Problem 10 - Algebra and Trigonometry

Question

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Verify each identity. $$ \frac{\cos (\alpha-\beta)}{\sin \alpha \sin \beta}=\cot \alpha \cot \beta+1 $$

Step-by-Step Solution

Answer

By substituting the formula for the cosine of the difference of two angles into the left side of the equation and then simplifying, it has been proven that \(\frac{\cos (\alpha-\beta)}{\sin \alpha \sin \beta} = \cot \alpha \cot \beta + 1 \)

Step 1: Substitute the formula for cos difference

Substitute the formula for the cosine of the difference of two angles into the left side of the equation. This gives: \[\frac{\cos a \cos b + \sin a \sin b}{\sin a \sin b}\]

Step 2: Simplify

Now, simplify the expression by dividing each term in the numerator by \(\sin a \sin b\). This turns the expression into: \[\cot a \cot b + 1\] which matches the right side of the original equation.

Key Concepts

Cosine of Difference Formula

The cosine of difference formula is an important identity in trigonometry. It allows us to find the cosine of the difference between two angles. This formula is given by: \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]This identity is helpful because it breaks down a more complex problem (the cosine of a difference) into simpler components using the cosine and sine of individual angles. You can think of it like taking a puzzle apart and examining each piece. This formula is often used to simplify expressions and solve trigonometric equations. In exercises, substituting this formula can help reveal new aspects of the problem, making it easier to address.

Cotangent

Cotangent, often abbreviated as \( \cot \), is another fundamental trigonometric function. It is defined as the reciprocal of the tangent function:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \]Here are some key points about cotangent:

• Cotangent in Right Triangles: If \( \theta \) is an angle in a right triangle, \( \cot \theta \) can be thought of as the ratio of the adjacent side to the opposite side.
• Cotangent and Slope: Like tangent, cotangent can be associated with slopes, as it relates the rise and run of a line segment.
• Uses of Cotangent: It is mainly used in solving problems where division of sine and cosine is beneficial, especially in identities and integrals.

Understanding cotangent and its properties is crucial, as it plays a significant role in simplifying trigonometric identities and equations.

Angle Subtraction

In trigonometry, dealing with the subtraction of angles is a critical concept. It helps in understanding how changes in angles affect trigonometric functions. When you see expressions like \( \alpha - \beta \), you're looking at angle subtraction.Here's why angle subtraction is useful:

• It allows us to derive other trigonometric identities, like the cosine and sine difference formulas.
• In geometry and real-world applications, angle subtraction is used to find angles in polygons or solve navigation problems.
• It simplifies equations that might otherwise seem too complex at first glance.

By practicing problems that involve angle subtraction, such as using the cosine of difference formula, one becomes better at interpreting and solving complex trigonometric expressions.

Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. It's not just useful for mathematicians, but also for engineers, architects, and anyone involved in design and construction. Trigonometry involves:

• Trigonometric Functions: These include sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent). They help to relate the angles of a triangle to the ratio of its sides.
• Periodic Nature: Trigonometric functions are periodic, which signifies that they repeat their values in regular intervals (helpful in modeling waves and oscillations).
• Applications: Beyond triangles, trigonometry is used in calculus, physics, and to solve real-world problems involving waves, sound, light, and more.

Understanding trigonometry and learning how to manipulate its functions and identities can be powerful, providing solutions to mathematical and practical challenges alike.