Solving for Side AC in Triangle ABC Using Trigonometric Methods

When given the angles and sides of a triangle, we can use various trigonometric methods to find the unknown side lengths. This article will demonstrate how to find the length of side AC in triangle ABC when ∠A 65°, AB 7 cm, and BC 12 cm.

Using the Law of Cosines

The Law of Cosines is a fundamental principle in trigonometry. It is used to find the length of a side in a triangle when two sides and the included angle are known. The formula for the Law of Cosines is:

[c^2 a^2 b^2 - 2ab cdot cos C]

In the context of our problem, we are given:

Substituting the given values into the Law of Cosines formula:

[AC^2 7^2 12^2 - 2 cdot 7 cdot 12 cdot cos 65°]

First, calculate the squares:

[AC^2 49 144 - 2 cdot 7 cdot 12 cdot cos 65°]

Next, calculate the value of (2 cdot 7 cdot 12):

[2 cdot 7 cdot 12 168]

Using the approximate value for (cos 65° approx 0.4226), substitute it into the equation:

[AC^2 49 144 - 168 cdot 0.4226]

Calculate (168 cdot 0.4226):

[168 cdot 0.4226 approx 71.1]

Now substitute this back into the equation:

[AC^2 49 144 - 71.1 193 - 71.1 approx 121.9]

Finally, take the square root to find the length of AC:

[AC approx sqrt{121.9} approx 11.03 text{ cm}]

Therefore, the length of side AC is approximately 11.03 cm.

Using the Sine Theorem

The Sine Theorem can also be used to find the unknown side length in a triangle. The theorem states that the ratio of the sine of an angle to the length of its opposite side is constant throughout the triangle. Using the same problem, we can find the length of AC as follows:

Starting with the Sine Theorem:

[frac{12}{sin 65°} frac{7}{sin C} frac{x}{sin B}]

Solve for (sin C):

[sin C frac{7 sin 65°}{12} approx 0.5286]

Find the angle (C):

[C arcsin 0.5286 approx 31.916°]

Now find the angle (B):

[B 180° - 65° - 31.916° approx 83.084°]

Using the Sine Theorem again to find AC:

[frac{12}{sin 65°} frac{AC}{sin 83.084°}]

Solve for AC:

[AC frac{12 sin 83.084°}{sin 65°} approx 13.144 text{ cm}]

Therefore, the length of side AC is approximately 13.14 cm.

Using the Cosine Formula

The Cosine Formula is another useful method to solve for the sides of a triangle. It is given by:

[cos hat{A} frac{AB^2 AC^2 - BC^2}{2AB cdot AC}]

Using the known values:

[2 cos 65° cdot 7 7^2 AC^2 - 12^2]

[ 14 cos 65° 49 AC^2 - 144 Rightarrow 14 cos 65° -95 AC^2]

[AC^2 14 cos 65° 95 Rightarrow AC^2 5.9167 95 100.9167]

[AC sqrt{100.9167} approx 10.046 text{ cm}]

After applying the quadratic formula to the equation (AC^2 - 5.9167AC - 95 0), we find:

[AC frac{5.917 pm sqrt{415.007}}{2} approx 13.144 text{ cm}]

Thus, the length of side AC is approximately 13.14 cm.

Conclusion

Using the Law of Cosines, Sine Theorem, and Cosine Formula, we can effectively determine the length of side AC in triangle ABC. The length of AC is approximately 11.03 cm, 13.14 cm, and 13.144 cm, respectively, demonstrating the importance of choosing the appropriate method based on the given information.