Problem 8
Question
Use the addition formulas for sine and cosine to derive the double-angle formulas $$ \begin{aligned} \sin (2 A) &=2 \sin A \cos A \\ \cos (2 A) &=\cos ^{2} A-\sin ^{2} A \\ &=2 \cos ^{2} A-1 \\ &=1-2 \sin ^{2} A \end{aligned} $$
Step-by-Step Solution
Verified Answer
Using addition formulas for sine and cosine, the double-angle formulas are derived as: \( \sin(2A) = 2 \sin A \cos A \), \( \cos(2A) = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A \).
1Step 1 - Understand the Addition Formulas
The addition formulas for sine and cosine are given by:\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]Our goal is to use these to derive the double-angle formulas.
2Step 2 - Derive the Double-Angle Formula for Sine
Set \(A = B\) in the addition formula for sine:\[\sin(2A) = \sin(A + A) = \sin A \cos A + \cos A \sin A\]Simplify the expression:\[\sin(2A) = 2 \sin A \cos A\]
3Step 3 - Derive the Double-Angle Formula for Cosine (Form 1)
Set \(A = B\) in the addition formula for cosine:\[\cos(2A) = \cos(A + A) = \cos A \cos A - \sin A \sin A\]Simplify the expression:\[\cos(2A) = \cos^2 A - \sin^2 A\]
4Step 4 - Derive the Double-Angle Formula for Cosine (Form 2)
Start with the result from Step 3 and use the Pythagorean identity \( \cos^2 A + \sin^2 A = 1\). Rearrange this identity to \( \cos^2 A = 1 - \sin^2 A \). Substitute this into the formula:\[\cos(2A) = (1 - \sin^2 A) - \sin^2 A = 1 - 2\sin^2 A\]
5Step 5 - Derive the Double-Angle Formula for Cosine (Form 3)
Similarly, start with the result from Step 3 and substitute \( \sin^2 A = 1 - \cos^2 A \) into the formula:\[\cos(2A) = \cos^2 A - (1 - \cos^2 A) = 2\cos^2 A - 1\]
Key Concepts
sine addition formulacosine addition formulapythagorean identity
sine addition formula
The sine addition formula helps to simplify and solve trigonometric identities and equations involving summed angles. It is expressed as:
\(\sin(A + B) = \sin A \cos B + \cos A \sin B \).
To understand this concept better, let's break it down:
\( \sin(2A) = \sin(A + A)\ = \sin A \cos A + \cos A \sin A \). Since \( \sin A \cos A = \cos A \sin A \), you can simplify this to:
\( \sin(2A) = 2 \sin A \cos A \).
This double-angle formula is very useful in various engineering and physics calculations.
\(\sin(A + B) = \sin A \cos B + \cos A \sin B \).
To understand this concept better, let's break it down:
- When you add two angles, you don't simply add their sines. Instead, you use a combination of sines and cosines of the individual angles.
\( \sin(2A) = \sin(A + A)\ = \sin A \cos A + \cos A \sin A \). Since \( \sin A \cos A = \cos A \sin A \), you can simplify this to:
\( \sin(2A) = 2 \sin A \cos A \).
This double-angle formula is very useful in various engineering and physics calculations.
cosine addition formula
The cosine addition formula, like its sine counterpart, is used to find the cosine of a sum of angles. It is given by:
\( \cos(A + B) = \cos A \cos B - \sin A \sin B \).
The cosine addition formula is essential for solving problems involving multiple angles.
\( \cos(2A) = \cos(A + A) = \cos A \cos A - \sin A \sin A \). Simplify this to:
\( \cos(2A) = \cos^2 A - \sin^2 A \).
This is the first form of the double-angle formula for cosine.
You can further manipulate the formula using the Pythagorean identity to get two more forms.
Replace \( \cos^2 A \) with \( 1 - \sin^2 A \):
\( \cos(2A) = (1 - \sin^2 A) - \sin^2 A = 1 - 2\sin^2 A \).
Likewise, replace \( \sin^2 A \) with \( 1 - \cos^2 A \):
\( \cos(2A) = \cos^2 A - (1 - \cos^2 A) = 2\cos^2 A - 1 \).
These multiple forms provide flexibility for various applications.
\( \cos(A + B) = \cos A \cos B - \sin A \sin B \).
The cosine addition formula is essential for solving problems involving multiple angles.
- When working with summed angles, you decompose the sum into a formula involving products of cosines and sines.
\( \cos(2A) = \cos(A + A) = \cos A \cos A - \sin A \sin A \). Simplify this to:
\( \cos(2A) = \cos^2 A - \sin^2 A \).
This is the first form of the double-angle formula for cosine.
You can further manipulate the formula using the Pythagorean identity to get two more forms.
Replace \( \cos^2 A \) with \( 1 - \sin^2 A \):
\( \cos(2A) = (1 - \sin^2 A) - \sin^2 A = 1 - 2\sin^2 A \).
Likewise, replace \( \sin^2 A \) with \( 1 - \cos^2 A \):
\( \cos(2A) = \cos^2 A - (1 - \cos^2 A) = 2\cos^2 A - 1 \).
These multiple forms provide flexibility for various applications.
pythagorean identity
The Pythagorean identity is one of the most fundamental relations in trigonometry. It states:
\[ \cos^2 A + \sin^2 A = 1 \].
This identity helps to express one trigonometric function in terms of another.
For instance, replacing \( \cos^2 A \) in the double-angle formula gives us:
\( \cos(2A) = (1 - \sin^2 A) - \sin^2 A = 1 - 2\sin^2 A \).
So mastering the Pythagorean identity is crucial for solving trigonometric problems efficiently.
\[ \cos^2 A + \sin^2 A = 1 \].
This identity helps to express one trigonometric function in terms of another.
- It arises from the Pythagorean theorem applied to a right-angled triangle.
- The sum of the squares of the sine and cosine of an angle always equals 1.
\( \cos^2 A = 1 - \sin^2 A \)
\( \sin^2 A = 1 - \cos^2 A \)
For instance, replacing \( \cos^2 A \) in the double-angle formula gives us:
\( \cos(2A) = (1 - \sin^2 A) - \sin^2 A = 1 - 2\sin^2 A \).
So mastering the Pythagorean identity is crucial for solving trigonometric problems efficiently.
Other exercises in this chapter
Problem 6
Find \(\csc \theta\) if \(\cot \theta=\frac{\sqrt{5}}{2}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).
View solution Problem 7
Starting with the addition formulas for the sine and cosine, derive these identities: \(\cos \left(\frac{\pi}{2}+\theta\right)=-\sin \theta \quad\) and \(\quad
View solution Problem 9
a. Use the double-angle formulas along with the Pythagorean identity \(\sin ^{2} A+\cos ^{2} A=1\) to show that \(\cos ^{2} \theta=\frac{1}{2}(1+\cos 2 \theta)
View solution Problem 11
Differentiate the given function. $$f(x)=\cos (1-5 x)$$
View solution