Problem 23 - Precalculus with Limits

Question

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ C=43^{\circ}, \quad a=\frac{4}{9}, \quad b=\frac{7}{9} $$

Step-by-Step Solution

Answer

The remaining side length is \(c=0.79\) and the angles are approximately \(A=36.97^\circ\) and \(B=100.03^\circ\).

Step 1: Using Law of Cosines to find side c

First, we need to find the remaining side length using the law of cosines. The formula is given by \(c= \sqrt{a^2 + b^2 - 2ab \cos C}\), substituting \(a=\frac{4}{9}\), \(b=\frac{7}{9}\), and \(C=43\) degrees, which is \(0.75\) in radians (since \(\cos C\) should be in radians).

Step 2: Calculation

Then, we do the calculation. c equals \(\sqrt{(\frac{4}{9})^2 + (\frac{7}{9})^2 - 2 * \frac{4}{9} * \frac{7}{9} * \cos 0.75}\) which gives \(c=0.79\) after rounding to two decimal places.

Step 3: Finding the remaining angles

Next, we can use the law of cosines again to find angles A and B. For angle A the formula is \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\) and for angle B it is \(\cos B = \frac{a^2 + c^2 - b^2}{2ac}\), substitute the values \(a=\frac{4}{9}\), \(b=\frac{7}{9}\), and \(c=0.79\), and calculate the angles.

Step 4: Calculate angles A and B

Angle A equals to \(\cos^{-1}(\frac{(\frac{7}{9})^2 + 0.79^2 - (\frac{4}{9})^2}{2*0.79*\frac{7}{9}})\) which is approximately \(36.97\) degrees after conversion from radians. Angle B equals to \(\cos^{-1}(\frac{(\frac{4}{9})^2 + 0.79^2 - (\frac{7}{9})^2}{2*0.79*\frac{4}{9}})\) which is approximately \(100.03\) degrees after conversion from radians.

Key Concepts

Triangle Solving

Solving a triangle involves finding all its unknown sides and angles. In this context, we're focusing on a triangle where we know two sides and one angle. To solve the triangle completely:

• Calculate the unknown side using given sides and angle.
• Find the remaining angles using the newly calculated side.

This process requires a firm grasp of trigonometry and related rules like the Law of Cosines, which allows us to piece together missing information depending on what is initially known. By step-by-step calculation, one can solve for all triangle dimensions in a systematic way.

Cosine Rule

The Law of Cosines, also known as the Cosine Rule, is essential when dealing with non-right triangles. This rule helps us solve for unknown parts of a triangle when we have:

• Two sides and the included angle (SAS situation).
• All three sides and needing an angle (SSS situation).

The formula is \( c = \sqrt{a^2 + b^2 - 2ab \cos C} \), and it is used to find a side, while its rearranged forms help find angles. For instance, to find angle A, the formula \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \) is used.
Understanding this rule helps us explore triangles without the limitation of needing a right angle.

Radians Conversion

Converting between degrees and radians is pivotal in trigonometry, especially when applying trigonometric rules using calculus or advanced math software that frequently uses radian mode.

• Degrees to Radians: Multiply the degree value by \( \frac{\pi}{180} \).
• Radians to Degrees: Multiply the radian value by \( \frac{180}{\pi} \).

In the given exercise, angle C is 43 degrees which equals approximately 0.75 radians. This conversion is crucial for accurate calculations when applying trigonometric functions like cosine.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It's fundamental for solving triangles and understanding the properties of different triangles.

• Trigonometric Ratios: Sine, Cosine, and Tangent are key functions that help relate angles to side ratios.
• The unit circle can further aid understanding of how these ratios extend beyond standard angle measures.

This area of math not only applies to triangles but also to waves, oscillations, and many real-world phenomena. Mastering these concepts equips you with tools to tackle a wide range of mathematical and physical problems.