Question

For questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=15 \mathrm{~cm}, \mathrm{AC}=23 \mathrm{~cm}, B=57^{\circ}\)

Short answer

Question: Using the given information of a triangle ABC with sides AB = 15cm and AC = 23cm, and angle B = 57 degrees, solve the triangle to find the length of side BC and the magnitudes of angles A and C. Answer: Side BC is approximately 17.85 cm, angle A is approximately 83.49 degrees, and angle C is approximately 39.51 degrees.

Step-by-step solution

Using the Law of Cosines to find BC

In order to find the length of side BC, we can use the Law of Cosines, given by the formula: \[c^2 = a^2 + b^2 - 2ab\cos(C)\] Here, a = 15 cm (side AB), b = 23 cm (side AC), and angle C = 57 degrees. So, plug in the given values and solve for c (side BC): \[c^2 = 15^2 + 23^2 - 2\times15\times23\times\cos(57^\circ)\] After calculating this, we find that c ≈ 17.85 cm.

Using the Law of Sines to find angle A

Now, we will use the Law of Sines to find angle A. This law states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. The Law of Sines is given by the formula: \[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\] We know the lengths of sides a, b, and c, and we also know the measure of angle B (57 degrees). So, we will use the ratio of side b to the sine of angle B to find angle A: \[\frac{23}{\sin(A)} = \frac{15}{\sin(57^{\circ})}\] Now, isolate sin(A) and use the inverse sine function to find the measure of angle A: \[\sin(A) = \frac{23\sin(57^\circ)}{15}\] \[A = \arcsin{\left(\frac{23\sin(57^\circ)}{15}\right)}\] After calculating this, we find that A ≈ 83.49 degrees.

Calculating angle C

Finally, we need to find the measure of angle C. Since we know that the sum of the angles in a triangle is always 180 degrees, we can simply subtract the measures of angles A and B to find angle C: \[C = 180^\circ - A - B\] Plugging in our calculated values for angles A and B: \[C = 180^\circ - 83.49^\circ - 57^\circ\] After calculating this, we find that C ≈ 39.51 degrees.

Summary

In summary, we have found all the missing information for the triangle ABC. Side BC is approximately 17.85 cm, angle A is approximately 83.49 degrees, and angle C is approximately 39.51 degrees.

Key concepts

Law of Cosines

Understanding the Law of Cosines is essential for solving triangles when we have a mix of sides and angles to find the missing parts. It extends the Pythagorean theorem to any triangle, not just right-angled ones. The Law of Cosines says for any triangle ABC, if you want to find the length of a side, let's call it 'c', and you know the lengths of the other two sides, 'a' and 'b', as well as the angle opposite to side 'c', which is angle C.

The formula looks like this: \[c^2 = a^2 + b^2 - 2ab\cos(C)\]
In the given problem, to find side BC we just insert the known values for AB, AC, and angle B and calculate the missing side. It's important to have your calculator set to the correct angle unit, like degrees in this case, to get the right answer. And after the computation, we learned that side BC is approximately 17.85 cm.

Law of Sines

The Law of Sines is another vital tool for solving for unknown parts of a triangle, particularly when dealing with angles and opposite sides. This law creates a relationship between all sides and angles of a triangle. The formula takes the form of a proportion:

\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]
For the given problem, we applied this formula to find angle A. We already knew side AB, side AC, and angle B. By creating a ratio of side AC to the sine of angle B and comparing it with side AB to the sine of angle A, we can isolate sin(A) and then use the inverse sine function to calculate angle A. The result is approximately 83.49 degrees. This step illustrates how the Law of Sines can unlock an angle when we know two sides and an opposite angle.

Triangle Angle Sum

The triangle angle sum is one of the fundamental properties of triangles. It states that the sum of the angles within a triangle will always equal 180 degrees. It is a useful check for finding missing angles and ensuring solutions are correct.

In the original exercise, after using the Law of Sines to find angle A and knowing angle B from the start, we applied the triangle angle sum property to find angle C. By subtracting the known angles A and B from 180 degrees, we found that angle C is approximately 39.51 degrees. This straightforward concept reinforces the relationship between the angles of any triangle and is a sound finishing step in validating the solution to a triangle problem.