Problem 21
Question
Solve the triangle. The Law of Cosines may be needed. $$a=6, b=12, c=16$$
Step-by-Step Solution
Verified Answer
Answer: The angles of the triangle are approximately A = 21.8 degrees, B = 48.19 degrees, and C = 110.01 degrees.
1Step 1: Calculate Angle A
Use the Law of Cosines to find the angle A:
$$cos A = \frac{b^2 + c^2 - a^2}{2bc}$$
Plug in the given values:
$$cos A = \frac{12^2 + 16^2 - 6^2}{2 \cdot 12 \cdot 16}$$
Perform the calculation and take the inverse cosine to find A in degrees:
$$A = cos^{-1}\left(\frac{12^2 + 16^2 - 6^2}{2 \cdot 12 \cdot 16}\right) \approx 21.8^{\circ}$$
2Step 2: Calculate Angle B
Similarly, use the Law of Cosines to find the angle B:
$$cos B = \frac{a^2 + c^2 - b^2}{2ac}$$
Plug in the given values:
$$cos B = \frac{6^2 + 16^2 - 12^2}{2 \cdot 6 \cdot 16}$$
Perform the calculation and take the inverse cosine to find B in degrees:
$$B = cos^{-1}\left(\frac{6^2 + 16^2 - 12^2}{2 \cdot 6 \cdot 16}\right) \approx 48.19^{\circ}$$
3Step 3: Calculate Angle C
Since we have already found both angles A and B, we can use the fact that the sum of all angles in a triangle is equal to 180 degrees to find angle C:
$$C = 180^{\circ} - A - B$$
Plug in the values for A and B:
$$C = 180^{\circ} - 21.8^{\circ} - 48.19^{\circ} \approx 110.01^{\circ}$$
4Step 4: Gather the results
Now we have the three angles and side lengths of the triangle:
- Side a = 6
- Side b = 12
- Side c = 16
- Angle A = 21.8 degrees
- Angle B = 48.19 degrees
- Angle C = 110.01 degrees
Key Concepts
Solving TrianglesInverse CosineTriangle Angle Sum
Solving Triangles
Solving triangles involves finding all the unknown angles and side lengths of a triangle given some initial information. It's often a matter of piecing a puzzle together, using different clues that the triangle provides. Triangles have three sides and three angles, and there are a few principles and formulas that can help solve them.
One of the key tools at your disposal is the Law of Cosines, which is used when you know at least two sides and one angle or three sides of the triangle. This law, which relates the lengths of the sides of a triangle to the cosine of one of its angles, is an extension of the Pythagorean theorem for any triangle, not just right triangles.
To apply the Law of Cosines effectively, one would typically follow these steps: identify the known sides and angles, use the appropriate Law of Cosines formula to find an unknown side or angle, and continue solving by finding the remaining angles or sides, often using the Law of Sines or again, the Law of Cosines, until all unknowns are determined.
One of the key tools at your disposal is the Law of Cosines, which is used when you know at least two sides and one angle or three sides of the triangle. This law, which relates the lengths of the sides of a triangle to the cosine of one of its angles, is an extension of the Pythagorean theorem for any triangle, not just right triangles.
To apply the Law of Cosines effectively, one would typically follow these steps: identify the known sides and angles, use the appropriate Law of Cosines formula to find an unknown side or angle, and continue solving by finding the remaining angles or sides, often using the Law of Sines or again, the Law of Cosines, until all unknowns are determined.
Inverse Cosine
Inverse cosine, often denoted as \(\cos^{-1}\) or arccos, is a function that helps you find an angle whose cosine value is known. This is particularly useful in triangle problems when you're given sides' lengths and need to find the angle between them.
The process of finding an angle using inverse cosine usually involves a calculator set to degree mode. Once you have the cosine value from the Law of Cosines, plug it into the inverse cosine function to get the angle's measure in degrees. For example, if you have \(\cos(A)\) already calculated, you would compute \(A = \cos^{-1}(\cos(A))\) to find the measure of angle A. It's important to remember that the range of the inverse cosine function is from \(0^\circ\) to \(180^\circ\), which aligns perfectly with the range of possible angles in a triangle.
The process of finding an angle using inverse cosine usually involves a calculator set to degree mode. Once you have the cosine value from the Law of Cosines, plug it into the inverse cosine function to get the angle's measure in degrees. For example, if you have \(\cos(A)\) already calculated, you would compute \(A = \cos^{-1}(\cos(A))\) to find the measure of angle A. It's important to remember that the range of the inverse cosine function is from \(0^\circ\) to \(180^\circ\), which aligns perfectly with the range of possible angles in a triangle.
Triangle Angle Sum
The Triangle Angle Sum is a fundamental concept in geometry stating that the sum of the interior angles of any triangle is always equal to \(180^\circ\). This property is extremely useful when you know some angles of a triangle and need to find the remaining ones.
For instance, once you've found two angles using the Law of Cosines or any other method, you can subtract their sum from \(180^\circ\) to find the third angle. It’s also helpful for error-checking; if your calculated angles don't add up to \(180^\circ\), it's clear that there's a mistake somewhere that needs to be addressed. Remember, no matter what the shape of the triangle is, the angle sum property always holds true, making it an indispensable tool in solving triangle problems.
For instance, once you've found two angles using the Law of Cosines or any other method, you can subtract their sum from \(180^\circ\) to find the third angle. It’s also helpful for error-checking; if your calculated angles don't add up to \(180^\circ\), it's clear that there's a mistake somewhere that needs to be addressed. Remember, no matter what the shape of the triangle is, the angle sum property always holds true, making it an indispensable tool in solving triangle problems.
Other exercises in this chapter
Problem 20
Solve the triangle. The Law of Cosines may be needed. $$a=50, c=80, C=45^{\circ}$$
View solution Problem 20
From the top of a 130 -foot-high lighthouse, the angle of depression to a boat in Lake Erie is \(2.5^{\circ} .\) How far is the boat from the lighthouse?
View solution Problem 22
Alice is flying a kite. Her hand is three feet above ground level and is holding the end of a 300 -foot-long kite string, which makes an angle of \(57^{\circ}\)
View solution Problem 23
Solve the triangle. The Law of Cosines may be needed. $$a=16.5, b=18.2, C=47^{\circ}$$
View solution