Solving Right Triangles: Videos & Practice Problems

To solve right triangles, use trigonometric functions and the Pythagorean theorem. For one side and one angle, find missing angles by subtracting from 90 degrees. Use SOHCAHTOA to relate sides and angles, such as $sin\left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. For two sides, apply the Pythagorean theorem $a²+b²=c²$ to find the hypotenuse, then use trigonometric ratios to find angles.

Finding Missing Side Lengths

Understanding how to find missing side lengths in right triangles is essential in trigonometry, especially when given one side and one angle. This process involves a few systematic steps that utilize trigonometric functions and the Pythagorean theorem.

To begin, if you have a right triangle with one known angle and one known side, the first step is to determine any missing angles. For instance, if you know one angle is 37 degrees, you can find the other non-right angle by subtracting from 90 degrees: 90^\circ - 37^\circ = 53^\circ. This gives you the second angle, which is crucial for further calculations.

Next, you will choose an appropriate trigonometric function to relate the known side to the unknown side. The mnemonic SOHCAHTOA can help you remember the relationships: SOH (sine = opposite/hypotenuse), CAH (cosine = adjacent/hypotenuse), and TOA (tangent = opposite/adjacent). In this case, to find the side opposite the 37-degree angle, you can use the sine function:

\(\sin(37^\circ) = \frac{x}{5}\), where \(x\) is the missing side and 5 is the hypotenuse. To solve for \(x\), multiply both sides by 5:

x = 5 \cdot \sin(37^\circ).

Using a calculator in degree mode, you would find that \(x\) is approximately 3.009, which can be rounded to 3.

After finding one side, the final step is to use the Pythagorean theorem to find the last missing side. The theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse:

a^2 + b^2 = c^2, where \(c\) is the hypotenuse. In this scenario, if we let \(a = 3\) and \(c = 5\), we can set up the equation:

3^2 + b^2 = 5^2, which simplifies to:

9 + b^2 = 25.

Subtracting 9 from both sides gives:

b^2 = 16.

Taking the square root of both sides results in:

b = 4.

Thus, the missing side \(b\) is 4. By following these steps—finding missing angles, applying trigonometric functions, and utilizing the Pythagorean theorem—you can effectively solve for all missing sides in any right triangle when given one side and one angle.

A right triangle with an angle of $31°$ has a hypotenuse of $10$. Calculate the side of the triangle opposite to the $31°$ angle (y), and the side adjacent to the $31°$ angle (x). Round your answer to 3 decimal places.

• A. $x=5.150,y=8.572$
• B. $x=8.572,y=5.150$
• C. $y=5.000,x=8.660$
• D. $y=8.660,x=5.000$

Answer: (B)

Example 1

In this problem, we analyze a scenario involving a lighthouse and a sailboat to determine the distance from the sailboat to the coastline using trigonometry. The lighthouse stands 288 meters tall and is located 2.3 kilometers (or 2310 meters) inland from the shore. A seafarer on a sailboat observes the top of the lighthouse, recording an angle of elevation of 3.4 degrees.

To solve this, we can visualize the situation as a right triangle where the height of the lighthouse represents the opposite side, the distance from the shore to the lighthouse is the adjacent side, and the distance from the sailboat to the shore is what we need to find. We denote this unknown distance as d.

Using the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle, we can set up the equation:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

Substituting the known values, we have:

$$\tan(3.4^\circ) = \frac{288}{2310 + d}$$

To isolate d, we first multiply both sides by the adjacent side (2310 + d):

$$288 = (2310 + d) \cdot \tan(3.4^\circ)$$

Next, we distribute the tangent:

$$288 = 2310 \cdot \tan(3.4^\circ) + d \cdot \tan(3.4^\circ)$$

Calculating \(2310 \cdot \tan(3.4^\circ)\) gives approximately 136.65. Thus, we can rewrite the equation as:

$$288 = 136.65 + d \cdot \tan(3.4^\circ)$$

Subtracting 136.65 from both sides results in:

$$151.35 = d \cdot \tan(3.4^\circ)$$

To solve for d, we divide both sides by \(\tan(3.4^\circ)\):

$$d = \frac{151.35}{\tan(3.4^\circ)}$$

Calculating this yields approximately 2547.6 meters. Since the problem requests the distance in kilometers, we convert this by dividing by 1000, resulting in:

$$d \approx 2.55 \text{ kilometers}$$

Thus, the distance from the sailboat to the shore is approximately 2.55 kilometers.

Finding Missing Angles

In the study of right triangles, understanding how to solve for unknown sides and angles is crucial, especially when given two or more side lengths. This process often begins with the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs (sides) equals the square of the hypotenuse. This can be expressed mathematically as:

$$a^2 + b^2 = c^2$$

Here, \(a\) and \(b\) represent the lengths of the two legs, while \(c\) is the length of the hypotenuse. For example, if one leg measures 12 units and the other measures 5 units, we can find the hypotenuse by calculating:

$$12^2 + 5^2 = c^2$$

$$144 + 25 = c^2$$

$$169 = c^2$$

Taking the square root gives us \(c = 13\) units.

