Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the trigonometric expression as an algebraic expression. $$\cos (2 \arcsin x)$$

Step-by-Step Solution

Answer

\(\cos (2 \arcsin x)\) can be written as an algebraic expression \(1 - 2x^2\)

Step 1: Apply the identity

Apply the trigonometric identity for \(\cos(2 \arcsin x)\) which is \(1 - 2\sin^2(\arcsin x)\)

Step 2: Simplify the expression

\(\sin(\arcsin x)\) simplifies to \(x\). So the expression becomes \(1 - 2( x^2)\)

Step 3: Simplify Numerical Terms

Simplify the term with the \(x^2\) to get the final simplified algebraic expression, which is \(1 - 2x^2\)

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics. They consist of variables, numbers, and operators such as addition, subtraction, multiplication, and division. When dealing with algebraic expressions, we often aim to simplify or rearrange them to make calculations easier or to see relationships more clearly.
For example, in our task, the trigonometric expression \( \cos(2 \arcsin x) \) is rewritten as an algebraic expression. This is done by using trigonometric identities and simplification techniques, converting it into \( 1 - 2x^2 \).
Here are some key aspects to consider about algebraic expressions:

• They can include constants (fixed values) and variables (symbols that can vary).
• Operations must be carried out in a specific order, often following the rules of operations (PEMDAS/BODMAS).
• Simplifying expressions often makes them easier to work with and understand.

Understanding how to convert and simplify trigonometric expressions into algebraic ones enhances problem-solving skills in mathematics.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable involved, usually angles. They are invaluable tools in simplifying expressions and solving trigonometric equations.
One of the main identities used in the exercise is the double-angle formula. In this case, for cosine, it is \( \cos(2\theta) = 1 - 2\sin^2 \theta \). Such identities can transform complex trigonometric expressions into simpler or more recognizable forms.

• These identities help in transforming and relating various trigonometric functions.
• Using identities for simplification can transform a trigonometric function into an algebraic one.
• Common identities include Pythagorean identities, angle sum identities, and double angle identities.

Mastering trigonometric identities enables mathematicians and students alike to solve complex problems with ease, as they reduce expressions and equations to more manageable forms.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine angles when given the ratio of the sides of a triangle. They are the inverse functions of the regular trigonometric functions (sine, cosine, tangent), allowing us to find angles from known values.
In the exercise, \( \arcsin(x) \) is an inverse trigonometric function. It provides us with the angle whose sine is \( x \). Once this angle is found, it is manipulated using other trigonometric properties and identities.
Important points about these functions include:

• The output of an inverse trigonometric function is an angle.
• They are essential for solving trigonometric equations and real-world problems involving right triangles.
• Knowing their ranges is crucial, as it affects the angle values they return (e.g., \( \arcsin \) returns values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)).

Understanding and applying inverse trigonometric functions is critical in various fields, from engineering and physics to computer science, where angle measurements are necessary to solve problems.