Problem 18 - Algebra 2 and Trigonometry

Question

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle R S T, R S=0.375, S T=1.29,\) and \(\mathrm{m} \angle S=167\)

Step-by-Step Solution

Answer

The third side of the triangle, \( RT \), is approximately 1.6.

Step 1: Understand the problem

We need to find the third side of triangle \( \triangle RST \) given that \( RS = 0.375 \), \( ST = 1.29 \), and \( \angle S = 167^\circ \).

Step 2: Recall the Law of Cosines

The Law of Cosines can be used to find the third side of a triangle when two sides and the included angle are known. The formula is: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \), where \( c \) is the side opposite angle \( C \), and \( a \) and \( b \) are the other two sides.

Step 3: Substitute the known values

In our triangle, let \( RT \) be the side we need to find. Thus, \( a = 0.375 \), \( b = 1.29 \), and \( C = 167^\circ \). Substitute these values into the law of cosines formula to get: \[ RT^2 = 0.375^2 + 1.29^2 - 2 \cdot 0.375 \cdot 1.29 \cdot \cos(167^\circ) \]

Step 4: Calculate each part of the formula

First, compute \( 0.375^2 = 0.140625 \) and \( 1.29^2 = 1.6641 \). Then calculate \(-2 \cdot 0.375 \cdot 1.29 = -0.9675\). Finally, find \( \cos(167^\circ) \) using a calculator, approximately \(-0.9135\).

Step 5: Simplify the equation

Substitute the values calculated: \[ RT^2 = 0.140625 + 1.6641 - 0.9675 \cdot (-0.9135) \] Compute the part involving the cosine: \( -0.9675 \cdot (-0.9135) \approx 0.884 \). Continue simplifying: \[ RT^2 = 0.140625 + 1.6641 + 0.884 \] Which gives: \[ RT^2 \approx 2.688725 \]

Step 6: Solve for RT

Take the square root of both sides to solve for \( RT \): \[ RT = \sqrt{2.688725} \] Use a calculator to find \( RT \approx 1.639 \). To the nearest tenth, \( RT \approx 1.6 \).

Key Concepts

Triangle Sides

The sides of a triangle are integral as they define its shape and size. Any triangle consists of three sides which denote its boundaries.
In a problem where two sides of a triangle are given along with an angle, like in this exercise, you get an incomplete triangle.

• The missing side challenge the learner to find out this side to fully understand the triangle.
• This often requires applying specific mathematical rules or theorems, such as the Law of Cosines.

This approach helps in piecing together the complete geometry of the triangle as in TRST, where RS and ST are known.

Included Angle

The included angle is the angle formed between two known sides of a triangle. In this context, it's crucial for applying the Law of Cosines. When you are given an angle, like measures 167^ C^ , it plays a central role in determining the unknown third side.
Understanding these angles:

• The angle helps in forming boundaries for the known and unknown sides.
• It directly influences calculations that follow in the law.

The inclusion of this angle insight helps produce a solvable equation. This makes it possible to compute unknown elements of the triangle.

Cosine Function

The cosine function is part of trigonometry and relates an angle in a right triangle to the adjacent side over the hypotenuse. Within the context of the Law of Cosines, it becomes essential.
In the exercise, we calculate cos(167^ ^ C^ ).

• This function helps us incorporate the included angle into computations.
• Being a fundamental part of the equation, cosine influences the final result for the unknown side.

It is especially highlighted by negative results, like -0.9135, showcasing how cosine also determines the direction of the angle computations.

Angle Computation

When you come across angles in a problem involving triangles, the computation often involves multiple steps: calculating power terms, multiplying terms, or using a calculator for trigonometric functions. For instance, in this scenario, angle S is 167°.

• Transform this information into numerical values that feed into the Law of Cosines equation.
• Each step can drastically alter outcomes if miss calculated.

The consistency and accuracy during each phase of these computations directly influence the final answer. This also underscores the significance of checking each part frequently to ensure precise results.