Question

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=80 \mathrm{~cm}, \mathrm{BC}=70 \mathrm{~cm}, \mathrm{AC}=62 \mathrm{~cm}\)

Short answer

Question: Given a triangle with side lengths a = 80 cm, b = 70 cm, and c = 62 cm, find all angles of the triangle. Answer: To find all the angles in the given triangle, follow the steps outlined in the solution. First, use the Law of Cosines to find angle C. Next, use the Law of Sines to find angle B. Finally, use the fact that the sum of angles in a triangle is 180 degrees to find angle A.

Step-by-step solution

Calculate angle C using the Law of Cosines

The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, we have: \[c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot cos(C)\] We can plug in the given values for a, b, and c, i.e., a = 80 cm, b = 70 cm, and c = 62 cm, and then solve for angle C: \[62^2 = 80^2 + 70^2 - 2 \cdot 80 \cdot 70 \cdot cos(C)\] Now, we can solve for cos(C): \[cos(C) = \frac{80^2 + 70^2 - 62^2}{2 \cdot 80 \cdot 70}\]

Find the angle C

Now compute the value of cos(C) and use inverse cosine to find the angle C: \[C = cos^{-1}\left(\frac{80^2 + 70^2 - 62^2}{2 \cdot 80 \cdot 70}\right)\]

Find angle B using the Law of Sines

The Law of Sines states that for any triangle with sides a, b, and c, and angles A, B and C, we have: \[\frac{a}{sin(A)} = \frac{b}{sin(B)} = \frac{c}{sin(C)}\] We can use this relationship to find angle B: \[\frac{70}{sin(B)} = \frac{62}{sin(C)}\] Now, solve for sin(B): \[sin(B) = \frac{70 \cdot sin(C)}{62}\]

Calculate angle B

Compute the value of sin(B) and use inverse sine to find the angle B: \[B = sin^{-1}\left(\frac{70 \cdot sin(C)}{62}\right)\]

Calculate angle A

Finally, we can compute angle A using the fact that the sum of angles in a triangle is always 180 degrees: \[A = 180 - B - C\] Now we have calculated all three angles of triangle ABC, and the triangle is solved.

Key concepts

Law of Cosines

The Law of Cosines is a crucial tool for solving triangles when we know the lengths of all three sides. It helps us determine one of the triangle's angles by relating it to the three sides. In a triangle with sides labeled as \(a\), \(b\), and \(c\), and an angle \(C\) opposite to side \(c\), the Law of Cosines formula is:

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

This equation lets us calculate \(\cos(C)\) by rearranging it:

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

Substituting the side lengths given in a problem lets us solve for the angle \(C\) using an inverse cosine function \(\cos^{-1}\). It’s essential when dealing with non-right triangles to make accurate calculations.

Law of Sines

The Law of Sines provides a different approach from the Law of Cosines, helping us solve triangles with known side lengths and angles (other than just side-side-side configurations). It states that in any given triangle, the ratio of a side length to the sine of its opposite angle is constant. The formula is:

• \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

This can be applied to find unknown angles or sides. For instance, if we know angle \(C\) and side \(c\), we can find angle \(B\) using:

• \(\sin(B) = \frac{b \cdot \sin(C)}{c}\).

Then, we calculate \(B\) using the inverse sine function \(\sin^{-1}\). The Law of Sines is incredibly useful when working with specific angle-side pairings, making it easier to solve for remaining triangle elements.

Triangle Solving

Triangle solving involves finding all unknown side lengths and angles of a triangle. The process is initiated by identifying known sides and angles. We then apply trigonometric laws such as the Law of Cosines and the Law of Sines accordingly.

• Given a problem where all sides are known, use the Law of Cosines to find an angle first.
• Choose either Law of Sines or the fact that angles sum up to 180 degrees to solve for remaining angles.

Breaking down the problem into smaller steps, each utilizing specific laws, renders complex problems manageable. With practice, triangle solving boosts mathematical understanding and visualization skills.

Angle Calculation

Angles in a triangle are key to understanding its form and behavior. Calculations often start with one label or measure, like when using the Law of Cosines to determine an angle from side measurements.

• After finding one angle with cosine, use known trigonometric rules like the sum of triangles' angles adding to 180° to calculate the rest.

Solving for angles could additionally involve applying the Law of Sines after securing a primary angle, making subsequent angle calculations straightforward. Techniques like computing angles with inverse trigonometric functions \(\cos^{-1}\) or \(\sin^{-1}\) follow from handling sines and cosines. Mastering these calculations supports broader applications in geometry and trigonometry.