Once the hypotenuse is determined, we can find the unknown angles using trigonometric functions, which relate the angles of a triangle to the ratios of its sides. The mnemonic SOHCAHTOA helps remember these relationships:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

For instance, to find an angle \(x\) using sine, we set up the equation:

$$\sin(x) = \frac{12}{13}$$

To solve for \(x\), we take the inverse sine:

$$x = \sin^{-1}\left(\frac{12}{13}\right)$$

Calculating this yields approximately \(67.38\) degrees, which we can round to \(67\) degrees. The other angle \(y\) can be found since the angles in a right triangle sum to \(90\) degrees:

$$y = 90 - 67 = 23 \text{ degrees}$$

In another scenario where all sides are known, we can directly apply trigonometric functions to find the angles. For example, using cosine to find angle \(b\):

$$\cos(b) = \frac{8}{17}$$

Taking the inverse cosine gives:

$$b = \cos^{-1}\left(\frac{8}{17}\right)$$

This results in approximately \(61.92\) degrees, rounded to \(62\) degrees. The remaining angle \(a\) is then calculated as:

$$a = 90 - 62 = 28 \text{ degrees}$$

Through these methods, one can effectively solve for all angles in a right triangle when provided with two or more side lengths, reinforcing the importance of both the Pythagorean theorem and trigonometric functions in geometry.

Given the right triangle below, calculate all missing angles in degrees (round your answer to 3 decimal places.

• A. $x=60.000°,y=30.000°$
• B. $x=26.565°,y=63.435°$
• C. $=63.435°,y=26.565°$
• D. $x=30.000°,y=60.000°$

Answer: (B)

Example 2

In this problem, we explore the application of trigonometry to determine the angle of inclination of a hiking path between two elevations. The mountain lodge is situated at an elevation of 6,500 feet, while the scenic viewpoint in the canyon is at 4,300 feet. The distance of the hiking path is given as 4,4100 feet. To find the angle of inclination, we first need to calculate the vertical height difference between the two elevations.

The height \( h \) can be calculated by subtracting the lower elevation from the higher elevation:

\[ h = 6500 - 4300 = 2200 \text{ feet} \]

With the height established, we can visualize the scenario as a right triangle where the height \( h \) is the opposite side, and the hiking path is the hypotenuse. To find the angle of inclination \( \theta \), we utilize the sine function, which relates the opposite side to the hypotenuse:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2200}{4410} \]

To isolate \( \theta \), we take the inverse sine of both sides:

\[ \theta = \sin^{-1}\left(\frac{2200}{4410}\right) \]

Upon simplifying the fraction, we find:

\[ \frac{2200}{4410} = \frac{1}{2} \]

Thus, we can express the angle as:

\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]

This results in an angle of:

\[ \theta = 30^\circ \]

Therefore, the angle of inclination of the hiking path is 30 degrees, demonstrating how trigonometric functions can be applied to solve real-world problems involving right triangles.

Here’s what students ask on this topic:

How do you find the missing side of a right triangle using trigonometric functions?

To find the missing side of a right triangle using trigonometric functions, you can use the SOHCAHTOA memory tool. For example, if you know one angle (θ) and the hypotenuse (c), you can find the opposite side (a) using the sine function: $\sin \left(\theta \right)=\frac{a}{c}$. Rearrange to solve for a: $a=c\cdot \sin \left(\theta \right)$. Similarly, you can use cosine for the adjacent side: $\cos \left(\theta \right)=\frac{b}{c}$, and rearrange to solve for b: $b=c\cdot \cos \left(\theta \right)$.

How do you find the missing angles in a right triangle?

To find the missing angles in a right triangle, start by using the fact that the sum of the angles in a triangle is 180 degrees. Since one angle is 90 degrees, the other two angles must sum to 90 degrees. If you know one of these angles (θ), the other angle (α) can be found by subtracting θ from 90 degrees: $\alpha =90°-\theta$. Additionally, you can use trigonometric functions to find angles if you know the sides. For example, if you know the opposite (a) and adjacent (b) sides, use the tangent function: $\tan \left(\theta \right)=\frac{a}{b}$, and then use the inverse tangent to find θ: $\theta =\tan ⁻¹\left(\frac{a}{b}\right)$.

How do you use the Pythagorean theorem to find the hypotenuse of a right triangle?

To find the hypotenuse of a right triangle using the Pythagorean theorem, use the formula: $c^2=a^2+b^2$, where c is the hypotenuse, and a and b are the other two sides. Rearrange to solve for c: $c=\sqrt{a^2+b^2}$. For example, if a = 3 and b = 4, then $c=\sqrt{3{²}^{}+4{²}^{}}=\sqrt{25}=5$.

What is SOHCAHTOA and how is it used in solving right triangles?

SOHCAHTOA is a mnemonic device to remember the relationships between the sides and angles of a right triangle using trigonometric functions. It stands for: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. To use SOHCAHTOA, identify the sides relative to the given angle (θ). For example, if you need to find the opposite side (a) and you know the hypotenuse (c) and angle (θ), use the sine function: $\sin \left(\theta \right)=\frac{a}{c}$. Rearrange to solve for a: $a=c\cdot \sin \left(\theta \right)$.

How do you solve a right triangle if you are given two sides?

To solve a right triangle when given two sides, first use the Pythagorean theorem to find the third side. For example, if you know sides a and b, find the hypotenuse c using: $c^2=a^2+b^2$. Then, use trigonometric functions to find the angles. For instance, to find angle θ, use the tangent function: $\tan \left(\theta \right)=\frac{a}{b}$, and then find θ using the inverse tangent: $\theta =\tan ⁻¹\left(\frac{a}{b}\right)$. Finally, find the remaining angle by subtracting θ from 90 degrees.