QuestionFind $\alpha$ in $[0^{\circ}, 90^{\circ}]$ such that $\csc \alpha = 1.3259174$ .

Studdy Solution

STEP 1

Assumptions1. The value of $\alpha$ is in the interval $\left[0^{\circ},90^{\circ}\right]$ . The given statement is $\csc \alpha=1.3259174$
3. We are looking for the value of $\alpha$ that satisfies the given statement

STEP 2

The cosecant function is defined as the reciprocal of the sine function. So, we can rewrite the given statement as follows $\sin \alpha = \frac{1}{\csc \alpha}$

STEP 3

Substitute the given value of $\csc \alpha$ into the equation.
$\sin \alpha = \frac{1}{1.325917}$

STEP 4

Calculate the value of $\sin \alpha$ .
$\sin \alpha = \frac{1}{1.3259174} \approx0.754200$

STEP 5

Now, we need to find the value of $\alpha$ that gives this sine value. We can do this by using the inverse sine function, also known as arcsine.
$\alpha = \arcsin(0.7542005)$

STEP 6

Calculate the value of $\alpha$ . Note that the arcsine function gives a result in radians, so we need to convert it to degrees.
$\alpha = \arcsin(0.7542005) \approx0.8480621 \, \text{radians}$ $\alpha =0.8480621 \times \frac{180}{\pi} \approx48.590176 \, \text{degrees}$ So, the value of $\alpha$ that satisfies the given statement is approximately $48.59^{\circ}$ .

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