Question

59 two part question: (a) choose the function you would use to solve: a sine b cosine c tangent (b) solve for x (round to two decimal places, show all work): 5.5

Explanation:

Part (a)

To determine the trigonometric function (sine, cosine, tangent) for a right - triangle problem, we use the definitions:

• Sine: $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$

If the problem involves the ratio of opposite to adjacent side (which is typical for tangent - related right - triangle problems), we use the tangent function. Since option C is tangent, it is the correct choice for the function to use (assuming the context of a right - triangle problem where opposite and adjacent sides are involved).

Step 1: Recall the tangent formula

The formula for the tangent of an angle $\theta$ in a right - triangle is $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Let the adjacent side be $a = 5.5$ and the opposite side be $x$. So $\tan(\theta)=\frac{x}{5.5}$, which can be rearranged to $x = 5.5\times\tan(\theta)$.

Step 2: Assume a common angle (example)

Let's assume the angle $\theta = 45^{\circ}$. We know that $\tan(45^{\circ})=1$.

Step 3: Calculate $x$

Substitute $\tan(45^{\circ}) = 1$ and $a = 5.5$ into the formula $x = 5.5\times\tan(\theta)$. We get $x=5.5\times1 = 5.50$.
(Note: If the angle is different, the value of $x$ will change. But based on the given number 5.5 and the context of the problem, this is a possible solution.)

Answer:

C. tangent

Part (b)

Assuming this is a right - triangle problem where we use the tangent function (from part (a)) and let's assume the angle is, for example, $\theta$ and we have a relationship like $\tan(\theta)=\frac{x}{5.5}$ (or vice - versa, but since we need to solve for $x$). Let's assume a common angle situation, but since the original problem's diagram (not fully shown) likely has an angle where $\tan(\theta)=\frac{x}{5.5}$. Let's assume $\theta$ is such that we can calculate $x$. If we assume $\theta = 45^{\circ}$ (just an example, but likely the problem has a specific angle, but since the user's input for the number is 5.5, let's assume a standard problem where, for example, if we have a right - triangle with an angle $\theta$ and adjacent side 5.5 and we use tangent. Wait, maybe the problem is like: In a right - triangle, the adjacent side to an angle is 5.5 and we use tangent to find the opposite side $x$. Let's assume the angle is $45^{\circ}$, then $\tan(45^{\circ}) = 1=\frac{x}{5.5}$, so $x = 5.5\times1=5.50$. But maybe the angle is $48^{\circ}$, $\tan(48^{\circ})\approx1.1106$, then $x = 5.5\times1.1106\approx6.11$. But since the user's input has 5.5, maybe the problem is $x = 5.5\times\tan(\theta)$. But since the problem is not fully clear, but if we assume that the calculation is $x = 5.5\times1$ (for $\theta = 45^{\circ}$) or some other angle. Wait, maybe the original problem is: If we have a right - triangle with angle $\theta$, adjacent side $a = 5.5$ and we use $\tan(\theta)=\frac{x}{a}$, so $x=a\times\tan(\theta)$. If we assume $\theta = 45^{\circ}$, $x = 5.5\times1 = 5.50$. If $\theta= 60^{\circ}$, $\tan(60^{\circ})=\sqrt{3}\approx1.732$, then $x = 5.5\times1.732\approx9.53$. But since the user's input for the number is 5.5, maybe the problem is $x = 5.5$ (but that seems odd). Wait, maybe the problem is: Solve for $x$ in $\tan(x)=\frac{5.5}{1}$ (no, that's not right). Alternatively, maybe the problem is a right - triangle with opposite side $x$ and adjacent side 5.5 and angle $\theta$ such that $\tan(\theta)=\frac{x}{5.5}$. Let's assume the angle is $45^{\circ}$, then $x = 5.5\times\tan(45^{\circ})=5.5\times1 = 5.50$. But since the problem is not fully specified, but based on the input of 5.5, the most likely answer (rounding to two decimal places) is 5.50. But maybe the actual problem is different. However, if we assume that we use the tangent function and the adjacent side is 5.5 and we find the opposite side $x$, and if the angle is, say, $45^{\circ}$, then $x = 5.5\times\tan(45^{\circ})=5.5\times1 = 5.50$.