Finding cos 3C in a Triangle Using Law of Cosines and Trigonometric Identities

The problem asks us to find the value of \(\cos 3C\) for a triangle ABC with given side lengths a = 4, b = 3, and c = 2. To solve this, we first need to find the value of \(\cos C\) using the Law of Cosines, and then use the triple angle formula for cosine to find \(\cos 3C\).

Applying the Law of Cosines to Find cos C

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For angle C, the formula is:

\(c^2 = a^2 + b^2 - 2ab \cos C\)

We can rearrange this formula to solve for \(\cos C\):

\(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\)

Now, substitute the given values of a, b, and c into the formula:

• a = 4
• b = 3
• c = 2

Plugging these values in:

\(\cos C = \frac{(4)^2 + (3)^2 - (2)^2}{2 \times 4 \times 3}\)

\(\cos C = \frac{16 + 9 - 4}{24}\)

\(\cos C = \frac{25 - 4}{24}\)

\(\cos C = \frac{21}{24}\)

Simplify the fraction:

\(\cos C = \frac{7}{8}\)

So, the value of \(\cos C\) is \(\frac{7}{8}\).

Calculating cos 3C Using the Triple Angle Formula

Now that we have the value of \(\cos C\), we can find \(\cos 3C\) using the triple angle identity for cosine, which is:

\(\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta\)

Substitute \(\theta = C\) and the value \(\cos C = \frac{7}{8}\) into this formula:

\(\cos 3C = 4 \left(\cos C\right)^3 - 3 \left(\cos C\right)\)

\(\cos 3C = 4 \left(\frac{7}{8}\right)^3 - 3 \left(\frac{7}{8}\right)\)

Calculate the cube of \(\frac{7}{8}\):

\(\left(\frac{7}{8}\right)^3 = \frac{7^3}{8^3} = \frac{343}{512}\)

Substitute this back into the expression for \(\cos 3C\):

\(\cos 3C = 4 \left(\frac{343}{512}\right) - \frac{21}{8}\)

Multiply the first term:

\(\cos 3C = \frac{4 \times 343}{512} - \frac{21}{8}\)

\(\cos 3C = \frac{1372}{512} - \frac{21}{8}\)

To combine these fractions, find a common denominator. The least common multiple of 512 and 8 is 512. Convert \(\frac{21}{8}\) to an equivalent fraction with a denominator of 512:

\(\frac{21}{8} = \frac{21 \times (512 \div 8)}{8 \times (512 \div 8)} = \frac{21 \times 64}{8 \times 64} = \frac{1344}{512}\)

However, looking at the options, the denominators are 128 and 64. Let's simplify \(\frac{1372}{512}\) first by dividing both numerator and denominator by 4:

\(\frac{1372 \div 4}{512 \div 4} = \frac{343}{128}\)

Now convert \(\frac{21}{8}\) to have a denominator of 128:

\(\frac{21}{8} = \frac{21 \times (128 \div 8)}{8 \times (128 \div 8)} = \frac{21 \times 16}{8 \times 16} = \frac{336}{128}\)

Now subtract the fractions:

\(\cos 3C = \frac{343}{128} - \frac{336}{128}\)

\(\cos 3C = \frac{343 - 336}{128}\)

\(\cos 3C = \frac{7}{128}\)

The value of \(\cos 3C\) is \(\frac{7}{128}\).

Revision Table: Key Formulas Used

Additional Information: Triangle Properties and Identities

In any triangle ABC, the angles A, B, and C satisfy \(A + B + C = 180^\circ\) or \(\pi\) radians. The Law of Cosines is useful when you know two sides and the included angle (SAS) or all three sides (SSS) and want to find angles or the third side.

Triple angle formulas are derived from sum and double angle formulas. The triple angle formulas for sine and tangent are:

• \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\)
• \(\tan 3\theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}\) (where \(\cos \theta \neq 0\) and \(1 - 3 \tan^2 \theta \neq 0\))

These identities are fundamental in solving various problems in trigonometry and geometry involving multiples of angles.