Calculate the third side of triangle ABC given a = 3.4, b = 7.8 and C = 80°

To calculate the third side of triangle \( ABC \) given sides \( a = 3.4 \), \( b = 7.8 \), and angle \( C = 80^\circ \), we can use the **Law of Cosines**. The Law of Cosines states:

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

Where:
- \( c \) is the side opposite angle \( C \),
- \( a \) and \( b \) are the other two sides,
- \( C \) is the included angle.

Let's calculate:

\[
c^2 = 3.4^2 + 7.8^2 - 2 \times 3.4 \times 7.8 \times \cos(80^\circ)
\]

First, calculate the square of \( a \) and \( b \):

\[
3.4^2 = 11.56
\]
\[
7.8^2 = 60.84
\]

Next, calculate the cosine term:

\[
\cos(80^\circ) \approx 0.1736
\]
\[
2 \times 3.4 \times 7.8 \times 0.1736 \approx 9.22
\]

Now, substitute everything into the equation:

\[
c^2 = 11.56 + 60.84 - 9.22
\]
\[
c^2 = 63.18
\]
\[
c = \sqrt{63.18} \approx 7.95
\]

So, the third side \( c \) is approximately \( 7.95 \) units.

Would you like more details or have any questions?

Here are some related questions:

1. How does the Law of Cosines differ from the Pythagorean theorem?
2. What is the sine rule, and when should it be used instead of the cosine rule?
3. Can you calculate the area of triangle ABC using the sides and angle given?
4. How would you find the other angles in triangle ABC using the sides and angle given?
5. What is the significance of knowing all three sides of a triangle in trigonometry?

**Tip:** The Law of Cosines is especially useful for non-right triangles, where the Pythagorean theorem doesn't apply.

Learn how to calculate the third side of triangle ABC using the Law of Cosines with sides a = 3.4, b = 7.8, and angle C = 80 degrees. Step-by-step solution including the formula application and calculations. Suitable for high school students and trigonometry enthusiasts.

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