Problem 24 - Precalculus Essentials

Question

In Exercises \(23-34\), verify each identity. $$\sin 2 \theta=\frac{2 \cot \theta}{1+\cot ^{2} \theta}$$

Step-by-Step Solution

Answer

Yes, \[sin 2θ = \frac{2cotθ}{1 + cot^2θ}\] is a verified identity.

Step 1: Express \(sin(2θ)\) in terms of \(sin(θ)\) and \(cos(θ)\)

Start with the left hand side (LHS) of the equation. Replace sin(2θ) with its equivalent expression \(2sin(θ)cos(θ)\). So, LHS = \[2sin(θ)cos(θ)\]

Step 2: Replace \(cot(θ)\) with \(cos(θ)/sin(θ)\)

On the right hand side (RHS) of the equation, replace \(cot(θ)\) with its equivalent expression \(cos(θ)/sin(θ)\). So, RHS = \[ \frac{2(cos(θ)/sin(θ))}{1 + (cos(θ)/sin(θ))^2}\] = \[\frac{2cos(θ)}{sin(θ) + cos^2(θ)/sin(θ)}\]

Step 3: Simplify the Right Hand Side

Simplify the RHS by multiplying the numerator and the denominator by \[sin(θ)\] to get rid of the fraction within the fraction. So, the RHS becomes: \[\frac{2cos(θ)sin(θ)}{sin^2(θ) + cos^2(θ)}\]Since \(sin^2(θ) + cos^2(θ) = 1\) (Pythagorean Identity), the RHS simplifies further to: \[2sin(θ)cos(θ)\]

Step 4: Compare LHS and RHS of the Equation

As seen above, the LHS of the equation: \[2sin(θ)cos(θ)\] is equal to the RHS of the equation: \[2sin(θ)cos(θ)\]. Hence, the identity is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental relationship in trigonometry. It states that for any angle \(\theta\), the sum of the square of the sine and the square of the cosine is always 1:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)

This identity arises from the Pythagorean theorem when considering a right triangle on the unit circle. In the given problem, the Pythagorean identity plays an essential role in simplifying the expression on the right-hand side of the equation.
When we reached the point where the expression became \(\sin^2(\theta) + \cos^2(\theta)\) in the denominator, this identity allowed us to replace it with 1, significantly simplifying the equation. Understanding this identity is crucial, as it is widely used in verifying and transforming trigonometric equations.

cotangent

The cotangent function, denoted as \(\cot(\theta)\), is a trigonometric ratio dealing with angles. It is defined as the reciprocal of the tangent function:

• \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\)

In our exercise, we needed to express \(\cot(\theta)\) in terms of sine and cosine to proceed with simplification. This step was crucial in modifying the right-hand side of the equation to align it with the desired identity.
By rewriting \(\cot(\theta)\) as \(\frac{\cos(\theta)}{\sin(\theta)}\), we could successfully work with the given expression and utilize other trigonometric identities to verify the equation. The cotangent is particularly useful in transformations and simplifying expressions, especially when we need to switch between different trigonometric functions.

double angle formulas

The double angle formulas are vital tools in trigonometry that make it easier to work with expressions involving angles that are twice another angle. For sine, the formula is:

• \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)

This formula simplifies the process of doubling angles, allowing us to rewrite and solve complex trigonometric identities.
In the given exercise, the double angle formula for sine was directly applied to express \(\sin(2\theta)\) in terms of \(\sin(\theta)\) and \(\cos(\theta)\). This step was pivotal in transforming the left-hand side of the equation, making it easier to compare and verify with the right-hand side.
Understanding and applying the double angle formulas is essential for efficiently solving trigonometric equations and verifying identities. They can also be used to derive other useful trigonometric relationships encountered in more complex mathematical problems.