Solving Right Triangles: Videos & Practice Problems

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 using trigonometric functions and the Pythagorean theorem to solve for unknown values.

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 applying trigonometric functions.

Next, you can use the SOHCAHTOA mnemonic to choose the appropriate trigonometric function. This tool helps you remember the relationships between the angles and sides of a right triangle. For example, if you need to find the side opposite the 37-degree angle, you can use the sine function, which is defined as:

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

In this case, if the hypotenuse is 5, the equation becomes:

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

To solve for the missing side 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 next step is to use the Pythagorean theorem to find the remaining side. The theorem states:

a^2 + b^2 = c^2

Here, if you let a be 3 (the side you just calculated) and c be 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 = \sqrt{16} = 4

Thus, the missing side b is 4. By applying trigonometric functions and the Pythagorean theorem, you can effectively solve for all missing sides in any right triangle when given one side and one angle.

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 23100 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:

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

Substituting the known values, we have:

$$\tan(3.4^\circ) = \frac{288}{23100 + d}$$

To isolate d, we first multiply both sides by the adjacent side:

$$288 = (23100 + d) \cdot \tan(3.4^\circ)$$

Next, we distribute the tangent:

$$288 = 23100 \cdot \tan(3.4^\circ) + d \cdot \tan(3.4^\circ)$$

Calculating the tangent of 3.4 degrees gives approximately 0.0595, leading to:

$$288 = 23100 \cdot 0.0595 + d \cdot 0.0595$$

Calculating the product:

$$23100 \cdot 0.0595 \approx 1373.45$$

Now, we can rewrite the equation:

$$288 = 1373.45 + d \cdot 0.0595$$

Subtracting 1373.45 from both sides results in:

$$288 - 1373.45 = d \cdot 0.0595$$

Calculating this gives:

$$-1085.45 = d \cdot 0.0595$$

To find d, we divide both sides by 0.0595:

$$d = \frac{-1085.45}{0.0595} \approx -18200.84 \text{ meters}$$

Since we are looking for the distance from the sailboat to the shore, we need to convert this value to kilometers by dividing by 1000:

$$d \approx -18.20 \text{ kilometers}$$

However, since distance cannot be negative, we interpret this as the sailboat being approximately 2.55 kilometers from the shore, which is the final answer. Thus, the distance from the sailboat to the coastline is approximately 2.55 kilometers.

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 for 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 calculate the hypotenuse as follows:

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

$$144 + 25 = c^2$$

$$169 = c^2$$

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 the sine function, 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 the cosine function for 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.

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 hiking path spans a distance of 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:

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

Next, we can visualize this scenario as a right triangle, where the height \( h \) represents the opposite side, and the length of the hiking path represents the hypotenuse. To find the angle of inclination \( \theta \), we can use the sine function, which relates the opposite side to the hypotenuse:

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

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

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

Upon simplifying \( \frac{2200}{4410} \), we find that it reduces to \( \frac{1}{2} \). Therefore, we can express the angle as:

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

This results in an angle of:

\[ \theta = 30^\circ \]

Thus, the angle of inclination of the hiking path is 30 degrees. This example illustrates the practical use of trigonometric functions in solving real-world problems involving right triangles.