Evaluate csc(theta) = square root of 2
The given problem asks for an evaluation of the cosecant function, denoted as "csc," at a particular angle theta such that the value of cosecant of theta equals the square root of 2. The task involves determining the specific angle(s) theta that would satisfy this trigonometric equation. The cosecant function is the reciprocal of the sine function, and so this problem fundamentally involves understanding relationships within the unit circle and the values of trigonometric functions at various angles.
$csc \left(\right. \theta\left.\right) = \sqrt{2}$
Answer
Solution:
Step 1:
Apply the inverse cosecant to both sides to isolate $\theta$: $\theta= \csc^{-1}(\sqrt{2})$
Step 2:
Resolve the expression on the right.
Step 2.1:
The precise value of $\csc^{-1}(\sqrt{2})$ is $\frac{\pi}{4}$, therefore $\theta= \frac{\pi}{4}$.
Step 3:
Cosecant is positive in the first and second quadrants. To find an additional solution in the second quadrant, subtract the reference angle from $\pi$: $\theta= \pi - \frac{\pi}{4}$.
Step 4:
Perform the subtraction $\pi - \frac{\pi}{4}$.
Step 4.1:
Express $\pi$ as a fraction with the same denominator: $\theta= \pi \times \frac{4}{4} - \frac{\pi}{4}$.
Step 4.2:
Merge the fractions.
Step 4.2.1:
Combine $\pi$ and $\frac{4}{4}$: $\theta= \frac{\pi \times 4}{4} - \frac{\pi}{4}$.
Step 4.2.2:
Add the numerators over the common denominator: $\theta= \frac{\pi \times 4 - \pi}{4}$.
Step 4.3:
Simplify the numerator.
Step 4.3.1:
Rearrange to $4$ times $\pi$: $\theta= \frac{4 \times \pi - \pi}{4}$.
Step 4.3.2:
Subtract $\pi$ from $4\pi$: $\theta= \frac{3\pi}{4}$.
Step 5:
Determine the period of $csc(\theta)$.
Step 5.1:
The period is found using the formula $\frac{2\pi}{|b|}$.
Step 5.2:
Insert $1$ for $b$ in the period equation: $\frac{2\pi}{|1|}$.
Step 5.3:
The absolute value of a number is its magnitude from zero. The magnitude of $1$ from $0$ is $1$: $\frac{2\pi}{1}$.
Step 5.4:
Divide $2\pi$ by $1$: $2\pi$.
Step 6:
Since the period of $csc(\theta)$ is $2\pi$, the function repeats every $2\pi$ radians. Thus, $\theta= \frac{\pi}{4} + 2\pi n$ and $\theta= \frac{3\pi}{4} + 2\pi n$ for any integer $n$.
Knowledge Notes:
Cosecant ($csc$) is the reciprocal of the sine function. The cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the opposite side.
The inverse cosecant ($\csc^{-1}$ or $arccsc$) is the function that returns the angle whose cosecant is a given number.
The exact values of trigonometric functions for specific angles (like $\frac{\pi}{4}$) are often known from trigonometric identities or the unit circle.
A function's period is the length of the interval over which the function's values repeat. For the sine and cosecant functions, the period is $2\pi$.
The general solution for trigonometric equations considers all angles where the function has the same value, which are found by adding integer multiples of the period to the reference angles.
The absolute value of a number, denoted as $|a|$, is the non-negative value of $a$ without regard to its sign.
