Problem 53
Question
Simplify the given expression. $$\frac{\sin 2 x}{2 \sin x}$$
Step-by-Step Solution
Verified Answer
Question: Simplify the expression \(\frac{\sin 2 x}{2 \sin x}\).
Answer: \(\cos x\)
1Step 1: Apply the double angle formula for sine
Use the following double-angle formula: \(\sin 2x = 2\sin x\cos x\).
$$\frac{\sin 2 x}{2 \sin x} = \frac{2 \sin x \cos x}{2 \sin x}$$
2Step 2: Simplify the fraction by canceling out common terms
Notice that we can now cancel out the \(2\sin x\) term in both the numerator and the denominator:
$$\frac{2 \sin x \cos x}{2 \sin x} = \frac{\cancel{2 \sin x} \cos x}{\cancel{2 \sin x}}$$
3Step 3: Write the simplified expression
Once the terms have been canceled, we obtain the simplified expression:
$$\cos x$$
So, after simplifying the given expression, we found that:
$$\frac{\sin 2 x}{2 \sin x} = \cos x$$
Key Concepts
Double Angle FormulasTrigonometric IdentitiesSimplifying Fractions
Double Angle Formulas
Understanding how to simplify trigonometric expressions often requires the use of double angle formulas. These formulas are specific cases of trigonometric identities that express trigonometric functions of double angles, that is, angles of the form \(2x\), in terms of functions involving the original angle \(x\).
For sine, the double angle formula is given by \( \text{sin}(2x) = 2 \text{sin}(x) \text{cos}(x) \). This particular formula demonstrates how a trigonometric function of a double angle can be rewritten as a product of simpler trigonometric functions. In our exercise, applying the double angle formula was crucial to break down the complex expression into a more manageable form that allowed for further simplification.
For sine, the double angle formula is given by \( \text{sin}(2x) = 2 \text{sin}(x) \text{cos}(x) \). This particular formula demonstrates how a trigonometric function of a double angle can be rewritten as a product of simpler trigonometric functions. In our exercise, applying the double angle formula was crucial to break down the complex expression into a more manageable form that allowed for further simplification.
Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables where both sides of the equation are defined. They act as tools to rewrite trigonometric expressions in different forms which can simplify the problem-solving process.
Common identities include the Pythagorean identity \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \), as well as sum and difference formulas, half-angle formulas, and the double angle formulas we used in our exercise. By appropriately utilizing these identities, we can often transform complex expressions into simpler ones or even compute specific values without the need for a calculator. They are foundational in understanding trigonometry and indispensable in many branches of mathematics and applied sciences.
Common identities include the Pythagorean identity \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \), as well as sum and difference formulas, half-angle formulas, and the double angle formulas we used in our exercise. By appropriately utilizing these identities, we can often transform complex expressions into simpler ones or even compute specific values without the need for a calculator. They are foundational in understanding trigonometry and indispensable in many branches of mathematics and applied sciences.
Simplifying Fractions
Simplifying fractions is a fundamental skill in algebra that often applies in dealing with trigonometric expressions. When faced with a complex fraction like the one in our exercise, we aim to reduce it to its simplest form by eliminating common factors in the numerator and denominator.
In the given problem, we canceled out the \(2 \text{sin}(x)\) term that appeared in both the top and the bottom of the fraction. This step is akin to reducing the fraction \( \frac{2}{4} \) to \( \frac{1}{2} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case.
This simplification technique is not only crucial in arithmetic but also in algebra and calculus where it can turn a seemingly daunting task into a manageable one. It is important to remember that this process is valid as long as we are not canceling out zero, as division by zero is undefined.
In the given problem, we canceled out the \(2 \text{sin}(x)\) term that appeared in both the top and the bottom of the fraction. This step is akin to reducing the fraction \( \frac{2}{4} \) to \( \frac{1}{2} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case.
This simplification technique is not only crucial in arithmetic but also in algebra and calculus where it can turn a seemingly daunting task into a manageable one. It is important to remember that this process is valid as long as we are not canceling out zero, as division by zero is undefined.
Other exercises in this chapter
Problem 52
$$\text { Prove that } \cot (x+y)=\frac{\cot x \cot y-1}{\cot x+\cot y}$$.
View solution Problem 52
State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2 \sec ^{2} x$$
View solution Problem 53
Write the expression as an algebraic expression in \(v\). $$\cos \left(\tan ^{-1} v\right)$$
View solution Problem 53
$$\text { Prove the identity.}$$ $$\sin (x-\pi)=-\sin x$$
View solution