Trig Ratios in Right Triangles

Trig Ratios in Right Triangles exercise

• Find the trigonometric ratio that describes the angle of the given right triangle.

  Hints

  The naming of the opposite and adjacent legs depends on the position of $\theta$.

  For every right triangle, there is only one correct ratio.

  Solution

  For right triangles, there are basically three trigonometric ratios. Let's see what this means by looking at the pictured triangle.

  The sine ratio is formed by dividing the opposite leg by the hypotenuse:

  • $\sin (\theta) = \frac{a}{c}$
  • $a$: opposite leg
  • $c$: hypotenuse

  The cosine ratio is formed by dividing the adjacent leg by the hypotenuse:

  • $\cos (\theta) = \frac{b}{c}$
  • $b$: adjacent leg
  • $c$: hypotenuse

  Last but not least, the tangent ratio is the opposite leg divided by the adjacent leg:

  • $\tan (\theta) = \frac{a}{b}$
  • $a$: opposite leg
  • $b$: adjacent leg

  You can solve a lot of problems using trigonometric ratios. You should remember, however, that the naming of the opposite and adjacent legs depends on the position of $\theta$.

• Evaluate the height of the Statue of Liberty using trigonometric ratios.

  Hints

  Every right triangle consists of a hypotenuse, opposite leg, and adjacent leg.

  You can evaluate an angle by using one of the trigonometric ratios.

  Solution

  We want to evaluate the height of the Statue of Liberty. But how can we do that?

  In a right triangle, the hypotenuse is always across from the right angle. The adjacent leg, along with the hypotenuse form $\angle A$. The opposite leg is always across from $\angle A$.

  The three most common trigonometric ratios are sine, cosine and tangent. Let's take a closer look:

  • $\sin A = \frac{\text{opposite leg}}{\text{hypotenuse}}$
  • $\cos A = \frac{\text{adjacent leg}}{\text{hypotenuse}}$
  • $\tan A = \frac{\text{opposite leg}}{\text{adjacent leg}}$

  Which side measurements of our triangle are given and which are unknown? First of all, we know that the distance from the viewer to the Statue of Liberty is our adjacent leg. The distance is $1,604.4$ feet. We are interested in the height of the statue, which corresponds to the opposite leg. The last piece of information we need to solve the problem is the measure of $\angle A$, which is $10.5^\circ$.

  The best trigonometric ratio to use is the tangent ratio. We want to evaluate the opposite leg, or the height of the Statue of Liberty, so we need to rearrange the equation. We can then plug these numbers into a calculator to solve for the $\tan A$:

  $\begin{align} \tan A& = \frac{\text{opposite leg}}{\text{adjacent leg}}\\ \tan A \times \text{adjacent leg} &= \text{opposite leg}\\ \tan 10.5^\circ \times 1,640.4~\text{feet}&= \text{opposite leg}\\ 0.1853... \times 1,640.4~\text{feet}&= \text{opposite leg}\\ 304~\text{feet} &= \text{opposite leg} \end{align}$

  Therefore, the height of the Statue of Liberty is $304$ feet.

• Solve for the height of each tree.

  Hints

  The adjacent leg of the right triangle corresponds to the distance to each tree.

  You are looking for the height of each tree, represented by the opposite leg of the right triangle.

  The only trigonometric ratio that includes both opposite and adjacent legs is:

  $\large{\tan A= \frac{\text{opposite leg}}{\text{adjacent leg}}}$

  Solution

  By using the trigonometric ratios, Kevin can determine the height of each tree. First, he must decide which ratio will help him with the calculation.

  He knows the approximate distance to each tree and the viewing angle, $\angle A$, for each tree.

  Because the adjacent leg corresponds to the distance to each tree, and the opposite leg represents the height of each tree, the tangent ratio is the most useful to solve this problem.

