Question

\(\csc \theta=\frac{15}{8}\)

Short answer

The solution to the exercise is \(θ = 32.172° or 32.172 \, in \, radians\)

Step-by-step solution

Convert to Sine Function

Convert the co-secant equation into a sine equation. All that needs to be done is taking the reciprocal of both sides as the co-secant of an angle is the reciprocal of sine of that angle. This gives the equation \(sin \theta = \frac{8}{15}\)

Find the Angle

Using a calculator, the sine inverse of \(\frac{8}{15}\) can be found. Enter sin-1(8/15) in the calculator and it gives a value of \(θ = 32.172° or 32.172 \, in \, radians\) which is the measure of the angle whose sine is \(8/15\).

Checking Validity

This step is important to confirm that the calculated angle is valid with the given equation. The original equation \(\csc \theta=\frac{15}{8}\) should be true when θ = 32.172°. Checking this, \( csc(32.172°) = \frac{1}{sin(32.172°)} = \frac{15}{8}\) . Hence θ = 32.172° is a solution.

Key concepts

Sine Function

The sine function is a fundamental component of trigonometry, and it relates an angle within a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse. In mathematical terms, we express this as
\( sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

When dealing with the sine function, it's crucial to understand that it oscillates between -1 and 1 as the angle \(\theta\) varies. Therefore, whenever you're given a cosecant function, which is the reciprocal of the sine function, converting it to sine can make solving for \(\theta\) simpler. This is because most calculators have a sine function, whereas the cosecant might need manual calculation.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine an angle when the value of a trigonometric ratio is known. For example, the inverse sine—denoted as \(sin^{-1}\) or arcsin—is used to find an angle when we know the sine of the angle.

Continuing from the sine function explanation, once we have \(sin(\theta)\), we can find the angle \(\theta\) by using the sine inverse function. In the given exercise, after converting cosecant to sine, the sine inverse is used to find the angle. It is important to remember that inverse trigonometric functions have specific ranges for their outputs to ensure the angles we get are unique. In the case of the sine inverse, the angle will fall between -90° and 90°, or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One such identity is the reciprocal relationship between sine and cosecant, which states that \(csc(\theta) = \frac{1}{sin(\theta)}\).

These identities are not only crucial to solving trigonometry problems but also to simplifying complex expression and verifying solutions. As seen in the final step of the solution, after finding \(\theta\), we use the original cosecant equation to check if the angle we found is valid, proving the practical utility of trigonometric identities in confirming our results.