Solve over the Interval csc(theta)=13/5 , (0,pi/2)
The question asks to find all values of the variable theta within the interval from 0 to pi/2 radians, for which the cosecant of theta equals 13/5. The cosecant function is the reciprocal of the sine function, so essentially you're being asked to solve for theta when sin(theta) is equal to the reciprocal of 13/5. The solution involves trigonometric concepts and potentially inverse trigonometric functions to find the angle theta that satisfies the equation within the specified interval.
$csc \left(\right. \theta\left.\right) = \frac{13}{5}$,$\left(\right. 0 , \frac{\pi}{2} \left.\right)$
Answer
Solution:
Step 1:
Isolate $\theta$ by applying the inverse cosecant to both sides: $\theta = \csc^{-1}\left(\frac{13}{5}\right)$.
Step 2:
Compute the value of $\theta$ on the right side.
Step 2.1:
Calculate $\csc^{-1}\left(\frac{13}{5}\right)$ to find $\theta \approx 0.39479111$.
Step 3:
Since the cosecant is positive in the first and second quadrants, determine the second solution by subtracting the reference angle from $\pi$: $\theta = \pi - 0.39479111$.
Step 4:
Calculate the exact value of $\theta$.
Step 4.1:
Eliminate the parentheses: $\theta = \pi - 0.39479111$.
Step 4.2:
Subtract $0.39479111$ from $\pi$: $\theta \approx 2.74680153$.
Step 5:
Determine the period of the cosecant function.
Step 5.1:
The period is given by $\frac{2\pi}{|b|}$.
Step 5.2:
Since $b=1$, the period formula simplifies to $\frac{2\pi}{|1|}$.
Step 5.3:
The absolute value of $1$ is $1$, so the period is $\frac{2\pi}{1}$.
Step 5.4:
Divide $2\pi$ by $1$ to get the period: $2\pi$.
Step 6:
The cosecant function repeats every $2\pi$ radians. Therefore, $\theta = 0.39479111 + 2\pi n$ and $\theta = 2.74680153 + 2\pi n$ for any integer $n$.
Step 7:
Test $n=0$ to see if the solution lies within $(0, \frac{\pi}{2})$.
Step 7.1:
Substitute $n=0$: $\theta = 0.39479111 + 2\pi(0)$.
Step 7.2:
Simplify the expression.
Step 7.2.1:
Perform the multiplication: $2\pi(0)$.
Step 7.2.1.1:
$0$ times $2$ is $0$: $\theta = 0.39479111 + 0\pi$.
Step 7.2.1.2:
$0$ times $\pi$ is $0$: $\theta = 0.39479111 + 0$.
Step 7.2.2:
Add $0.39479111$ and $0$: $\theta = 0.39479111$.
Step 7.3:
Since $0.39479111$ is within the interval $(0, \frac{\pi}{2})$, it is a valid solution: $\theta = 0.39479111$.
Knowledge Notes:
To solve the equation $\csc(\theta) = \frac{13}{5}$ over the interval $(0, \frac{\pi}{2})$, we follow these steps:
Inverse Trigonometric Functions: To isolate $\theta$, we use the inverse cosecant function, $\csc^{-1}$, which is the inverse operation of the cosecant function.
Reference Angles: In trigonometry, reference angles are used to find the angle associated with a given trigonometric value in different quadrants. Since the cosecant function is positive in the first and second quadrants, we consider the reference angle and its supplement.
Periodicity of Trigonometric Functions: Trigonometric functions are periodic, meaning they repeat values at regular intervals. The period of the cosecant function is $2\pi$ radians, which means the function repeats its values every $2\pi$ radians.
Solving Trigonometric Equations: When solving trigonometric equations, we often find multiple solutions due to the periodic nature of these functions. We use the formula $\theta = \theta_0 + 2\pi n$ (where $\theta_0$ is a known solution and $n$ is any integer) to find all possible solutions.
Interval Checking: After finding the general solutions, we must check which of them lie within the given interval. In this case, the interval is $(0, \frac{\pi}{2})$, so we only consider solutions that fall within this range.
Radians and Degrees: In this solution, we work with radians, which is the standard unit of angular measure used in mathematics. $\pi$ radians is equivalent to 180 degrees.
Calculators and Precision: When using a calculator to find the value of an inverse trigonometric function, the result is an approximation. The precision of this approximation depends on the calculator's settings and capabilities.
