Evaluate sin(x)=cos(x)
The problem is asking for an evaluation of the trigonometric equality \( sin(x) = cos(x) \). To address this, one would typically look for the specific angle(s) \( x \) for which the sine and cosine functions yield the same value. This typically involves understanding the unit circle, the relationship between sine and cosine, and employing the properties of the trigonometric functions to find the solution(s).
$sin \left(\right. x \left.\right) = cos \left(\right. x \left.\right)$
Answer
Solution:
Step 1:
Isolate the trigonometric functions by dividing the entire equation by $\cos(x)$.
$\frac{\sin(x)}{\cos(x)} = \frac{\cos(x)}{\cos(x)}$
Step 2:
Transform the left side of the equation into the tangent function.
$\tan(x) = \frac{\cos(x)}{\cos(x)}$
Step 3:
Eliminate the common $\cos(x)$ terms.
Step 3.1:
Remove the common $\cos(x)$.
$\tan(x) = \frac{\cancel{\cos(x)}}{\cancel{\cos(x)}}$
Step 3.2:
Rewrite the simplified equation.
$\tan(x) = 1$
Step 4:
Solve for $x$ by applying the arctangent to both sides.
$x = \arctan(1)$
Step 5:
Determine the exact value of the arctangent.
Step 5.1:
The arctangent of 1 is known to be $\frac{\pi}{4}$.
$x = \frac{\pi}{4}$
Step 6:
Identify the additional solutions based on the periodicity of the tangent function.
$x = \pi + \frac{\pi}{4}$
Step 7:
Combine like terms to simplify the expression.
Step 7.1:
Express $\pi$ with a denominator of 4.
$x = \pi \cdot \frac{4}{4} + \frac{\pi}{4}$
Step 7.2:
Add the fractions together.
Step 7.2.1:
Combine $\pi$ and $\frac{4}{4}$.
$x = \frac{\pi \cdot 4}{4} + \frac{\pi}{4}$
Step 7.2.2:
Sum the numerators over the common denominator.
$x = \frac{\pi \cdot 4 + \pi}{4}$
Step 7.3:
Simplify the numerator.
Step 7.3.1:
Rearrange the terms.
$x = \frac{4 \cdot \pi + \pi}{4}$
Step 7.3.2:
Add the terms.
$x = \frac{5\pi}{4}$
Step 8:
Calculate the period of the tangent function.
Step 8.1:
The period is given by $\frac{\pi}{|b|}$.
Step 8.2:
Since $b = 1$, plug it into the period formula.
$\frac{\pi}{|1|}$
Step 8.3:
The absolute value of 1 is 1.
$\frac{\pi}{1}$
Step 8.4:
Divide $\pi$ by 1.
$\pi$
Step 9:
Recognize that the tangent function repeats every $\pi$ radians.
$x = \frac{\pi}{4} + \pi n , \frac{5\pi}{4} + \pi n$, where $n$ is any integer.
Step 10:
Combine the solutions into a general form.
$x = \frac{\pi}{4} + \pi n$, where $n$ is any integer.
Knowledge Notes:
The problem involves solving a trigonometric equation where the sine and cosine functions are equal. The key points to understand in solving this problem are:
Trigonometric Identities: Understanding that $\tan(x) = \frac{\sin(x)}{\cos(x)}$ is a fundamental trigonometric identity.
Inverse Trigonometric Functions: Knowing how to use inverse trigonometric functions, such as $\arctan(x)$, to find the angle whose tangent is $x$.
Trigonometric Function Periodicity: Recognizing that trigonometric functions repeat their values in regular intervals, known as their periods. For the tangent function, the period is $\pi$ radians.
Simplifying Expressions: Being able to simplify expressions by combining like terms and using common denominators.
Absolute Value: Understanding that the absolute value of a number is its distance from zero on the number line.
General Solution for Trigonometric Equations: Knowing that the solutions to trigonometric equations can be expressed in a general form that accounts for their periodicity. For the tangent function, this means adding integer multiples of the period to the specific solutions found.
