Question

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate BC given \(\mathrm{AB}=27 \mathrm{~cm}\) and \(B=32^{\circ}\).

Short answer

Answer: The length of side BC in the right triangle ABC is approximately 14.31 cm.

Step-by-step solution

Write down the given information

We are given the following information: - Right triangle ABC, with right angle at C - AB = 27 cm (hypotenuse) - B = 32 degrees

Apply the sine function

Since we have the hypotenuse and the angle, B, we can use the sine function to find the length of BC. The sine function can be written as: sin(B) = opposite side (BC) / hypotenuse (AB)

Plug in the given values

Now, plug in the given values (angle B and hypotenuse AB) into the sine function equation: sin(32°) = BC / 27 cm

Solve for BC

To find BC, we can multiply both sides of the equation by 27 cm: BC = 27 cm × sin(32°) Now just compute the value: BC ≈ 27 cm × 0.5299 BC ≈ 14.31 cm

Write down the answer

The length of side BC in the right triangle is approximately 14.31 cm.

Key concepts

Trigonometric Functions

Trigonometric functions are vital tools in mathematics, especially when dealing with right angled triangles. They relate the angles of a triangle to the lengths of its sides and are essential in a wide range of applications, from physics to engineering. There are three primary trigonometric functions: sine (sin), cosine (cos), and tangent (tan). Each function corresponds to a specific ratio of sides in a right-angled triangle.

For example, the sine of an angle in a right triangle is the ratio of the length of the opposite side to the angle to the length of the hypotenuse. Similarly, cosine is the adjacent over hypotenuse, and tangent is the opposite over adjacent. These functions allow for the calculation of unknown sides or angles in triangles when certain values are known, providing a bridge between angles and side lengths.

Sine Function

The sine function plays a pivotal role in understanding the relationship within a right triangle. In a formal definition, for a given angle \theta, the sine function is the ratio of the length of the side that is opposite to the angle to the length of the longest side of the triangle, the hypotenuse. Mathematically, this is expressed as \( sin(\theta) = \frac{opposite}{hypotenuse} \).

Furthermore, the sine function is periodic, meaning it repeats its values in a predictable pattern, which is particularly useful when solving problems involving wave motion or oscillations. When working with right triangles, the sine function allows us to find an unknown side length when we have the measure of an angle and the length of the hypotenuse, as demonstrated in the original exercise. By isolating the desired side in the equation, we're able to solve for its length with elementary algebra.

Right Triangle Geometry

The geometry of right triangles is a fundamental aspect of trigonometry. A right triangle is defined as having one angle that is exactly 90 degrees, which significantly constrains its properties and makes calculations more straightforward. In addition to the three sides—hypotenuse, opposite, and adjacent—right triangles have two acute angles, which always add up to 90 degrees.

The Pythagorean theorem is a central principle in understanding right triangle geometry, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem, in conjunction with trigonometric functions, helps solve for unknown measures within the triangle. Knowing how these elements interplay is crucial for solving many real-life problems that involve this type of geometry.