Question

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\mathrm{AC}\) given \(\mathrm{BC}=12 \mathrm{~cm}\) and \(B=53^{\circ}\)

Short answer

Solution: The length of side AC in the given right triangle is 16 cm.

Step-by-step solution

Analyze the given information

We have a right triangle with a right angle at C. We are given the length of side BC as \(12 \mathrm{~cm}\) and angle B as \(53^{\circ}\). Our goal is to find the length of side AC. Let AC be denoted as 'a'.

Define the sine function in the context of the triangle

In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In our triangle, sin(B) would be the ratio of the side opposite angle B (AC or 'a') to the hypotenuse (AB). We can write this as: \(\sin{B}=\frac{a}{\mathrm{AB}}\)

Solve for the length of side AC

We want to find the length of side AC, or 'a'. We were given the value for angle B and the length of side BC. We can use the sine function to find 'a': \(\sin{53^{\circ}}=\frac{a}{\mathrm{AB}}\) Since \(\sin{53^{\circ}}\) is a known value, we can replace it with its decimal equivalent (approximately 0.8): \(0.8=\frac{a}{\mathrm{AB}}\) We can solve for 'a' by multiplying both sides by AB: \(a=0.8 \times \mathrm{AB}\) The relationship between the sides of a right triangle is given by the Pythagorean theorem: \(a^{2}+\mathrm{BC}^2=\mathrm{AB}^2\) We were given the length of side BC (\(12\,\mathrm{cm}\)), so we can plug that into the equation: \(a^{2}+144=\mathrm{AB}^2\) Since we found the value of 'a' in terms of AB in an earlier step, we can plug that expression into the equation: \((0.8 \times \mathrm{AB})^{2}+144=\mathrm{AB}^2\) Solve for AB: \((0.64)\mathrm{AB}^{2}+144=\mathrm{AB}^2\) \(144 = (1 - 0.64) \mathrm{AB}^2\) \(\frac{144}{0.36} = \mathrm{AB}^2\) \(\mathrm{AB} = \sqrt{400}=20\) Now that we have the value of AB, we can plug that back into the equation we found earlier for 'a': \(a=0.8 \times 20\) \(a=16\) Therefore, the length of side AC is \(16 \mathrm{~cm}\).

Key concepts

Sine Function

The sine function is an essential part of trigonometry, especially when dealing with right angle triangles. It helps us relate the angles of a triangle to its sides, allowing us to solve for unknown lengths. In a right triangle, the sine function is defined as the ratio of the length of the side opposite a given angle to the length of the triangle's hypotenuse. This means that if you know one angle and one side, you can find the other sides.

• The sine of angle B can be written as: \( \sin(B) = \frac{\text{opposite side to } B}{\text{hypotenuse}} \).
• In our given triangle, angle B is \(53^{\circ}\), the opposite side is AC, and the hypotenuse is AB.
• The known value \( \sin(53^{\circ}) \) is approximately 0.8, which is crucial for solving our exercise.

Understanding the sine function allows us to translate a part of a triangle into a workable equation. It is the backbone of finding unknown sides when angles are known, making it indispensable in geometry and real-world applications.

Right Angle Triangle

A right angle triangle is a triangle where one angle is exactly 90 degrees. This type of triangle is unique due to its predictable properties and the simplicity it brings to calculations in trigonometry. Several key characteristics define right angle triangles:

• One of the angles, known as the right angle, measures exactly \(90^{\circ}\).
• The side opposite the right angle is called the hypotenuse, and it's always the longest side.
• The other two sides are referred to as the adjacent side and the opposite side, depending on the angle in question.

In the context of our exercise, the right angle at \(C\) creates a framework allowing the use of formulas like the sine function and the Pythagorean theorem. Recognizing a right angle triangle facilitates easy solving of various mathematical problems by opening the door to useful theorems and functions.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in trigonometry concerning right angle triangles. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This means that in every right angle triangle, this relationship holds true:

• The formula is expressed as \( a^2 + b^2 = c^2 \), where \(c\) is the hypotenuse.
• In our problem, by substituting known values into the theorem, we can determine unknown side lengths.
• For \( \triangle ABC \), with \(AC = 16 \text{ cm} \), \(BC = 12 \text{ cm} \), and \(AB = 20 \text{ cm} \), it satisfies the theorem as: \(16^2 + 12^2 = 20^2\), reaffirming the correctness of our calculations.

The Pythagorean theorem not only confirms our earlier results derived from the sine function but also highlights an essential technique in verifying solutions. Its reliability and simplicity make it a go-to strategy for solving various geometric and trigonometric problems.