Solving Triangles

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Trigonometry › Solving Triangles

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Artemis wants to build a ramp to make the entrance to their home more accessible. The angle between the ramp and the ground cannot be more than $7^\circ$ steep. Artemis has feet of space in their yard that the ramp can take up, and the distance between the ground and the house entrance is feet high. Will Artemis be able to build a ramp that complies with the $\leq7^\circ$ standard?

Yes

Explanation

Begin the problem by visualizing a diagram of the situation:

Screen shot 2020 08 09 at 3.36.03 pm

We can use inverse trig to solve for the unknown angle $\theta$ .

$\tan\theta=\frac{3}{12}$

$\theta=\tan^-^1\left (\frac{3}{12} \right )$

$\theta=14.04^\circ$

Because this angle is larger than $7^\circ$ , this ramp would not comply with standards.

If the hypotenuse of a right triangle has a length of 42.29 meters, and the length of a leg is 12.88 meters, what is the angle between the hypotenuse and the leg?

$14.64^\circ$

$15.15^\circ$

$74.85^\circ$

$99.99^\circ$

Explanation

The leg must be an adjacent side to the hypotenuse.

Therefore, we can use inverse cosine to solve for the angle.

First write the equation for sine of an angle.

$cos(\theta)=\frac{\textup{Adjacent}}{\textup{Hypotenuse}}$

$\theta=cos^{-1}\left(\frac{\textup{Adjacent}}{\textup{Hypotenuse}}\right)$

Substitute the lengths given and solve for the angle.

$\theta=cos^-^1\left (\frac{12.88}{49.29} \right )=74.85^\circ$

If the hypotenuse of a right triangle has a length of 42.29 meters, and the length of a leg is 12.88 meters, what is the angle between the hypotenuse and the leg?

$14.64^\circ$

$15.15^\circ$

$74.85^\circ$

$99.99^\circ$

Explanation

The leg must be an adjacent side to the hypotenuse.

Therefore, we can use inverse cosine to solve for the angle.

First write the equation for sine of an angle.

$cos(\theta)=\frac{\textup{Adjacent}}{\textup{Hypotenuse}}$

$\theta=cos^{-1}\left(\frac{\textup{Adjacent}}{\textup{Hypotenuse}}\right)$

Substitute the lengths given and solve for the angle.

$\theta=cos^-^1\left (\frac{12.88}{49.29} \right )=74.85^\circ$

Yes

Explanation

Begin the problem by visualizing a diagram of the situation:

Screen shot 2020 08 09 at 3.36.03 pm

We can use inverse trig to solve for the unknown angle $\theta$ .

$\tan\theta=\frac{3}{12}$

$\theta=\tan^-^1\left (\frac{3}{12} \right )$

$\theta=14.04^\circ$

Because this angle is larger than $7^\circ$ , this ramp would not comply with standards.

A plank has one end on the ground and one end off the ground. What is the measure of the angle formed by the plank and the ground?

$\approx 24^{\circ}$

$\approx 10^{\circ}$

$\approx 50^{\circ}$

$\approx 70^{\circ}$

Explanation

The length of the plank becomes the hypotenuse of the triangle, while the distance between the plank and the ground becomes the length of one side. To solve for the angle between the plank and the ground, you must find the value of $sin (\angle )$ . The sine of the angle is the value of the opposite side over the hypotenuse, which are values that we know.