Problem 10 - Precalculus with Calculus Previews

Question

Use the Law of Cosines to solve the triangle. $$ a=6, b=5, c=7 $$

Step-by-Step Solution

Answer

Angles: \(A \approx 56.25^\circ\), \(B \approx 45.29^\circ\), \(C \approx 78.46^\circ\).

Step 1: Identify the Law of Cosines

The Law of Cosines states: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\] where \(a\), \(b\), and \(c\) are the sides of the triangle and \(C\) is the angle opposite side \(c\). We will use this formula to find angle \(C\).

Step 2: Plug in the Known Values

We know that \(a = 6\), \(b = 5\), and \(c = 7\). Substitute these values into the Law of Cosines formula:\[7^2 = 6^2 + 5^2 - 2 \cdot 6 \cdot 5 \cdot \cos(C)\]

Step 3: Simplify and Solve for \(\cos(C)\)

Calculate the squares of the sides and simplify:\[49 = 36 + 25 - 60 \cdot \cos(C)\]Combine the terms on the right:\[49 = 61 - 60 \cdot \cos(C)\]Rearrange to solve for \(\cos(C)\):\[60 \cdot \cos(C) = 61 - 49\]\[60 \cdot \cos(C) = 12\]\[\cos(C) = \frac{12}{60} = \frac{1}{5}\]

Step 4: Find Angle \(C\) Using Inverse Cosine

Use the inverse cosine function to find the measure of angle \(C\):\[C = \cos^{-1}\left(\frac{1}{5}\right)\]After evaluating, we find:\(C \approx 78.46^\circ\)

Step 5: Use Law of Cosines for Angle \(A\)

Choose another angle and use the Law of Cosines again. For angle \(A\), the formula is:\[a^2 = b^2 + c^2 - 2bc \cdot \cos(A)\]Plug in the known values:\[6^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(A)\]Simplify and solve for \(\cos(A)\):\[36 = 25 + 49 - 70\cdot \cos(A)\]\[36 = 74 - 70\cdot \cos(A)\]\[70\cdot \cos(A) = 74 - 36\]\[70\cdot \cos(A) = 38\]\[\cos(A) = \frac{38}{70} = \frac{19}{35}\]

Step 6: Find Angle \(A\) Using Inverse Cosine

Calculate \(A\) using the inverse cosine function:\[A = \cos^{-1}\left(\frac{19}{35}\right)\]After evaluating, we find:\(A \approx 56.25^\circ\)

Step 7: Calculate Angle \(B\) by Subtraction

Finally, use the fact that the sum of angles in a triangle is \(180^\circ\). Find angle \(B\) by subtracting the other angles from \(180^\circ\):\[B = 180^\circ - 78.46^\circ - 56.25^\circ\]\(B \approx 45.29^\circ\)

Key Concepts

Angle Calculation

Calculating angles in a triangle can be approached in different ways, but one of the most reliable methods is using the Law of Cosines. This method is especially useful in situations where you cannot use simpler methods like the Law of Sines.
When you're given all three sides of a triangle, like in our exercise with sides 6, 5, and 7, calculating the angles becomes straightforward using this law.
The Law of Cosines states that for a triangle with sides

• \( a, b, \) and \( c \)
• opposite angles \( A, B, \) and \( C \)

You can find angle \( C \) using: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Substitute the known values into this equation, substitute for \( \cos(C) \), and then compute the inverse cosine to find the angle. This process helps us avoid the trial and error of guesswork in angle calculation.

Triangle Sides

Understanding the relationship between the sides of a triangle is crucial when approaching problems involving the Law of Cosines.
Each side of a triangle is linked to the angle opposite it. This is particularly clear when you look at the formula for the Law of Cosines:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] In this formula,

• \( c \) is directly opposite \( C \)
• \( a \) and \( b \) are the other two sides.

Subsequently, the length of side \( c \) and the values obtained from \( a \) and \( b \) are pivotal.
These sides dictate the size of angle \( C \) since \( C \) is derived from \( \cos(C) \), an integral of the expression linking side lengths to cosine values.

Inverse Cosine

Once you have calculated \( \cos(C) \) or \( \cos(A) \), you'll need to find the actual angle values. This is where the inverse cosine function (\( \cos^{-1} \)) comes into play.
The inverse cosine function allows us to convert a cosine value back to an angle. In our exercise, we discover that

• \(\cos(C) = \frac{1}{5}\)

To calculate the angle \( C \), simply use \( \cos^{-1} \left( \frac{1}{5} \right) \).
This function essentially helps to "backtrack" from a

• cosine "result"
• and determines the angle that would produce it.

Understanding inverse trigonometric functions is vital to solving triangles when side lengths and angles are involved.

Trigonometry

Trigonometry, the study of triangle measurements, is fundamentally about relationships in triangles.
It uses angles and side lengths along with trigonometric functions such as sine, cosine, and tangent to elucidate these relationships. The Law of Cosines is one of the key relationships that trigonometry provides.
This law helps solve triangles in scenarios not possible with basic algebra or the simpler Law of Sines. Like in our problem, trigonometry can clarify hidden angles when all side lengths are known.

• It offers a systematic approach
• to find both angles and side lengths,
• ultimately grounding abstract concepts into tangible measurements.

For students delving into angle calculations and side determinations, grasping these trigonometric principles is crucial. Understanding this framework equips them with powerful tools for resolving complex geometric problems.