Let's solve each triangle problem one by one.

Problem 1

For the first triangle $\triangle TUV$:

Given:

• $\angle V = 15^\circ$
• Hypotenuse $TU = 69$
• Adjacent side $UV = x$

We need to solve for $x$ using trigonometry.

Since $x$ is adjacent to $\angle V$ and $69$ is the hypotenuse, we can use the cosine function: $\cos(15^\circ) = \frac{x}{69}$

Solving for $x$: $x = 69 \cdot \cos(15^\circ)$

Using a calculator to find $\cos(15^\circ) \approx 0.9659$: $x \approx 69 \cdot 0.9659$ $x \approx 66.7$

So, $x \approx 66.7$ (rounded to the nearest tenth).

Problem 2

For the second triangle $\triangle PON$:

Given:

• $\angle P = 60^\circ$
• Side $PN = 9$
• Side $PO = x$ (adjacent to $\angle P$)

We will use the cosine function again because $x$ is adjacent to $\angle P$, and $9$ is the hypotenuse.

$\cos(60^\circ) = \frac{x}{9}$

Solving for $x$: $x = 9 \cdot \cos(60^\circ)$

Since $\cos(60^\circ) = 0.5$: $x = 9 \cdot 0.5$ $x = 4.5$

So, $x = 4.5$.

Summary of Solutions

1. In $\triangle TUV$, $x \approx 66.7$.
2. In $\triangle PON$, $x = 4.5$.

Would you like a deeper explanation of any of these steps or further clarification?

Here are 5 related questions to extend your understanding:

1. How would you solve for the opposite side if only the adjacent side and angle were known?
2. What is the relationship between sine, cosine, and tangent in right triangles?
3. How does the Pythagorean theorem relate to trigonometry in right triangles?
4. What is the effect on $x$ if the angle $\angle V$ is increased in $\triangle TUV$?
5. How would the solution change if $TU$ was given as 70 instead of 69 in $\triangle TUV$?

Tip: Remember that the cosine function relates the adjacent side and hypotenuse in right triangles, making it very useful for solving problems where these sides are given.