Problem 21 - Trigonometry

Question

$$ \text { Solve each of the following triangles. } $$ $$ a=48 \mathrm{yd}, b=75 \mathrm{yd}, c=63 \mathrm{yd} $$

Step-by-Step Solution

Answer

Find angle using Law of Cosines, then use Sine Law for other angles. Approx angles are A=39.8°, B=86.2°, and C=54.0°.

Step 1: Use the Law of Cosines to Find an Angle

We need to find one of the angles first. We can use the Law of Cosines for this. Let's find angle \( C \). Recall the formula: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).Substitute the given values: \( 63^2 = 48^2 + 75^2 - 2 \cdot 48 \cdot 75 \cdot \cos(C) \).This simplifies to: \( 3969 = 2304 + 5625 - 7200 \cdot \cos(C) \).Combine like terms: \( 3969 = 7929 - 7200 \cdot \cos(C) \).Rearrange to solve for \( \cos(C) \): \( 7200 \cdot \cos(C) = 7929 - 3969 \).Thus, \( 7200 \cdot \cos(C) = 3960 \), leading to \( \cos(C) = \frac{3960}{7200} = 0.55 \).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

Find angle using Law of Cosines, then use Sine Law for other angles. Approx angles are A=39.8°, B=86.2°, and C=54.0°.

Key Concepts

Solving Triangles

When faced with a triangle where you know the lengths of all three sides, solving the triangle typically means finding all the internal angles. This can be approached by using the Law of Cosines, which is especially useful for non-right triangles. The Law of Cosines generalizes the Pythagorean theorem and allows us to calculate angles when we only know sides. Specifically, it provides a formula to determine an angle when three sides are given.By using the formula\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]we can determine one angle at a time. Initially, replace the variables with the known side lengths. After simplifying, rearrange the equation to solve for the cosine of the angle. By finding this cosine value, you're able to find the angle using an inverse cosine function, usually employing a calculator.It’s a step-by-step process of finding one angle, and then you can find others by similar steps or using the Law of Sines once one angle is known. This method is powerful in solving various scenarios involving triangles.

Angle Calculation

Once you have effectively applied the Law of Cosines to calculate the cosine of an angle, the next step is to actually determine the angle itself. This is called angle calculation. It's a critical part of solving triangles, and it's important to understand how to transition from knowing a cosine value to having the actual angle.After computing the cosine of an angle, such as finding \( \cos(C) = 0.55 \) in the exercise, you will use an inverse trigonometric function to get the angle's value in degrees or radians: the arccosine.Using a calculator, typically you would input the value of \( 0.55 \) and apply the inverse cosine function to find the angle \( C \). This value will often be a decimal that represents degrees, so be sure your calculator is set to the correct mode (degrees or radians) that matches what you need.Angle calculation is pivotal since it lays the groundwork for confirming the accuracy of remaining angles, leveraging relationships like the sum of angles in a triangle being 180 degrees, or facilitating further solving with the Law of Sines.

Trigonometry Problem

Trigonometry tackles a broad range of problems involving triangles, extending beyond merely right triangles. In particular, problems involving non-right triangles often employ the Law of Cosines and the Law of Sines. This makes trigonometry much more versatile in solving real-world problems like navigation, engineering, and physics by determining not just missing sides, but also unknown angles.Considering a trigonometry problem such as the one tackled here leads to a methodical approach: first, outline what you know – such as side lengths. Use the Law of Cosines to begin finding angles, as was done to find angle \( C \).It's crucial, as trigonometry problems go, to simplify equations, patiently solve for cosines, and proceed methodically with inverse trigonometric functions. This method’s application ensures a consistent solution process, applicable to several types of triangle configurations, not merely those fitting neatly into right-triangle scenarios.Achieving a clearer understanding of trigonometry concepts like these equips you better to tackle complex geometry problems both in academic settings and real-life situations.