We have a right triangle $\triangle WXY$ with the following information:

• $\angle X$ is $90^\circ$ (right angle),
• $\angle W$ is $51^\circ$,
• The side $XY = 5$ units.

We are tasked with finding the following:

1. $WY$ (hypotenuse),
2. $WX$ (leg opposite $\angle W$),
3. $m \angle Y$ (the measure of angle $Y$).

Step 1: Find $m \angle Y$

Since the sum of the angles in any triangle is $180^\circ$, we can find $m \angle Y$:

$m \angle Y = 180^\circ - 90^\circ - 51^\circ = 39^\circ$

Step 2: Find $WY$ (Hypotenuse)

We can use the trigonometric function cosine to find $WY$. Since $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, we use the given angle $\angle W = 51^\circ$ and side $XY = 5$ (adjacent to $\angle W$):

$\cos(51^\circ) = \frac{5}{WY}$

Solving for $WY$:

$WY = \frac{5}{\cos(51^\circ)} \approx \frac{5}{0.6293} \approx 7.9$

Step 3: Find $WX$ (Opposite Side)

We can use the sine function to find $WX$. Since $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$:

$\sin(51^\circ) = \frac{WX}{WY}$

Using $WY \approx 7.9$:

$WX = 7.9 \times \sin(51^\circ) \approx 7.9 \times 0.7771 \approx 6.1$

Final Answers:

• $WY \approx 7.9$
• $WX \approx 6.1$
• $m \angle Y = 39^\circ$

Do you want further details or have any questions?

Here are 5 related questions to explore:

1. How do you calculate the remaining angle in a right triangle if you know one of the acute angles?
2. How do sine and cosine relate to each other in a right triangle?
3. What is the importance of knowing the hypotenuse in right triangle trigonometry?
4. Can you use tangent to solve for sides in this right triangle?
5. How does rounding affect the accuracy of trigonometric calculations?

Tip: Always make sure your calculator is set to degrees, not radians, when working with angles measured in degrees.