Problem 41
Question
Use words to describe the given formula. $$\sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$$
Step-by-Step Solution
Verified Answer
The given formula states that the sum of the sines of two angles is equal to twice the product of the sine of half their sum and the cosine of half their difference.
1Step 1: Identify the given formula
The provided formula is a trigonometric identity that connects the sum of sine of two angles with their half angles: \( \sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} \)
2Step 2: Express sine addition in terms of cosine
Expanding the left side of the equation using the sine addition formula which is given by: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\), we can rewrite the given expression as: \(\sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}\).
3Step 3: Verbalize the Identity
This identity tells us that the sum of the sine of two angles (alpha and beta) is equal to twice the product of sine of the half sum of these two angles and the cosine of half of their difference.
Key Concepts
Sine Addition FormulaTrigonometryHalf-Angle Identities
Sine Addition Formula
Understanding the sine addition formula is crucial for various applications in trigonometry. It's one of the basic tools in our trigonometric toolkit for breaking down more complex problems.
At its heart, the sine addition formula provides a way to simplify the sum of the sine of two angles. Specifically, the formula states that for any two angles \( \alpha \) and \( \beta \) the formula is given by: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \.\] This identity is immensely powerful as it allows us to relate the sine of a sum to products of sines and cosines of the two angles separately.
Here's a simplified interpretation: when you hear a phrase like 'the sine of the sum,' the formula provides a transition into a product of individual trigonometric functions. This transformation can make other mathematics operations easier. For illustration, consider trying to integrate the sine of a sum; without the sine addition formula, this task becomes much more complex.
At its heart, the sine addition formula provides a way to simplify the sum of the sine of two angles. Specifically, the formula states that for any two angles \( \alpha \) and \( \beta \) the formula is given by: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \.\] This identity is immensely powerful as it allows us to relate the sine of a sum to products of sines and cosines of the two angles separately.
Here's a simplified interpretation: when you hear a phrase like 'the sine of the sum,' the formula provides a transition into a product of individual trigonometric functions. This transformation can make other mathematics operations easier. For illustration, consider trying to integrate the sine of a sum; without the sine addition formula, this task becomes much more complex.
Trigonometry
Trigonometry is a branch of mathematics that delves into the relationships between the angles and sides of triangles. It's not just about solving for unknown angles and sides, though; trigonometry is a foundational element in calculus, physics, engineering, and many other fields.
The discipline is built on the concept of trigonometric functions, which include sine, cosine, and tangent, to name just a few. These functions relate an angle of a right triangle to the ratios of two of its sides. Through the use of various trigonometric identities, like the sine addition formula, more complex relationships and patterns can be solved. For instance, trigonometry allows us to determine the height of a building using a simple tool called a clinometer, and understand waves in physics by examining their sinusoidal representations.
Trigonometry underpins many principles in the real world – it helps in designing mechanical structures, understanding sound waves, and navigating by the stars. For students to truly grasp its applications, they need to build a strong foundation in basic trigonometric functions and identities.
The discipline is built on the concept of trigonometric functions, which include sine, cosine, and tangent, to name just a few. These functions relate an angle of a right triangle to the ratios of two of its sides. Through the use of various trigonometric identities, like the sine addition formula, more complex relationships and patterns can be solved. For instance, trigonometry allows us to determine the height of a building using a simple tool called a clinometer, and understand waves in physics by examining their sinusoidal representations.
Trigonometry underpins many principles in the real world – it helps in designing mechanical structures, understanding sound waves, and navigating by the stars. For students to truly grasp its applications, they need to build a strong foundation in basic trigonometric functions and identities.
Half-Angle Identities
Half-angle identities are a subset of trigonometric identities that specifically deal with the sines, cosines, and tangents of half-angles. These are especially useful when dealing with problems that involve powers of trigonometric functions or transforming expressions to facilitate easier integration or differentiation.
The general form of a half-angle identity for sine and cosine is given by: \[ \sin(\frac{\alpha}{2}) = \pm\sqrt{\frac{1 - \cos(\alpha)}{2}} \] and \[ \cos(\frac{\alpha}{2}) = \pm\sqrt{\frac{1 + \cos(\alpha)}{2}} \]. The plus-or-minus sign depends on the quadrant in which the half-angle resides. These identities are derived from the double-angle formulas and are fundamentally important in solving trigonometry problems where half-angles are present.
For students dealing with these identities, a firm understanding of the unit circle and the properties of square roots are essential. Real-world applications of half-angle identities are numerous, including but not limited to, computer graphics, where they help in calculating angles for light reflection and refraction, and signal processing, where they simplify the analysis of oscillatory functions.
The general form of a half-angle identity for sine and cosine is given by: \[ \sin(\frac{\alpha}{2}) = \pm\sqrt{\frac{1 - \cos(\alpha)}{2}} \] and \[ \cos(\frac{\alpha}{2}) = \pm\sqrt{\frac{1 + \cos(\alpha)}{2}} \]. The plus-or-minus sign depends on the quadrant in which the half-angle resides. These identities are derived from the double-angle formulas and are fundamentally important in solving trigonometry problems where half-angles are present.
For students dealing with these identities, a firm understanding of the unit circle and the properties of square roots are essential. Real-world applications of half-angle identities are numerous, including but not limited to, computer graphics, where they help in calculating angles for light reflection and refraction, and signal processing, where they simplify the analysis of oscillatory functions.
Other exercises in this chapter
Problem 40
In Exercises \(39-46,\) use a half-angle formula to find the exact value of each expression. $$\cos 22.5^{\circ}$$
View solution Problem 40
Verify each identity. $$\cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta$$
View solution Problem 41
Verify each identity. $$\frac{\tan 2 \theta+\cot 2 \theta}{\csc 2 \theta}=\sec 2 \theta$$
View solution Problem 41
Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$2 \cos ^{2} x+3 \cos x+1=0$$
View solution