Solving Triangles - Trigonometry

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Question

Two angles in a triangle are $30^\circ$ and $70^\circ$ . What is the measure of the 3rd angle?

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Answer

The sum of the angles of a triangle is 180˚.

Thus, since the sum of our two angles is 100˚, our missing angle must be,

.

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Question

If the hypotenuse of a right triangle has a length of 6, and the length of a leg is 2, what is the angle between the hypotenuse and the leg?

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Answer

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.

$\theta=cos^{-1}\left(\frac{\textup{2}}{\textup{6}}\right)=70.53^\circ$

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Question

A skateboard ramp made so that the rider can gain sufficient speed before a jump is 15 feet high and the ramp is 17 feet long. What is the measure of the angle $\theta$ between the ramp and the ground?

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Answer

For the angle in question we have the opposite side and the hypotenuse given to us. We can use the sine function.

$\sin\theta= \frac{opp}{hyp}\rightarrow\sin\theta=(\frac{15}{17})$

Use the inverse sin to find the measure of an angle between these sides:

$\theta =\sin^{-1}(\frac{15}{17})=62^{\circ}$

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Question

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?

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Answer

Begin the problem by visualizing a diagram of the situation:

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.

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Question

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?

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Answer

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.

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

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Question

If the height of the stair is 2 ft, and the length of the stair is 3 ft, how long must the ramp be to cover the stair?

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Answer

Use the Pythagorean triangle to solve for the third side of the triangle.

$\therefore c^2=13$

$c=\sqrt{13}$

Simplify and you have the answer:

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Question

A surveyor looks up to the top of a mountain at an angle of 35 degrees. If the surveyor is 2400 feet from the base of the mountain, how tall is the mountain (to the nearest tenth of a foot)?

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Answer

This is a typical angle of elevation problem.

From the question, we can infer that the ratio of the mountain height to the surveyor's horizontal distance from the mountain is equal to the tangent of 35 degrees - the mountain height is the "opposite side", while the horizontal distance is the "adjacent side". In other words:

$tan (35^{\circ}) = \frac{mountain\ height}{ 2400\ ft}$

Solving for the mountain height gives a vertical distance of 1680.5 feet.

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Question

A zipline is attached from the edge of the roof of a 23 foot high building and meets the ground 23 feet from the base of the building. Find the angle $\theta$ of elevation the cable makes with the ground.

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Answer

Concerning the angle of elevation which we are to find, we know the side length opposite this angle and the side length adjacent to it.

We use

$\tan=\frac{opp}{adj}$ .

$\tan\theta=\frac{opp}{adj}=\frac{23}{23}=1$

Take the inverse of tan to find the corresponding angle:

$\tan^{-1}(1)=45^{\circ}$

A short cut for solving is to realize right triangles with equal sides must have two 45° angles!

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Question

In a right triangle, the legs are and . What is the length of the hypotenuse?

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Answer

Here, we can use the Pythagorean Theorem.

This states that the sum of the squares of the legs is equal to the square of the hypotenuse.

Thus, we plug into the formula to get our hypotenuse.

$7^{^{2}}+24^{2} = c^{2}$

$c=\sqrt{49+576}=\sqrt{625}=25$

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Question

If the altitude of a triangle has a height of 5, and also bisects a 60 degree angle, what must be the perimeter of the full triangle?

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Answer

The altitude of the triangle splits the 60 degree angle into two 30 degree angles. Notice that this will create two right triangles with 30-60-90 angles. This indicates that the full triangle is an equilateral triangle.

The altitude represents the adjacent side of the split triangles. The bisected angle is 30 degrees. Find the hypotenuse of either split triangle. This is the side length of the equilateral triangle.

$cos(\theta)=\frac{A}{H}$

$H=\frac{5}{cos(30)}= \frac{5}{\frac{\sqrt3}{2}}= \frac{5 \times 2}{\sqrt{3}} = \frac{10}{\sqrt3}$

Rationalize the denominator.

