Evaluate sin(2arccos(( square root of 2)/2))
The question provided is a trigonometric problem where you are asked to evaluate the sine of twice the arc cosine of the square root of 2 divided by 2. This involves understanding and applying trigonometric identities and inverse trigonometric functions. The problem requires you to first determine the angle whose cosine is (sqrt(2)/2), then find that angle's double, and finally calculate the sine of the resulting angle.
$sin \left(\right. 2 arccos \left(\right. \frac{\sqrt{2}}{2} \left.\right) \left.\right)$
Answer
Solution:
Step 1:
Determine the value of $arccos\left(\frac{\sqrt{2}}{2}\right)$, which is $\frac{\pi}{4}$. Therefore, we need to find the sine of twice this angle: $sin\left(2 \cdot \frac{\pi}{4}\right)$.
Step 2:
Simplify the expression by removing the common factor.
Step 2.1:
Extract the factor of 2 from the denominator: $sin\left(2 \cdot \frac{\pi}{2 \cdot 2}\right)$.
Step 2.2:
Eliminate the common factor of 2: $sin\left(\frac{\pi}{2}\right)$.
Step 2.3:
Express the simplified result: $sin\left(\frac{\pi}{2}\right)$.
Step 3:
Recognize that the sine of $\frac{\pi}{2}$ is exactly 1: $1$.
Knowledge Notes:
The problem involves evaluating a trigonometric expression that contains an inverse trigonometric function. Here are the relevant knowledge points:
Inverse Trigonometric Functions: These functions provide the angle that corresponds to a given trigonometric ratio. For instance, $arccos(x)$ gives the angle whose cosine is $x$.
Trigonometric Identities: There are several identities that can simplify trigonometric expressions. One such identity is the double angle formula for sine: $sin(2\theta) = 2sin(\theta)cos(\theta)$.
Exact Values: Some trigonometric functions have exact values for specific angles, like $0$, $\frac{\pi}{2}$, $\pi$, etc. For example, $sin\left(\frac{\pi}{2}\right) = 1$ and $cos\left(\frac{\pi}{2}\right) = 0$.
Simplification: Part of solving trigonometric problems can involve simplifying expressions by canceling common factors or using algebraic manipulation.
Radians vs Degrees: Trigonometric functions can be evaluated in either radians or degrees. In this problem, the angle is given in radians, where $\pi$ radians equals 180 degrees.
Understanding these concepts is crucial for solving trigonometric problems involving inverse functions and their compositions with other trigonometric functions.
