Question

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=32 \mathrm{~cm}, \mathrm{BC}=30 \mathrm{~cm}, \mathrm{AC}=41 \mathrm{~cm}\)

Short answer

Question: Given a triangle ABC with side lengths AB = 32 cm, BC = 30 cm, and AC = 41 cm, find the angles of the triangle. Answer: The angles of the triangle ABC are ∠A ≈ 75.87°, ∠B ≈ 62.05°, and ∠C ≈ 42.08°.

Step-by-step solution

Use the Cosine Law to find angle A

Using the Cosine Law, which states that \(c^2 = a^2 + b^2 - 2ab \cos C\), we can find angle A (∠A) by plugging in the provided side lengths: \(a = 30\mathrm{~cm}, b = 41\mathrm{~cm}, c = 32\mathrm{~cm}\) \(32^2 = 30^2 + 41^2 - 2(30)(41) \cos A\) Solve for angle A: \(\cos A = \frac{30^2 + 41^2 - 32^2}{2(30)(41)}\) Now calculate the value of angle A: \(A \approx \arccos{(0.2411)} \approx 75.87^\circ\)

Use the Cosine Law to find angle B

Similarly, we can find angle B (∠B) using the Cosine Law: \(a = 32\mathrm{~cm}, b = 41\mathrm{~cm}, c = 30\mathrm{~cm}\) \(30^2 = 32^2 + 41^2 - 2(32)(41) \cos B\) Solve for angle B: \(\cos B = \frac{32^2 + 41^2 - 30^2}{2(32)(41)}\) Now calculate the value of angle B: \(B \approx \arccos{(0.4695)} \approx 62.05^\circ\)

Find angle C using the Triangle Angle Sum theorem

The Triangle Angle Sum theorem states that the sum of the angles in a triangle is \(180^\circ\). So, to find angle C (∠C), we can subtract the other two angles from \(180^\circ\): \(C = 180^\circ - A - B \approx 180^\circ - 75.87^\circ - 62.05^\circ \approx 42.08^\circ\)

Compile the results

The solved triangle ABC has the following angles and side lengths: ∠A ≈ 75.87°, ∠B ≈ 62.05°, ∠C ≈ 42.08° AB = 32 cm, BC = 30 cm, AC = 41 cm

Key concepts

Cosine Law

The Cosine Law, also known as the Law of Cosines, is an essential tool in trigonometry for solving triangles when you know the length of all three sides but none of the angles. It comes in handy for non-right-angled triangles and extends the Pythagorean theorem to all triangles. The formula is:\[ c^2 = a^2 + b^2 - 2ab \cos C \]Here, \( c \) is the side opposite angle \( C \), while \( a \) and \( b \) are the other two sides. This equation helps calculate an unknown angle when all side lengths are known. In practice:

• Solve for the cosine of the desired angle.
• Use the inverse cosine function to find the angle itself.

For example, in our exercise, we used the Cosine Law twice to determine angles \( A \) and \( B \) of triangle \( \triangle ABC \). This allowed us to then use another method, the Triangle Angle Sum theorem, to find the third angle.

Triangle Angle Sum theorem

The Triangle Angle Sum theorem is a fundamental property of Euclidean geometry, stating that the sum of the internal angles in any triangle is always \( 180^\circ \). This is crucial for solving triangles when you know an angle or two alongside certain side lengths.For example:\[ A + B + C = 180^\circ \]In our triangle problem, once angles \( A \) and \( B \) were calculated using the Cosine Law, angle \( C \) could be easily found by:\[ C = 180^\circ - A - B \]This method highlights the importance of knowing fundamental properties of triangles, like the triangle angle sum, which simplifies the calculation process when one or two angles are known.

Trigonometry

Trigonometry is an important branch of mathematics dealing with the relationships between the angles and sides of triangles. It is particularly useful in various real-world applications like navigation, physics, engineering, and architecture, among others. Key concepts include:

• Sine, cosine, and tangent functions for right and non-right triangles.
• Laws of sines and cosines to solve for unknown angles or sides.
• Trigonometric identities and equations for simplifying expressions.

In solving our triangle problem, trigonometry was vital. We employed it through the Law of Cosines to determine specific angles from given sides, emphasizing the power and flexibility of trigonometric functions in analyzing non-right angles. Mastery of these concepts can significantly bolster mathematical confidence and capability in both academic and practical contexts.