Question

In a right-angled triangle, the hypotenuse is 4 -times as long as the perpendicular from the opposite vertex. One of the acute angles of the triangle is (a) \(45^{\circ}\) (b) \(30^{\circ}\) (c) \(15^{\circ}\) (d) \(22.5^{\circ}\)

Short answer

(a) \(45^{\circ}\) (b) \(30^{\circ}\) (c) \(15^{\circ}\) (d) \(22.5^{\circ}\) Answer: (c) \(15^{\circ}\)

Step-by-step solution

Define the variables

Let's denote the sides of the triangle as follows: - Hypotenuse (H) = 4P (since it is 4 times the perpendicular) - Perpendicular (P) = P - Base (B)

Use the given information

We know that in a right-angled triangle, the relationship between the sides and angles can be described using trigonometric ratios. The fundamental ratios are sine, cosine, and tangent. Since we are given information about the hypotenuse and perpendicular, we can use the sine function. The sine of an angle in a right-angled triangle is defined as the ratio of the opposite side (perpendicular) to the hypotenuse. Let's denote the acute angle we want to find as 'θ'. Using the sine function, we can write: \(\sin{\theta} = \frac{P}{4P}\)

Simplify the sine function

Simplify the sine function: \(\sin{\theta} = \frac{P}{4P} = \frac{1}{4}\)

Find the acute angle

Now, we need to find the angle 'θ' such that the sine of the angle is 1/4. To do this, we can use the inverse sine function. \(\theta = \sin^{-1}{\frac{1}{4}}\) Calculate the angle: \(\theta \approx 14.48^{\circ}\)

Identify the correct option

Now that we have the approximate value of the angle, let's check which option from the given choices is closest to the calculated value: (a) \(45^{\circ}\) (b) \(30^{\circ}\) (c) \(15^{\circ}\) (d) \(22.5^{\circ}\) The correct answer is: (c) \(15^{\circ}\), as it is closest to the calculated angle value of \(14.48^{\circ}\).

Key concepts

Trigonometric Ratios

Understanding trigonometric ratios is crucial when solving right-angled triangle problems. In such triangles, these ratios provide a direct link between the angles and the lengths of the sides.

Sine, cosine, and tangent are the primary trigonometric ratios. For a given acute angle in a right-angled triangle:

• The sine (sin) of the angle is the ratio between the length of the opposite side and the hypotenuse.
• The cosine (cos) of the angle is the ratio of the adjacent side to the hypotenuse.
• The tangent (tan) of the angle is the ratio of the opposite side to the adjacent side.

It's essential to know these relationships as they are frequently used to find missing angles or side lengths in right-angled triangles. For example, if a problem provides the ratio of the hypotenuse to a leg, as seen in the step-by-step solution provided, the sine ratio is the most suitable to use. By simplifying the ratio and using the known values of the sides, one can determine the angle using the inverse of the ratio.

Inverse Sine Function

When we know the value of the sine of an angle but need to find the angle itself, we turn to the inverse sine function, denoted as \(\sin^{-1}\) or arcsin. This function does the reverse of the sine function: given the value of the sine, it yields the measure of the angle in degrees or radians.

In the provided solution, once the sine of the unknown angle \(\theta\) was found to be \(\frac{1}{4}\), the inverse sine function was employed to calculate the measure of that angle. This calculation can often be executed on a scientific calculator or using mathematical software.

Limits of Inverse Sine Function

It's important to note that the output of \(\sin^{-1}\) is limited to angles between \( -90^\circ \) and \( 90^\circ \), or \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) in radians. Because of this, when working with \(\sin^{-1}\), one can expect the angle to be an acute angle, as it falls within the first quadrant of the Cartesian plane.

Acute Angles in Triangles

In a right-angled triangle, the two angles other than the right angle are always acute, meaning each is less than \(90^\circ\). These angles are of particular interest since they are the ones associated with the trigonometric ratios when it comes to calculations.

It's these acute angles that we often need to find or evaluate in problems involving right-angled triangles. The step-by-step solution exemplifies this, as it required finding the measure of an acute angle when given the ratio of two sides of the triangle.

Characteristics of Acute Angles

One characteristic of acute angles to remember is that they never exceed \(90^\circ\). Therefore, when using trigonometric functions or their inverses to solve for angles, solutions greater than \(90^\circ\) are not valid for acute angles in right-angled triangles. Additionally, knowing one acute angle allows us to easily determine the other, as they must sum to \(90^\circ\) due to the third angle being a right angle. This fundamental understanding of acute angles in right-angled triangles is a powerful tool in the arsenal of anyone dealing with geometric calculations and proofs.