Problem 91
Question
In Exercises 91-98, use the sum-to-product formulas to write the sum or difference as a product. \( \sin 3 \theta + \sin \theta \)
Step-by-Step Solution
Verified Answer
The expression \( \sin 3 \theta + \sin \theta \) can be written as a product using the sum-to-product formulas as \( 2 \sin(2 \theta) \cos (\theta)\).
1Step 1: Identify the variables in the formula
In the sum-to-product formula, \(a\) and \(b\) correspond to the arguments of the sine functions in the given expression. Here, \(a = 3 \theta\) and \(b = \theta\).
2Step 2: Apply the sum-to-product formula
Substitute \(a = 3 \theta\) and \(b = \theta\) into the sum-to-product formula: \[ \sin a + \sin b = 2 \sin \left( \frac{a + b}{2} \right) \cos \left( \frac{a - b}{2} \right) \] to get \[ \sin 3 \theta + \sin \theta = 2 \sin \left( \frac{3 \theta + \theta}{2} \right) \cos \left( \frac{3 \theta - \theta}{2} \right) \].
3Step 3: Simplify the expression
Simplify the expression to get the final result. This results in \[ 2 \sin(2 \theta) \cos (\theta)\].
Key Concepts
TrigonometrySine FunctionTrigonometric Identities
Trigonometry
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. At its core are trigonometric functions, which are functions of an angle, commonly used to model periodic phenomena.
One of the primary applications of trigonometry is to solve problems involving right-angled triangles, where one angle is 90 degrees. However, trigonometry extends far beyond right triangles to any triangle and can be applied to any shape by dividing it into triangles.
To understand trigonometry, it's essential to familiarize ourselves with concepts like the unit circle, which is a circle with a radius of one unit centered at the origin of a coordinate system. Angles in trigonometry are usually measured in radians, where one complete revolution is equal to 2π radians, providing a way to relate circular and linear motion.
Moreover, trigonometry is also about transforming and manipulating equations to make them easier to understand or solve. The sum-to-product formulas are a perfect example of these transformations.
One of the primary applications of trigonometry is to solve problems involving right-angled triangles, where one angle is 90 degrees. However, trigonometry extends far beyond right triangles to any triangle and can be applied to any shape by dividing it into triangles.
To understand trigonometry, it's essential to familiarize ourselves with concepts like the unit circle, which is a circle with a radius of one unit centered at the origin of a coordinate system. Angles in trigonometry are usually measured in radians, where one complete revolution is equal to 2π radians, providing a way to relate circular and linear motion.
Moreover, trigonometry is also about transforming and manipulating equations to make them easier to understand or solve. The sum-to-product formulas are a perfect example of these transformations.
Sine Function
The sine function, denoted as \(\sin\), is one of the fundamental trigonometric functions. It is defined as the ratio of the length of the side of the triangle opposite the angle to the length of the hypotenuse in a right-angled triangle.
In the context of the unit circle, \(\sin\) of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Mathematically, if \( \theta \) is an angle, then \(\sin \theta \) represents the y-coordinate of this intersection point.
The sine function is periodic with a period of 2π, meaning that it repeats its values every 2π radians. It is also an odd function, satisfying the property that \(\sin(-\theta) = -\sin(\theta)\). Knowing the sine function is crucial for understanding other trigonometric concepts, as it relates to various phenomena, such as waves and oscillations, in physics and engineering.
In the context of the unit circle, \(\sin\) of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Mathematically, if \( \theta \) is an angle, then \(\sin \theta \) represents the y-coordinate of this intersection point.
The sine function is periodic with a period of 2π, meaning that it repeats its values every 2π radians. It is also an odd function, satisfying the property that \(\sin(-\theta) = -\sin(\theta)\). Knowing the sine function is crucial for understanding other trigonometric concepts, as it relates to various phenomena, such as waves and oscillations, in physics and engineering.
Trigonometric Identities
Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. These identities are useful in simplifying expressions and solving trigonometric equations.
There are several types of trigonometric identities, such as reciprocal identities, Pythagorean identities, and sum-to-product formulas. Sum-to-product formulas are particularly handy when dealing with the addition or subtraction of two trigonometric functions, allowing us to express sums or differences as products of functions.
For example, the sum-to-product identities can be written as follows:
\[\sin a + \sin b = 2 \sin \left( \frac{a + b}{2} \right) \cos \left( \frac{a - b}{2} \right)\]
\[\sin a - \sin b = 2 \cos \left( \frac{a + b}{2} \right) \sin \left( \frac{a - b}{2} \right)\]
These identities are derived from the angle sum and difference identities and provide a method to simplify mathematical problems in trigonometry.
There are several types of trigonometric identities, such as reciprocal identities, Pythagorean identities, and sum-to-product formulas. Sum-to-product formulas are particularly handy when dealing with the addition or subtraction of two trigonometric functions, allowing us to express sums or differences as products of functions.
For example, the sum-to-product identities can be written as follows:
\[\sin a + \sin b = 2 \sin \left( \frac{a + b}{2} \right) \cos \left( \frac{a - b}{2} \right)\]
\[\sin a - \sin b = 2 \cos \left( \frac{a + b}{2} \right) \sin \left( \frac{a - b}{2} \right)\]
These identities are derived from the angle sum and difference identities and provide a method to simplify mathematical problems in trigonometry.
Other exercises in this chapter
Problem 90
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View solution Problem 91
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In Exercises 89 - 92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression.Verify your answer algebraically.
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