Question

In questions 1-11 $\mathrm{\Delta }\mathrm{ABC}$ has a right angle at $\mathrm{C}$. Calculate $\mathrm{AC}$ given $\mathrm{BC}=10\mathrm{cm}$ and $A={40}^{\circ }$.

Short answer

Answer: The length of the side adjacent to the 40° angle is approximately 11.92 cm.

Step-by-step solution

Identify the known values and the required side

In this problem, we know that: - Angle A = 40° - Side BC (opposite to angle A) = 10 cm - Right angle C The required side is AC, which is adjacent to angle A.

Set up the Trigonometric Function

Since we are given an angle (40°) and need to find a side adjacent to that angle, we can use the tangent function. The formula for the tangent function in a right triangle is: $\tan (A)=\frac{\text{opposite side}}{\text{adjacent side}}$ In our problem, we can replace the Angle A and the opposite side length with the given values: $\tan (40°)=\frac{10}{AC}$

Solve for AC

Now, we just need to solve for AC: $AC=\frac{10}{\tan (40°)}$ Use a calculator to find the value of $\tan (40°)$: $AC=\frac{10}{0.8391}$ Divide the numerator by the denominator: $AC\approx 11.92$

Write down the answer

The length of side AC is approximately 11.92 cm.

Key concepts

Right Triangle

A right triangle is a special kind of triangle that has one angle measuring exactly 90 degrees. This angle is known as a right angle. Right triangles are fundamental in trigonometry as they allow us to define trigonometric ratios using the lengths of the sides. They consist of three sides:

• Hypotenuse: The longest side, opposite the right angle.
• Opposite: The side opposite the angle you are focusing on.
• Adjacent: The side next to the angle you are focusing on, but not the hypotenuse.

In a right triangle, the Pythagorean theorem, trigonometric functions, and angle properties are often used to find unknown information. Understanding each side's position relative to the angles allows you to effectively use these mathematical tools.

Tangent Function

The tangent function is one of the primary trigonometric functions and is crucial in solving problems involving right triangles. Defined for a given angle in a right triangle, tangent compares the lengths of the sides:

• Tangent of an angle ($an(heta)$) is the ratio of the length of the opposite side to the adjacent side.

Using tangent, you can find either an angle or a side if the other two corresponding elements are known. For example, if the opposite side and angle are known, you can easily calculate the adjacent side. This makes it a handy tool for solving a wide range of geometric problems and is fundamental to understanding trigonometry.

Triangle Calculations

Calculating missing sides or angles in triangles is a common task in trigonometry. For right triangles, three main steps are generally involved:

• Identify: Determine the known sides and angles along with what needs to be found. Decide which trigonometric function applies.
• Set Up: Write down the proper trigonometric ratio. Substitute known values into the equation to form a mathematical statement.
• Solve: Rearrange and simplify the equation to solve for the unknown. Use a calculator for precise calculations of trigonometric values where necessary.

These steps help structure your approach and ensure accurate results. Knowing which trigonometric rule or theorem to apply is key.

Angle Measurements

Angles in triangles, including right triangles, are usually measured in degrees. The sum of all angles in any triangle is always 180 degrees. In a right triangle, this means:

• One angle is 90 degrees.
• The other two angles must add up to 90 degrees (since 180 - 90 = 90).

Measuring angles accurately is essential for calculating sides or other angles in trigonometric functions. Tools such as protractors, or software, provide the means to measure angles. In trigonometry, angles help set up trigonometric functions, allowing for calculations related to the triangle's dimensions.