Question

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\sin B\) given \(\mathrm{AC}=6 \mathrm{~cm}\) and \(\mathrm{BC}=9 \mathrm{~cm}\)

Short answer

Answer: The sine of angle B is $$\frac{2\sqrt{117}}{39}$$.

Step-by-step solution

Write down the given information

We're given a right-angled triangle ΔABC, with a right angle at C, and the side lengths AC = 6 cm and BC = 9 cm.

Pythagorean theorem

To find the length of side AB, we'll use the Pythagorean theorem: $$a^2 + b^2 = c^2$$ where a and b are the legs of the triangle (AC and BC) and c is the hypotenuse (AB). Substitute the given values into the equation: $$6^2 + 9^2 = AB^2$$

Solve for AB

Calculate the square of the given side lengths and add them: $$AB^2 = 36 + 81 = 117$$ Now, take the square root of the result to find the length of AB: $$AB = \sqrt{117}$$

Calculate sin(B) using the sine formula

In a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In our case: $$\sin(B) = \frac{AC}{AB}$$ Substitute the values of AC and AB into the equation: $$\sin(B) = \frac{6}{\sqrt{117}}$$

Simplify the result

To simplify the result, multiply the numerator and denominator by the square root of 117: $$\sin(B) = \frac{6 \cdot \sqrt{117}}{117}$$ Now, divide 6 by 117 to get the final answer: $$\sin(B) = \frac{2\sqrt{117}}{39}$$ So, the sine of angle B is $$\frac{2\sqrt{117}}{39}$$.

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry, especially when dealing with right triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is expressed as:

\[ c^2 = a^2 + b^2 \]
In the context of our problem, side AC and side BC are the legs of the triangle, while side AB is the hypotenuse. By plugging the known values into the theorem:

\[ 6^2 + 9^2 = AB^2 \]\[ AB^2 = 36 + 81 = 117 \]\[ AB = \(\sqrt{117}\) \]
By applying the Pythagorean theorem, we can easily determine the length of any side of a right triangle if the lengths of the other two are known. This method is not just a cornerstone in solving right triangles but is also widely used across various fields, including architecture, engineering, and physics.

Solving Right Triangles

Solving right triangles involves finding the unknown lengths and angles of the triangle. The process typically involves a combination of the Pythagorean theorem for lengths and trigonometric ratios for angles. A right-angled triangle has one angle of 90 degrees, and the lengths of the sides are related by the Pythagorean theorem.

When it comes to angles, various trigonometric functions are used based on the known and required sides. For instance, when we know the lengths of the adjacent side (next to the angle) and the hypotenuse, we can find the angle using the cosine function. Conversely, if we know the opposite side and the hypotenuse, as in our example, we can find the angle using the sine function.

Key Trigonometric Ratios

• Sine (sin) - opposite/hypotenuse
• Cosine (cos) - adjacent/hypotenuse
• Tangent (tan) - opposite/adjacent

These ratios are essential for solving for the unknown sides or angles in right triangles, making it possible to analyze various real-world problems that can be modeled with right triangles.

Sine of an Angle

The sine of an angle in a right triangle is a trigonometric ratio, representing the relationship between the length of the side opposite the angle and the length of the hypotenuse. It's one of the three primary trigonometric functions used to solve right triangles. The sine of angle B, in our problem, is calculated by the formula:

\[ \text{sin}(B) = \frac{\text{opposite side to angle B}}{\text{hypotenuse}} \]
Given that AC is 6 cm (the opposite side to angle B), and AB is the hypotenuse which we found using the Pythagorean theorem (\(\sqrt{117}\) cm), we plug these values into the formula:

\[ \text{sin}(B) = \frac{6}{\(\sqrt{117}\)} \]To simplify, we rationalize the denominator by multiplying the top and bottom by \(\sqrt{117}\):

\[ \text{sin}(B) = \frac{6 \(\sqrt{117}\)}{117} = \frac{2\sqrt{117}}{39} \]Understanding the sine function and its applications is critical not only in geometry but also in physics, engineering, and even in fields like music and acoustics where wave properties are analyzed.