Problem 7 - Contemporary Precalculus

Question

Standard notation for triangle ABC is used throughout. Use a calculator and round off your answers to one decimal place at the end of the computation. Solve triangle ABC under the given conditions. $$A=102.3^{\circ}, B=36.2^{\circ}, a=16$$

Step-by-Step Solution

Answer

Question: Given triangle ABC with angle A = 102.3 degrees, angle B = 36.2 degrees, and side a = 16, find the missing angle C and the other two sides b and c. Solution: The missing angle C is approximately $$41.5^{\circ}$$, side b is approximately 9.8, and side c is approximately 10.8.

Step 1: Find angle C using angle sum property of a triangle

We know that the sum of the angles in a triangle is 180 degrees. Using this property, we can find the missing angle C by subtracting the given angles from 180. $$C = 180^{\circ} - A - B$$ $$C = 180^{\circ} - 102.3^{\circ} - 36.2^{\circ}$$ $$C = 41.5^{\circ}$$

Step 2: Use the sine rule to find side b

Using the sine rule, we have $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ Rearrange the equation to find b: $$b = \frac{a \sin B}{\sin A}$$ Substitute the given values and make sure the calculator is in degree mode: $$b = \frac{16 \sin 36.2^{\circ}}{\sin 102.3^{\circ}}$$ $$b \approx 9.8$$

Step 3: Use the sine rule to find side c

Similarly, using the sine rule, we have: $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ Rearrange the equation to find c: $$c = \frac{a \sin C}{\sin A}$$ Substitute the given values: $$c = \frac{16 \sin 41.5^{\circ}}{\sin 102.3^{\circ}}$$ $$c \approx 10.8$$ Our final solution for triangle ABC is angle $$C = 41.5^{\circ}$$, side $$b \approx 9.8$$, and side $$c \approx 10.8$$.

Key Concepts

Sine Rule

Understanding the sine rule is crucial when dealing with non-right-angled triangles. It is a trustworthy technique that interrelates the sides and angles within a triangle. The sine rule states that the ratio of each side of a triangle to the sine of its opposite angle is equal for all three sides and angles. In formula terms, it is expressed as:
\[\begin{equation}\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\end{equation}\]
where letters \[\begin{equation}(a, b, c)\end{equation}\] correspond to the sides and letters \[\begin{equation}(A, B, C)\end{equation}\] to the opposite angles of the triangle. This comes handy when you need to find an unknown side or angle, as was shown in the step-by-step solution where we solved for sides \[\begin{equation}(b)\end{equation}\] and \[\begin{equation}(c)\end{equation}\] of the triangle ABC.

Triangle Angle Sum

A foundational concept in geometry is the triangle angle sum theorem. This theorem simplifies many problems by leveraging the fact that the interior angles of a triangle always add up to \[\begin{equation}180^\circ\end{equation}\]. Understanding this principle allows us to solve for an unknown angle when the other two are known, as illustrated in the solution of angle \[\begin{equation}(C)\end{equation}\] of triangle ABC. It's a simple subtraction problem in essence:
\[\begin{equation}C = 180^\circ - A - B\end{equation}\]
It not only helps in solving triangles but also lays the groundwork for more complex geometrical concepts.

Trigonometry in Precalculus

Trigonometry plays a pivotal role in precalculus and is a field of mathematics that studies the relationships between the sides and angles of triangles. The foundation of trigonometry in precalculus extends beyond right-angled triangles, exploring the properties and applications of sine, cosine, and tangent functions. These functions are necessary for solving triangles, analyzing periodic phenomena, and understanding rotary motion. Seeing trigonometry in action, as in our exercise with triangle ABC, aids students in recognizing how precalculus lays the groundwork for calculus by introducing them to crucial mathematical concepts.

Using Calculator for Trigonometry

Today's calculators are incredibly powerful and play a significant role in solving trigonometry problems. However, it is essential to set them to the correct mode - degree or radian - depending on the problem at hand. For the given exercise, the degree mode is used as the angles are given in degrees. It is important to note, when punching numbers into the calculator:

• Ensure the mode matches the angle units provided.
• Use the inverse trigonometric functions carefully when solving for angles.
• Double-check your entries to avoid input errors.

In our exercise, the calculator facilitated finding the lengths of \[\begin{equation}b\end{equation}\] and \[\begin{equation}c\end{equation}\] by handling the necessary sine calculations.