves) and in optics to calculate the path of light through prisms.
- In Engineering, especially mechanics, they help in resolving forces and calculating the net effect of multiple forces acting on a body from different directions.
- In Navigation and Astronomy, they are used to pinpoint the location of objects in the sky or on Earth by combining angular measurements from different reference points.
8. Are the formulas for sin(A - B) and cos(A - B) derived independently of the sum formulas?
No, they are not derived independently. The difference formulas can be found efficiently from the sum formulas by a simple substitution. To find the formula for sin(A - B), you can write it as sin(A + (-B)) and apply the sum formula. By using the trigonometric identities that cos(-B) = cos(B) and sin(-B) = -sin(B), the sum formula directly simplifies into the difference formula. This demonstrates the elegant and interconnected nature of these trigonometric identities.
