Chapter 9: Problem 2
True or False $$\cos ^{2} \frac{\theta}{2}=\frac{1+\sin \theta}{2}$$
Short Answer
Expert verified
False
Step by step solution
01
- Recall the half-angle identity for cosine
The half-angle identity for cosine states that \ \ \[\cos^{2} \frac{x}{2} = \frac{1 + \cos x}{2}\] \ \ This identity will help verify the given statement.
02
- Express the given statement using the sine identity
Recall that \(\sin \theta = \cos (90^\textdegree - \theta)\). Consequently, we know that \(\sin \theta\) can be expressed in terms of \(\cos \theta\). Let's substitute this into our given equation to find if they match.
03
- Verify the equality
Substitute the value from the half-angle identity of cosine into the given statement. \ \ Given: \(\cos^{2} \frac{\theta}{2} = \frac{1 + \sin \theta}{2}\) \ Using the identity, substitute \(\cos^{2} \frac{\theta}{2} = \frac{1 + \cos \theta}{2}\) \ Get: \( \frac{1 + \cos \theta}{2} eq \frac{1 + \sin \theta}{2} \) \ This shows that the left side does not equal the right side.
04
- Conclusion
The equation \(\cos^{2} \frac{\theta}{2} = \frac{1 + \sin \theta}{2} \) does not hold true. Therefore, the given statement is False.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Half-angle identities
In trigonometry, half-angle identities are very useful for simplifying expressions involving trigonometric functions. These identities relate the value of trigonometric functions of an angle to the value of the functions of half that angle. The half-angle identity for cosine is one of the key formulas: \[ \cos^2 \left( \frac{x}{2} \right) = \frac{1 + \cos x}{2} \] This identity allows us to express the square of the cosine of half an angle in terms of the cosine of the original angle. Using these identities can simplify complex trigonometric expressions, making them easier to handle.
Knowing and understanding these identities is crucial when solving trigonometric problems or verifying equations. They can help you transform an equation into a more manageable form.
In our specific problem, we need to recall the half-angle identity for cosine from the start, as given in the steps of the original solution. This fundamental understanding will guide us in checking the correctness of the given equation.
Knowing and understanding these identities is crucial when solving trigonometric problems or verifying equations. They can help you transform an equation into a more manageable form.
- Half-angle identities are derived from double-angle identities.
- They provide a way to break down angles into smaller components.
- Understanding these identities can help in solving integrals and other calculus problems.
In our specific problem, we need to recall the half-angle identity for cosine from the start, as given in the steps of the original solution. This fundamental understanding will guide us in checking the correctness of the given equation.
Cosine functions
Cosine functions are one of the basic trigonometric functions, alongside sine and tangent. They describe the relationship between the angle and ratios of sides in a right-angled triangle. The cosine function, often written as \cos(\theta), gives the ratio of the adjacent side to the hypotenuse.
Cosine functions have various useful properties and identities:
In trigonometric equations, cosine functions can often be rewritten using identities involving sine. For instance, \( \sin(\theta) = \cos(90^{\textdegree} - \theta) \). This relationship is helpful for converting between sine and cosine in problems.
Connect this with our problem: Given the statement \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1+\sin\theta}{2} \), we originally compare it to the identity using cosine: \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1 + \cos\theta}{2} \). By verifying equations, we can check the correctness and see whether cosine substitutions and simplifications lead to a valid outcome.
Cosine functions have various useful properties and identities:
- Cosine values range from -1 to 1.
- The cosine wave is an even function, meaning \cos(-\theta) = \cos(\theta).
- Fundamental Pythagorean identity: \cos^2(\theta) + \sin^2(\theta) = 1.
In trigonometric equations, cosine functions can often be rewritten using identities involving sine. For instance, \( \sin(\theta) = \cos(90^{\textdegree} - \theta) \). This relationship is helpful for converting between sine and cosine in problems.
Connect this with our problem: Given the statement \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1+\sin\theta}{2} \), we originally compare it to the identity using cosine: \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1 + \cos\theta}{2} \). By verifying equations, we can check the correctness and see whether cosine substitutions and simplifications lead to a valid outcome.
Verifying equations
Verifying trigonometric equations involves showing that both sides of the given equation are equal under the constraints of known identities. It requires a good grasp of fundamental trigonometric identities and relationships. Here's how you can approach verifying trigonometric equations:
In the given exercise, we aim to verify whether \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1 + \sin\theta}{2} \). By leveraging the half-angle identity for cosine, we know that \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1 + \cos\theta}{2} \).
Now, substituting this identity into the given statement:
\( \frac{1 + \cos\theta}{2} \e \frac{1 + \sin\theta}{2} \). This clearly shows a discrepancy, leading to the conclusion that the given statement is False.
Mastery of verifying equations balances understanding identities and aware algebraic manipulation practices.
- Identify known identities that are relevant to the equation. These can include basic Pythagorean identities, angle sum and difference identities, and half-angle identities.
- Rewrite each term using these identities to simplify the equation. Look for opportunities to make the form of both sides of the equation match more closely.
- Perform algebraic manipulations carefully to avoid mistakes.
- Simplify both sides of the equation to their simplest forms and compare. If both sides are equal, the equation is verified.
In the given exercise, we aim to verify whether \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1 + \sin\theta}{2} \). By leveraging the half-angle identity for cosine, we know that \( \cos^{2} \left( \frac{\theta}{2} \right) = \frac{1 + \cos\theta}{2} \).
Now, substituting this identity into the given statement:
\( \frac{1 + \cos\theta}{2} \e \frac{1 + \sin\theta}{2} \). This clearly shows a discrepancy, leading to the conclusion that the given statement is False.
Mastery of verifying equations balances understanding identities and aware algebraic manipulation practices.