Question

Find the missing side. 42° Y y 18 y = [?] Round to the nearest tenth.​

Answer 1

Step-by-step explanation: a = o/t

a = 18/tan(42)

a = 19.99

Answer 2

The base is approximately 19.9910 units when calculated to four decimal places.

To find the base of a right-angled triangle given the perpendicular (opposite side to θ) and the angle θ, we can use the tangent trigonometric function, which is the ratio of the opposite side to the adjacent side (base).

The tangent of angle θ is defined as:

`\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]`

Given that the opposite side is 18 and θ is 42 degrees, we can rearrange the formula to solve for the adjacent side (base):

`\[ \text{Adjacent} = \frac{\text{Opposite}}{\tan(\theta)} \]`

Now we plug in the values:

`\[ \text{Adjacent} = (18)/(\tan(42^\circ)) \]`

Let's calculate the length of the base using this formula.

Certainly, here's the step-by-step calculation to find the base of the right triangle:

1. Define the relationship using the tangent function:

Since we have a right-angled triangle with one angle \( \theta \) and the length of the side opposite this angle (perpendicular side), we can use the tangent function, which is the ratio of the opposite side to the adjacent side (which in this case is the base we want to find).

`\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]`

2. Substitute the known values:

We know the opposite side is 18 units and \( \theta = 42^\circ \), so we place these values into the tangent function to solve for the adjacent side (base).

`\[ \tan(42^\circ) = \frac{18}{\text{Base}} \]`

3. Rearrange the equation to solve for the base:

`\[ \text{Base} = (18)/(\tan(42^\circ)) \]`

4. Calculate the tangent of 42 degrees:

To do this, we first convert the angle from degrees to radians because the trigonometric functions in most calculators and programming languages use radians. The conversion is done using the formula:

`\[ \text{Radians} = \frac{\text{Degrees} * \pi}{180} \]`

`\[ \text{Radians} = (42 * \pi)/(180) \]`

5. Calculate the tangent:

`\[ \tan(42^\circ) \text{ in radians} \]`

6. Divide the length of the opposite side by the tangent of the angle:

`\[ \text{Base} = \frac{18}{\tan(42^\circ \text{ in radians})} \]`

After performing the calculations with a calculator or programming:

`\[ \text{Radians} = (42 * \pi)/(180) \approx 0.733 \text{ (approximately)} \]`

`\[ \tan(42^\circ) \approx \tan(0.733) \approx 0.9004 \text{ (approximately)} \]`

`\[ \text{Base} \approx (18)/(0.9004) \approx 19.9910 \]`

The length of the base of the right-angled triangle is approximately 19.99 units (assuming the units are the same as those for the perpendicular side, which is 18 units). This result is using the tangent of a 42-degree angle to find the adjacent side when the opposite side is 18 units.