/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 97 Use the identity for \(\sin (\al... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 97

Use the identity for \(\sin (\alpha+\beta)\) to rewrite \(\sin 2 \alpha .[5.2]\)

Short Answer

Expert verified
\(\sin 2\alpha = 2\sin \alpha \cos \alpha\)

Step by step solution

01

Identify the relevant formula

Recall the double angle formula for sine, which is \(\sin 2\alpha = 2\sin \alpha \cos \alpha \). This formula is an extension of the formula for \(\sin(\alpha+\beta)\) which allows us to simplify expressions involving the sine of a double angle.
02

Apply the formula

\(\sin 2 \alpha\) can be rewritten using the double angle formula for sine, so we get \(\sin 2 \alpha = 2\sin \alpha \cos \alpha\). This simplifies the term \(\sin 2\alpha\).

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are fundamental relationships between trigonometric functions. They help us simplify and transform trigonometric expressions and are crucial for solving equations involving angles. In this exercise, we deal with the double angle formula. Knowing such identities is vital for various fields like engineering and physics since they enable us to work with angles in a precise manner.
  • The double angle identity for sine: \( \sin 2\alpha = 2\sin \alpha \cos \alpha \).
  • These identities often arise from well-known angle addition formulas.
  • Understanding these identities allows us to transform and simplify expressions.

Recognizing these identities and knowing how to apply them can turn complex problems into more manageable ones. It鈥檚 like having a mathematical toolkit that enables you to navigate through diverse problems with ease.
Sine Function
The sine function, a staple of trigonometry, relates the opposite side to the hypotenuse in a right triangle. More generally, it is a crucial component in describing wave-like patterns and phenomena in various scientific disciplines.
  • The sine function is periodic, meaning it repeats its values in regular intervals of \(2\pi\).
  • It is defined for every real number, which makes it very versatile.
  • Beyond simple triangles, the sine function is used in oscillatory motion and signal processing.

Understanding how the sine function behaves allows us to harness it in formulas such as the double angle formula. Transforming the sine function using identities helps push the boundaries of how we solve and simplify mathematical problems.
Angle Addition Formula
The angle addition formula is essential in trigonometry for breaking down complex angles into simpler components. It allows us to derive other important trigonometric identities, like the double angle formulas. The general formula for sine is:
\(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)
  • This formula facilitates calculations by simplifying angles into known values.
  • It is especially useful when dealing with compound angles in various applications, like physics.
  • The angle addition formula is the foundation upon which the double angle formulas are built.

By understanding the angle addition formula, we can tackle more complex trigonometric expressions by breaking them down into simpler, more manageable parts, making our calculations efficient and effective.

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Most popular questions from this chapter

A right triangle has sides of lengths \(3,4,\) and 5 inches. a. Find the perimeter of the triangle. b. Find the area of the triangle.

Solve each equation for exact solutions in the interval \(0 \leq x<2 \pi\) $$\sin x-\cos x=1$$

Is the dot product an associative operation? That is, given any nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w},\) does $$ (\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w}) ? $$

In Exercises 73 to \(88,\) verify the identity. $$\cos (\alpha-\beta)-\cos (\alpha+\beta)=2 \sin \alpha \sin \beta$$

The drag (resistance) on a fish when it is swimming is two to three times the drag when it is gliding. To compensate for this, some fish swim in a sawtooth pattern, as shown in the accompanying figure. The ratio of the amount of energy the fish expends when swimming upward at angle \(\beta\) and then gliding down at angle \(\alpha\) to the energy it expends swimming horizontally is given by $$E_{R}=\frac{k \sin \alpha+\sin \beta}{k \sin (\alpha+\beta)}$$ where \(k\) is a value such that \(2 \leq k \leq 3,\) and \(k\) depends on the assumptions we make about the amount of drag experienced by the fish. Find \(E_{R}\) for \(k=2, \alpha=10^{\circ},\) and \(\beta=20^{\circ}\)
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