Problem 40
Question
Use words to describe the given formula. $$\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha-\beta)+\cos (\alpha+\beta)]$$
Step-by-Step Solution
Verified Answer
The product of the cosine of alpha and the cosine of beta equals half the sum of the cosine of the difference between alpha and beta, and the cosine of the sum of alpha and beta.
1Step 1: Identify formula
Identify the formula to be expressed in words. The formula given here is a standard trigonometric identity, specifically the product-to-sum formula for cosine.
2Step 2: Explain the formula
Break down the formula into its components. The cosine of \(\alpha\) multiplied with the cosine of \(\beta\) is equal to one-half the sum of [the cosine of the difference of \(\alpha\) and \(\beta\), and the cosine of the sum of \(\alpha\) and \(\beta\)].
Key Concepts
CosineProduct-to-Sum FormulaTrigonometric Functions
Cosine
The cosine function is a fundamental trigonometric function that describes the relationship between the lengths of sides of a right triangle and the angles within the triangle. Specifically, cosine is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
For an angle \( \alpha \), denoted as \( \cos(\alpha) \), it reveals how 'wide' the angle is by evaluating the horizontal distance when projected onto a unit circle. The function oscillates between -1 and 1, corresponding to the full range of possible right triangle ratios or positions on the unit circle.
In the context of the product-to-sum formula, the cosine function interacts with another cosine function through multiplication. Understanding this function by itself is key because it helps students grasp how it behaves under multiplication and how it relates to the formula as a whole.
For an angle \( \alpha \), denoted as \( \cos(\alpha) \), it reveals how 'wide' the angle is by evaluating the horizontal distance when projected onto a unit circle. The function oscillates between -1 and 1, corresponding to the full range of possible right triangle ratios or positions on the unit circle.
In the context of the product-to-sum formula, the cosine function interacts with another cosine function through multiplication. Understanding this function by itself is key because it helps students grasp how it behaves under multiplication and how it relates to the formula as a whole.
Product-to-Sum Formula
The product-to-sum formula is a type of trigonometric identity that expresses the product of two cosines in terms of the sum of two cosine functions. Trigonometric identities are various formulas that relate trigonometric functions to one another. They are essential for simplifying expressions and solving equations involving trigonometric functions.
The specific product-to-sum formula given in the exercise is \( \cos(\alpha)\cos(\beta) = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)] \). This means that instead of working directly with the product of two cosines, you can rewrite it as the average of two cosines where one involves the difference of the angles, and the other involves the sum of the angles.
By using the product-to-sum formula, complex trigonometric expressions can be converted into simpler forms that are easier to integrate, differentiate, or solve. For students, mastering this formula is critical for success in advanced mathematics and physics problems.
The specific product-to-sum formula given in the exercise is \( \cos(\alpha)\cos(\beta) = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)] \). This means that instead of working directly with the product of two cosines, you can rewrite it as the average of two cosines where one involves the difference of the angles, and the other involves the sum of the angles.
By using the product-to-sum formula, complex trigonometric expressions can be converted into simpler forms that are easier to integrate, differentiate, or solve. For students, mastering this formula is critical for success in advanced mathematics and physics problems.
Trigonometric Functions
Trigonometric functions are a set of functions that are fundamental to the study of triangles, periodic phenomena, and many other areas in mathematics and science. These functions include sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), along with their reciprocals cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).
Each of these functions helps to describe a unique aspect of the angle or triangle being studied. The sine function, for example, defines the ratio of the opposite side to the hypotenuse of a right triangle. While the tangent represents the ratio of the opposite side to the adjacent side.
Understanding how these functions work, and how they relate to each other through identities like the product-to-sum formula, equips students with the tools to navigate through a host of problems in calculus, geometry, physical sciences, and engineering. Appreciating these relationships and their applications is what draws the beautiful picture of mathematics where everything is interconnected.
Each of these functions helps to describe a unique aspect of the angle or triangle being studied. The sine function, for example, defines the ratio of the opposite side to the hypotenuse of a right triangle. While the tangent represents the ratio of the opposite side to the adjacent side.
Understanding how these functions work, and how they relate to each other through identities like the product-to-sum formula, equips students with the tools to navigate through a host of problems in calculus, geometry, physical sciences, and engineering. Appreciating these relationships and their applications is what draws the beautiful picture of mathematics where everything is interconnected.
Other exercises in this chapter
Problem 39
In Exercises \(39-46,\) use a half-angle formula to find the exact value of each expression. $$\sin 15^{\circ}$$
View solution Problem 39
Verify each identity. $$\sin (\alpha+\beta)+\sin (\alpha-\beta)=2 \sin \alpha \cos \beta$$
View solution Problem 40
Verify each identity. $$\cot ^{2} 2 x+\cos ^{2} 2 x+\sin ^{2} 2 x=\csc ^{2} 2 x$$
View solution Problem 40
Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$2 \sin ^{2} x+\sin x-1=0$$
View solution