Problem 11 Evaluate each expression below w... [FREE SOLUTION]

Chapter 5: Problem 11

Evaluate each expression below without using a calculator. (Assume any variables represent positive numbers.) \(\sin \left(2 \cos ^{-1} \frac{\sqrt{5}}{5}\right)\)

Short Answer

Expert verified

The answer is \( \frac{4}{5} \).

Step by step solution

Recall the Double Angle Formula for Sine

To solve this problem, we will use the double angle formula for sine: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). Before using the formula, identify \( \theta \) as the angle such that \( \cos(\theta) = \frac{\sqrt{5}}{5} \).

Determine the Sine of Theta

Since \( \theta \) is in a right triangle where \( \cos(\theta) = \frac{\sqrt{5}}{5} \), we can use the Pythagorean identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \) to find \( \sin(\theta) \). Calculate \( \sin(\theta) = \sqrt{1 - \left(\frac{\sqrt{5}}{5}\right)^2} = \sqrt{1 - \frac{5}{25}} = \sqrt{\frac{20}{25}} = \frac{2\sqrt{5}}{5} \).

Apply the Double Angle Formula

Now use the double angle formula: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). Substitute the values found: \( \sin(2\theta) = 2 \times \frac{2\sqrt{5}}{5} \times \frac{\sqrt{5}}{5} \).

Simplify the Expression

Simplify the expression by performing the multiplications: \( 2 \times \frac{2\sqrt{5}}{5} \times \frac{\sqrt{5}}{5} = \frac{2 \times 2 \times \sqrt{5} \times \sqrt{5}}{5 \times 5} = \frac{4 \times 5}{25} = \frac{20}{25} = \frac{4}{5} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double Angle Formula

The double angle formula for sine is a handy tool in trigonometry. It simplifies calculations involving angles that are double the measure of another angle. The formula is expressed as: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). This formula allows us to convert a complex angle into expressions involving simpler trigonometric functions.

Understanding the formula: It states that the sine of twice an angle equals twice the product of the sine and cosine of the angle.
Application: This is particularly useful when paired with inverse trigonometric functions like \( \cos^{-1} \), which means we can transform an arc cosine value back into usable sine and cosine functions.
Example: In the context of the problem, \( \theta \) was the angle for which the cosine was already known. Using the formula translated this into a straightforward computation of \( \sin(2\theta) \).

Using the double angle formula can make solving trigonometric problems much faster by transforming complex expressions into manageable calculations.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry, and it's a great tool to link sine and cosine. It states that for any angle \( \theta \):
\( \cos^2(\theta) + \sin^2(\theta) = 1 \).

Basic use: This identity shows the intrinsic relationship between \( \sin \theta \) and \( \cos \theta \). No matter what the angle is, their squares add up to 1.
Why it matters: When one of the trigonometric values is known, the other can always be computed using this identity. This is incredibly helpful for simplifying expressions and solving triangles.
Example in practice: In the given problem, we knew \( \cos \theta = \frac{\sqrt{5}}{5} \). Using the Pythagorean identity, we determined \( \sin \theta \) without any complicated calculations.

The Pythagorean identity not only offers insight into the right triangle properties but is a strong foundation for more advanced trigonometric manipulations.

Trigonometric Expression Simplification

Simplifying trigonometric expressions involves rewriting them to make problem-solving easier and to see relationships more clearly. This is valuable when dealing with complex terms.

Process of simplification: Often, simplification involves expressing equations in terms of simpler forms or substituting known identities, like the double angle formula or Pythagorean identity.
Techniques: Methods include breaking down expressions into known identities, factoring, and using reciprocal identities.
Example from the solution: Utilizing the values \( \sin(\theta) \) and \( \cos(\theta) \) alongside the double angle formula, we transformed \( \sin(2\theta) \) into a simplified fraction, \( \frac{4}{5} \).

Expression simplification allows for clearer and quicker mathematical understanding, turning complex problems into comprehensible solutions.

91影视

Evaluate each expression below without using a calculator. (Assume any variables represent positive numbers.) \(\sin \left(2 \cos ^{-1} \frac{\sqrt{5}}{5}\right)\)

Short Answer

Step by step solution

Recall the Double Angle Formula for Sine

Determine the Sine of Theta

Apply the Double Angle Formula

Simplify the Expression

Key Concepts

Double Angle Formula

Pythagorean Identity

Trigonometric Expression Simplification

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