Problem 22 - Trigonometry

Question

$$ \text { Solve each of the following triangles. } $$ $$ a=0.48 \mathrm{yd}, b=0.63 \mathrm{yd}, c=0.75 \mathrm{yd} $$

Step-by-Step Solution

Answer

The angles of the triangle are approximately \( A = 39.54^\circ \), \( B = 56.61^\circ \), \( C = 83.85^\circ \).

Step 1: Identify the Type of Triangle

Given the sides \( a = 0.48 \text{ yd} \), \( b = 0.63 \text{ yd} \), and \( c = 0.75 \text{ yd} \), check if these sides form a right triangle using the Pythagorean theorem \( a^2 + b^2 = c^2 \). Calculate: \( a^2 = 0.2304 \), \( b^2 = 0.3969 \), and \( c^2 = 0.5625 \). Add \( a^2 + b^2 = 0.2304 + 0.3969 = 0.6273 \), which is not equal to \( c^2 = 0.5625 \). Therefore, this is not a right triangle.

Step 2: Use the Law of Cosines

Since this is not a right triangle, we use the Law of Cosines to find one angle. Let's find angle \( C \) first using: \[ c^2 = a^2 + b^2 - 2ab \cos C \].Plugging in the values: \( 0.5625 = 0.2304 + 0.3969 - 2(0.48)(0.63) \cos C \), then solve for \( \cos C \).

Step 3: Solve for \( \cos C \)

Simplify the equation from Step 2: \( 0.5625 = 0.6273 - 0.6048 \cos C \), rearranging gives: \( 0.6048 \cos C = 0.6273 - 0.5625 \).\( 0.6048 \cos C = 0.0648 \). Thus, \( \cos C = \frac{0.0648}{0.6048} \approx 0.1072 \). Find \( C \) using \( C = \cos^{-1}(0.1072) \approx 83.85^\circ \).

Step 4: Use the Law of Sines

Once you know angle \( C \), use the Law of Sines to find another angle, \( A \), using: \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] \[ \sin A = \frac{a \sin C}{c} \]Plugging in \( a = 0.48 \), \( c = 0.75 \), and \( C = 83.85^\circ \), find \( \sin C = \sin(83.85^\circ) \approx 0.9952 \).Therefore, \( \sin A = \frac{0.48 \times 0.9952}{0.75} \approx 0.6361 \), and hence \( A \approx \sin^{-1}(0.6361) \approx 39.54^\circ \).

Step 5: Calculate the Remaining Angle

With angles \( C \approx 83.85^\circ \) and \( A \approx 39.54^\circ \), use the angle sum property of triangles to find angle \( B \):\[ B = 180^\circ - A - C = 180^\circ - 39.54^\circ - 83.85^\circ \approx 56.61^\circ \].

Step 6: Verify Solution

Use the Law of Sines to verify \( \frac{b}{\sin B} = \frac{c}{\sin C} \)\( \sin B = \sin(56.61^\circ) \approx 0.834 \) \( \frac{0.63}{0.834} \approx \frac{0.75}{0.9952} \)Both are approximately equal, confirming the solution is consistent.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle used to solve right-angled triangles. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, this is expressed as:

• \( a^2 + b^2 = c^2 \)

This formula enables you to find the length of one side of a triangle if you know the other two. However, the current problem involves side lengths that do not satisfy this formula.
The calculated sums, \( a^2 + b^2 = 0.6273 \) and \( c^2 = 0.5625 \), do not match. Thus, this is not a right triangle, and therefore you must use another method, such as the Law of Cosines, to solve it effectively.

Law of Cosines

When you are dealing with a non-right triangle, you'll often use the Law of Cosines to find angles or sides. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. Here's how it's formulated:

• \( c^2 = a^2 + b^2 - 2ab \cos C \)

This equation helps you find the angle if you know all three side lengths, or find a side if you know two sides and the enclosed angle. In our problem, we applied it to find angle \( C \) using sides \( a = 0.48 \), \( b = 0.63 \), and \( c = 0.75 \). The formula rearranges to find \( \cos C \), leading to precise calculations for angles where the Pythagorean theorem isn't applicable.
So, to solve for \( C \), we computed: \( 0.6048 \cos C = 0.0648 \), yielding \( \cos C \approx 0.1072 \). Find the angle using the inverse cosine function to get \( C \approx 83.85^\circ \).

Law of Sines

After determining an initial angle using the Law of Cosines, the Law of Sines is useful for finding the remaining angles. This law, unlike the Pythagorean theorem, can be applied to any triangle. It states:

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

Using this helps you find angles and side lengths by setting up a proportion. In this exercise, after finding angle \( C \), we used the Law of Sines to find angle \( A \):

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

Substitute the known values, and calculate \( \sin A \). Solving gives \( A \approx 39.54^\circ \). This method is handy because once you know two angles, finding the third using the angle sum property becomes straightforward.

Angle Sum Property

The angle sum property of triangles is a simple yet essential rule stating that the sum of the interior angles of any triangle is always \( 180^\circ \). This property allows us to easily find the unknown angle when two angles are known.
For the current problem, after finding \( A \approx 39.54^\circ \) and \( C \approx 83.85^\circ \), using the angle sum property is straightforward:

• \( B = 180^\circ - A - C \)

Calculate to get \( B \approx 56.61^\circ \). This principle effectively ties up the problem by ensuring that all possible angles in a triangle are accounted for.
Verifying the solution is also made simpler with this tactic, because confirming that the angles add up to \( 180^\circ \) serves as a check for any possible calculation errors earlier in the process.