Question

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(A\) given \(\mathrm{AC}=14 \mathrm{~cm}\) and \(\mathrm{BC}=9 \mathrm{~cm}\)

Short answer

Answer: The measure of angle A in the right-angled triangle ΔABC is approximately 32.73°.

Step-by-step solution

Identify the given measurements

We are given the measurements of two sides in a right-angled triangle: - AC = 14 cm (adjacent side) - BC = 9 cm (opposite side)

Use the inverse tangent function

The inverse tangent (arctan) function is used to find the angle A, using the formula: A = arctan(opposite side / adjacent side) In our case, we have: A = arctan(BC / AC)

Plug the values and calculate angle A

Now, we can substitute the given side measurements into the formula: A = arctan(9 cm / 14 cm) Calculate the angle using a calculator: A ≈ 32.73°

Write the final answer

The measure of angle A in the right-angled triangle ΔABC is approximately 32.73°.

Key concepts

Inverse Tangent

Understanding the inverse tangent function, also known as arctan, is crucial when dealing with right-angled triangles. This function helps us find an angle when we know the lengths of the opposite and adjacent sides.

The basic formula for using inverse tangent is:

• A = \( \text{arctan} \left( \frac{\text{opposite side}}{\text{adjacent side}} \right) \)

The inverse tangent function applies to a right-angled triangle where you know two sides: one opposite to the angle you are finding and one adjacent to it.

Calculators usually have dedicated buttons for arctan or you access it through the tan or trig function menu. Simply input the ratio of the two sides and the calculator will output the angle's measure. This is especially useful in fields like engineering and architecture where precise angles are essential. Remember that the result is often given in degrees, so ensure your calculator is set to the correct mode.

Right-Angled Triangle

A right-angled triangle is a triangle where one of its angles is exactly 90 degrees. It's one of the most fundamental shapes in trigonometry and geometry.

Here are some key features of right-angled triangles:

• The three sides are referred to as the opposite, adjacent, and hypotenuse.
• The hypotenuse is the longest side and is opposite the right angle.
• The other two sides form a perpendicular intersection at the right angle.

The properties of right-angled triangles make them ideal for applying trigonometric functions, such as sine, cosine, and tangent. These functions relate the angles of the triangle to the ratios of its sides, allowing us to calculate unknown angles or lengths with ease.

Knowing which side corresponds to which term is crucial for correctly using trigonometric formulas. For instance, in our scenario, AC is adjacent to angle A, and BC is the side opposite angle A.

Angle Calculation

Calculating angles in right-angled triangles is a practical application of trigonometry that can aid in many real-world scenarios. Here's a simple guide on how to approach these calculations:

• First, determine which sides you have measured. Here, we have an opposite side and an adjacent side.
• Next, identify the correct trigonometric function to use. In this instance, it's the tangent function since we're using opposite and adjacent sides.

Using the formula \( \text{arctan} \left( \frac{\text{opposite}}{\text{adjacent}} \right) \), plug in your known measurements to find the angle. In our example, plugging in BC = 9 cm and AC = 14 cm gives us \( A = \text{arctan} \left( \frac{9}{14} \right) \).

Finally, use a calculator to solve for A. Ensure the calculator is set to degree mode for clearer real-world application results. Once calculated, you'll have the measure of angle A. Always double-check your work for accuracy, especially when practical applications depend on these calculations.