  $\tan A = \frac{\text{opposite leg}}{\text{adjacent leg}}$

  In order to solve for the unknown height of the opposite leg, he modifies the equation. This is what he gets:

  $\tan A \times \text{adjacent leg} = \text{opposite leg}$

  Now he can plug in the given information. Let's solve for the height of the tallest tree in Kevin's garden, which stands at a distance of $164$ feet and an angle of $20.1^\circ$:

  $\tan 20.1^\circ \times 164 \approx 60$

  The tallest tree has a height of $60$ feet.

• Determine the distance to the Eiffel Tower.

  Hints

  Which sides do you know? Which angle is given?

  Solution

  Kevin is passionate traveler. Today he's in Paris. He remembers when he was in NYC he calculated the height of the Statue of Liberty, now that he's in Paris, he wants to calculate his distance from the Eiffel Tower.

  He knows he can use one of the trigonometric ratios. From a guide book, he knows the height of the Eiffel Tower is $986$ feet. Thanks to his binoculars, he knows the viewing angle, $\angle A=20.6^\circ$. Considering the given information, he decides to use tangent to find out how far he is from the Eiffel Tower:

  $\begin{align} \tan A & = \frac{\text{opposite leg}}{\text{adjacent leg}}\\ \tan A \times \text{adjacent leg} & = \text{opposite leg}\\ \text{adjacent leg} & = \frac{\text{opposite leg}}{\tan A}\\ \text{adjacent leg} & = \frac{986}{\tan 20.6^\circ}\\ & = 2,623.2 \end{align}$

  Kevin is $2,623.2$ feet away from the Eiffel Tower.

• Identify the properties of right triangles.

  Hints

  The hypotenuse is always the longest leg of a right triangle.

  Adjacent leg and opposite leg depend on the position of $\angle A$.

  Solution

  Understanding the characteristics right triangles can teach us a lot about geometry.

  Every right triangle has exactly one right angle. Two right angles are not possible. Try it out! (...but don't spend too much time trying.)

  The side opposite the right angle is called the hypotenuse, and this is always the longest side of a right triangle. The opposite leg is across from $\angle A$, and the adjacent leg is always between $\angle A$ and the right angle.

  As you maybe have noticed, you can change the position of $\angle A$. Then, the opposite leg becomes the adjacent leg, and the adjacent leg becomes the opposite leg. The sides are always named according to the location of angle $A$.

• Calculate the side lengths and angles of the right triangle.

  Hints

  Did you know that the interior angles of every triangle add up to the same number?

  $\angle A + \angle B + \angle C = 180^\circ$

  Solution

  We want to find the missing information in order to complete our right triangle. We know that $\angle A = 37^\circ$ and that the hypotenuse has a length of $5$ feet.

  The given information is enough to calculate what we don't know. Also, there are several ways to solve this problem.

  First, we can find $\angle B$ since we know a triangle's internal angles sum to $180^\circ$.

  $\begin{align} \\ 180^\circ - \angle A - \angle C &= \angle B \\ 180^\circ - 37^\circ - 90^\circ &= \angle B \\ 53^\circ &= \angle B \end{align}$.

  Now, we can use either the sine or cosine ratios to solve for the unknown side measures. Let's use cosine first. The correct ratio for cosine is:

  $\cos B = \frac{\text{adjacent leg}}{\text{hypotenuse}}$

  When we plug in our values, this is what we get:

  $\begin{align} \cos 53^\circ & = \frac{a}{5}\\ \cos 53^\circ \times 5 & = a\\ 4 & \approx a \end{align}$

  Now we have all lengths except for side $b$. Let's use the sine ratio to calculate the length of side $b$.

  $\sin B = \frac{\text{opposite leg}}{\text{hypotenuse}}$

  $\begin{align} \sin 53^\circ & = \frac{b}{5}\\ \sin 53^\circ \times 5 & = b\\ 3 & \approx b \end{align}$

  Side $b$ has a length of $3$ feet.