$\frac{10}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{ 10\sqrt3}{3}$

Since the full triangle has three sides, multiply this value by three for the perimeter.

$3\left(\frac{ 10\sqrt3}{3}\right)=10\sqrt3$

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Question

What angle does the ramp make with the bottom of the stair?

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Answer

$\small tan(c)=\frac{c}{a}=\frac{2}{3}$

$\small tan(c)=\frac{2}{3}$

$\small c=tan^{-1}(\frac{2}{3})$

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Question

Given that angle A is 23.0o, side a is 1.43 in., and side b is 3.62 in., what is the angle of B?

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Answer

$\small \small \small \frac{sin(23.0^o)}{1.43: in.}=\frac{sin(B)}{3.62: in.}$

$\small \small \small \sin(B)=\frac{3.62 in.\times \sin(23.0^o)}{1.43 in.}$

$\small \small \small \sin(B)=0.99$

$\small B=sin^{-1}(0.99)$

$\small B=81.5^o$

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Question

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?

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Answer

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.

$sin(\angle )= \frac{opposite}{hypotenuse} = \frac{6}{15}$

$\angle =sin^{-1} \frac{6}{15} = 24^{\circ}$

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Question

A horizontal ramp of unknown length meets the ground at angle of 20°. The height of the ramp is 6 feet and makes a right angle with the ground. What is the length of the ramp?

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Answer

The question is looking for the length of the ramp which would be the length of the part one would walk on and is also synonymous with the hypotenuse of our right triangle. With the given angle and opposite side we can use the sine function to solve for the length of the ramp because

$\sin\theta =\frac{opp}{hyp}$ .

We have all but the length of the hypotenuse.

Solve for the hypotenuse:

$hyp=\frac{opp}{\sin\theta }$

Plug in the angles and sides given:

$hyp=\frac{6}{\sin20 }=17.5:\uptext{ft}$

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Question

In a right triangle, one of the legs has a length of and a hypotenuse of . Find the length of the other side...

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Answer

Step 1: To find the missing side of a right triangle using the Pythagorean Theorem:

.

Step 2: Substitute in the known values.

$\a^2+b^2=c^2 \8^2+b^2=17^2$

Step 3: Solve for the missing variable by manipulating the equation to isolate the variable.

$\b=\sqrt{17^2-8^2} \b=\sqrt{289-64} \b=\sqrt{225} \b=15$

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Question

If the two legs of a right triangle are 5 and 7 respectively, what is the length of the hypotenuse?

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Answer

Step 1: Recall the Pythagorean Theorem:

, where $a \and\ b$ are the legs of the triangle.

Step 2: Plug in the values that are given for the legs into the theorem:

Evaluate:

Add:

Step 3: To find , take the square root of both sides:

$\sqrt{74}=\sqrt{c^2}$

Reduce (if possible):

$\sqrt{74}=c$

The length of the hypotenuse is $\sqrt {74}$ .

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Question

If is between and degrees, and $\small \cos x = \frac{3}{5}$ , what is $\small \tan x$ ?

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Answer

Using Pythagoras theory, you can see that is . If is , than $\small \tan x$ is $\small \frac{4}{3}$ , because $\small \tan x$ is $\small \frac{opposite}{adjacent}$

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Question

Find the length of .

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Answer

Recall the Law of Sines for a generic triangle, as shown above:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{14}{\sin 72}=\frac{b}{\sin 35}$

Rearrange the equation to solve for :

$b=(\sin 35)\frac{14}{\sin 72}=8.44$

Make sure to round to two places after the decimal.

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Question

Find the length of side .

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Answer

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{6}{\sin 85}=\frac{b}{\sin 27}$

Rearrange the equation to solve for :

$b=(\sin 27)\frac{6}{\sin 85}=2.73$

Make sure to round to two places after the decimal.

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Question

Find the length of side .

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Answer

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{7}{\sin 62}=\frac{b}{\sin 36}$

Rearrange the equation to solve for :

$b=(\sin 36)\frac{7}{\sin 62}=4.66$

Make sure to round to two places after the decimal.

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