Solving Right Triangles: Videos & Practice Problems

To solve any right triangle with one side and one angle, first find the missing angle by subtracting the known angle from 90 degrees. Use the sine function, where $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$, to find the missing side. For two sides given, apply the Pythagorean theorem $a^2+b^2=c^2$ to find the hypotenuse, then use trigonometric functions to find the angles.

Finding Missing Side Lengths

Understanding how to find missing side lengths in right triangles is a crucial skill in trigonometry. When given one side and one angle, you can apply trigonometric functions to solve for the unknowns. The first step is to identify any missing angles. In a right triangle, if you know one angle (let's say 37 degrees), you can find the other non-right angle by subtracting it from 90 degrees. For example, 90 - 37 = 53 degrees, which gives you the second angle.

Next, you can use the SOHCAHTOA mnemonic to determine which trigonometric function to apply. SOHCAHTOA stands for:

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

In this case, to find the side opposite the 37-degree angle, you would use the sine function. The equation can be set up as:

\[ \sin(37^\circ) = \frac{x}{5} \]

Here, \(x\) 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.

To find the final missing side, you can apply the Pythagorean theorem, which states:

\[ a^2 + b^2 = c^2 \]

In this scenario, if \(a\) is 3 (the side we just calculated) and \(c\) is 5 (the hypotenuse), you can set up the equation:

\[ 3^2 + b^2 = 5^2 \]

This 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 lengths of the triangle are 3 and 4, demonstrating how trigonometric functions and the Pythagorean theorem can be effectively used to solve for all sides of a 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)

Finding Missing Side Lengths 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 Grand Lighthouse stands 288 meters tall and is located 2.3 kilometers (or 2,300 meters) inland from the shore. A seafarer on a sailboat directly in front of the lighthouse measures the angle of elevation to the top of the lighthouse as 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:

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

In this case, we have:

\[\tan(3.4^\circ) = \frac{288 \text{ m}}{2300 \text{ m} + d}\]

To isolate d, we first multiply both sides by the adjacent side (2300 + d):

\[288 = (2300 + d) \cdot \tan(3.4^\circ)\]

Next, we distribute the tangent:

\[288 = 2300 \cdot \tan(3.4^\circ) + d \cdot \tan(3.4^\circ)\]

Calculating \(2300 \cdot \tan(3.4^\circ)\) gives approximately 36.65. Thus, we can rewrite the equation as:

\[288 = 36.65 + d \cdot \tan(3.4^\circ)\]

Subtracting 36.65 from both sides results in:

\[251.35 = d \cdot \tan(3.4^\circ)\]

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

\[d = \frac{251.35}{\tan(3.4^\circ)}\]

Calculating this yields approximately 2,547.6 meters. To convert this distance into kilometers, we divide by 1,000, resulting in:

\[d \approx 2.55 \text{ km}\]

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 two or more sides are given. 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 (the longest side). The formula can be expressed as:

\[a^2 + b^2 = c^2\]

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

\[12^2 + 5^2 = c^2\]

\[144 + 25 = c^2\]

\[169 = c^2\]

\[c = \sqrt{169} = 13\]

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 the sine function, we set up the equation:

\[\sin(x) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{12}{13}\]

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

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

Calculating this gives 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:

\[y = 90 - x = 90 - 67 = 23\]

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

\[\cos(b) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{17}\]

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:

\[a = 90 - b = 90 - 62 = 28\]

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)

Finding Missing Angles Example 2

In this scenario, we are tasked with determining the angle of inclination of a hiking path that connects a mountain lodge at an elevation of 6,500 feet to a scenic viewpoint in a canyon at an elevation of 4,300 feet. The path spans a distance of 4,400 feet. To visualize this, we can represent the situation as a right triangle where the vertical leg represents the change in elevation and the hypotenuse represents the hiking path.

First, we calculate the height difference between the two elevations. The height \( h \) can be found by subtracting the lower elevation from the higher elevation:

\( h = 6500 \, \text{feet} - 4300 \, \text{feet} = 2200 \, \text{feet} \)

Now, we have a right triangle where the opposite side (the height) is 2,200 feet and the hypotenuse (the hiking path) is 4,400 feet. To find the angle of inclination \( \theta \), we can use the sine function, which relates the opposite side to the hypotenuse:

\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2200}{4400} \)

This simplifies to:

\( \sin(\theta) = \frac{1}{2} \)

To find the angle \( \theta \), we take the inverse sine:

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

The inverse sine of \( \frac{1}{2} \) corresponds to an angle of 30 degrees. Therefore, the angle of inclination of the hiking path is 30 degrees. This process illustrates how to apply trigonometric functions 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. First, identify the given side and angle. For example, if you know the hypotenuse and an angle, you can use the sine function: $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. Rearrange the equation to solve for the missing side: $\mathrm{opposite}=\mathrm{hypotenuse}\times \sin \left(\theta \right)$. Plug the values into your calculator, ensuring it is in degree mode, to find the missing side length.

How do you calculate the missing angle in a right triangle?

To calculate the missing angle in a right triangle, use the complementary angle property or trigonometric functions. If one angle is given, subtract it from 90°: $\mathrm{missing}\mathrm{angle}=90-\mathrm{given}\mathrm{angle}$. If two sides are given, use inverse trigonometric functions. For example, if you know the opposite and hypotenuse, use the sine function: $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. Take the inverse sine: $\theta =\sin ⁻¹\left(\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\right)$. Use a calculator to find the angle.

What is the Pythagorean theorem and how is it used to solve right triangles?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: $a^2+b^2=c^2$. To find a missing side, rearrange the equation. For example, if the hypotenuse is given, solve for a leg: $b^2=c^2-a^2$. Take the square root: $b=\sqrt{c^2-a^2}$. This theorem is essential for solving right triangles when two sides are known.

How do you use SOHCAHTOA to solve for missing sides and angles in a right triangle?

SOHCAHTOA is a mnemonic for trigonometric functions: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent). To solve for a missing side, choose the function based on the given side and angle. For example, if the hypotenuse and an angle are given, use sine: $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. Rearrange to find the opposite side: $\mathrm{opposite}=\mathrm{hypotenuse}\times \sin \left(\theta \right)$. To find an angle, use the inverse function. For example, if opposite and hypotenuse are known, use: $\theta =\sin ⁻¹\left(\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\right)$. Plug values into a calculator to solve.

What are complementary angles in a right triangle, and how are they calculated?

Complementary angles in a right triangle are the two non-right angles that sum to 90°. If one angle is known, the other can be calculated by subtracting the known angle from 90°: $\mathrm{missing}\mathrm{angle}=90-\mathrm{given}\mathrm{angle}$. For example, if one angle is 37°, the other is $90-37=53$. This property simplifies calculations when solving right triangles.