Problem 270 - Precalculus

Question

For the following exercises, find the lengths of the missing sides if side \(a\) is opposite angle \(A,\) side \(b\) is opposite angle \(B,\) and side \(c\) is the hypotenuse. $$ c=12, \quad \& A=45^{\circ} $$

Step-by-Step Solution

Answer

Missing sides are \( a = 6\sqrt{2} \) and \( b = 6\sqrt{2} \).

Step 1: Understanding the Problem

We need to find the lengths of the missing sides in a right triangle where the hypotenuse \( c = 12 \) and angle \( A = 45^{\circ} \). The sides opposite the angles are \( a \) for \( A \) and \( b \) for \( B \).

Step 2: Using Trigonometric Ratios

In a right triangle, we use sine and cosine to find the missing sides. For angle \( A = 45^{\circ} \):- \( \sin(45^{\circ}) = \frac{a}{c} \)- \( \cos(45^{\circ}) = \frac{b}{c} \)Given \( c = 12 \), we use \( \sin(45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \).

Step 3: Calculate Length of Side a

Use the sine function:\[ \sin(45^{\circ}) = \frac{a}{12} \]\[ \frac{\sqrt{2}}{2} = \frac{a}{12} \]Solve for \( a \):\[ a = 12 \times \frac{\sqrt{2}}{2} = 6\sqrt{2} \]

Step 4: Calculate Length of Side b

Use the cosine function:\[ \cos(45^{\circ}) = \frac{b}{12} \]\[ \frac{\sqrt{2}}{2} = \frac{b}{12} \]Solve for \( b \):\[ b = 12 \times \frac{\sqrt{2}}{2} = 6\sqrt{2} \]

Step 5: Verify the Solution

Since both angles \( A \) and \( B \) are \( 45^{\circ} \), this forms a 45-45-90 triangle, where both legs \( a \) and \( b \) should be equal, which confirms our calculations as correct.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are crucial when solving problems involving right triangles. They connect the angles with the side lengths, making it easier to find unknown values. In any right triangle, the main trigonometric ratios—sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \))—are defined as follows:

• Sine (\( \sin \theta \)) is the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)) is the ratio of the opposite side to the adjacent side.

For the specific problem in the original exercise, the relationship starts with knowing the angle of \( 45^\circ \) and the hypotenuse length (\( c = 12 \)). Thus, to find the legs (\( a \) and \( b \)), we employ the sine and cosine ratios:

• \( \sin(45^\circ) = \frac{a}{c} \)
• \( \cos(45^\circ) = \frac{b}{c} \)

Both \( \sin(45^\circ) \) and \( \cos(45^\circ) \) equal \( \frac{\sqrt{2}}{2} \), which keeps calculations straightforward since the values simplify symmetrically.
This principle helps navigate through triangle problems with ease, as these ratios provide a direct connection between angles and sides.

45-45-90 Triangle

The 45-45-90 triangle is a special type of right triangle where two angles measure \( 45^\circ \) each. This symmetry implies that the two legs of the triangle have equal lengths. The unique property of the 45-45-90 triangle is its consistent ratio:

• The legs (\( a \) and \( b \)) are equal.
• The hypotenuse (\( c \)) is \( \sqrt{2} \) times longer than each leg.

In the context of the exercise, given the hypotenuse \( c = 12 \), finding the lengths of the two legs involves using this ratio:

• If \( a = b \), then \( a = \frac{c}{\sqrt{2}} \)
• Simplified, \( a = 6\sqrt{2} \)

This predictable pattern makes solving for missing sides quite straightforward. The 45-45-90 triangle ensures calculations are less intensive, making it an efficient method for finding side lengths without complex equations.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in mathematics, particularly useful in right triangle calculations. It states that for any right triangle, the sum of the squares of the two shorter sides (legs) equals the square of the longest side (hypotenuse). Mathematically, this is expressed as:

• \( a^2 + b^2 = c^2 \)

In the context of our exercise, although the Pythagorean Theorem wasn't used directly due to the special properties of the 45-45-90 triangle, it still underpins our understanding:

• Given \( a = b = 6\sqrt{2} \)
• The hypotenuse \( c = 12 \)
• Verify with \( (6\sqrt{2})^2 + (6\sqrt{2})^2 = 12^2 \)

Thus, calculations confirm the theorem holds true:\[72 + 72 = 144\]The theorem acts as a check for the accuracy of side lengths, ensuring that no mistakes were made in calculating \( a \) and \( b \). This consistency reaffirms the logic used in solving right triangle problems.