Select The Correct Answer.A Right Angle Triangle Where ABC Angle Is 90 And Angle ACB Is 45.Which Trigonometric

Select The Correct Answer. A Right Angle Triangle Where ABC Angle Is 90 And Angle ACB Is 45. Which Trigonometric

Understanding the fundamentals of right-angled triangles is essential for mastering trigonometry, a branch of mathematics focused on the relationships between the angles and sides of triangles. In this article, we will analyze a specific right-angled triangle where angle ABC is 90 degrees, and angle ACB is 45 degrees. Our goal is to determine the correct trigonometric ratios associated with this triangle and explore their applications.

Understanding the Triangle Configuration

Before delving into trigonometric ratios, let's clarify the triangle's structure and the significance of the given angles.

Triangle ABC Description

• Vertices: The triangle is labeled as ABC, with points A, B, and C.
• Angles:
• Angle ABC: 90 degrees (a right angle at point B).
• Angle ACB: 45 degrees.
• Sides:
• The side opposite angle A is BC.
• The side opposite angle B is AC.
• The side opposite angle C is AB.

Implications of the Given Angles

• Since angle ABC is 90°, triangle ABC is a right-angled triangle with the right angle at B.
• Angle ACB is 45°, which indicates that the remaining angle, at point A, can be calculated as 45°, making the triangle an isosceles right triangle.

Calculating the third angle:

• Total sum of angles in a triangle = 180°
• Given:
• ∠ABC = 90°
• ∠ACB = 45°
• Therefore,
• ∠CAB = 180° - 90° - 45° = 45°

This reveals that triangle ABC is a 45°-45°-90° right triangle, which has specific properties and ratios.

Properties of a 45°-45°-90° Triangle

The 45°-45°-90° triangle holds a special place in trigonometry due to its unique ratios.

Key Characteristics

• Isosceles right triangle: Two angles are equal (both 45°), so the sides opposite these are equal.
• Side ratios:
• The legs (the sides opposite the 45° angles) are equal in length.
• The hypotenuse (the side opposite the right angle) is √2 times the length of each leg.

Expressed mathematically:

• If each leg = \( x \),
• Then hypotenuse = \( x\sqrt{2} \).

This ratio simplifies many trigonometric calculations and makes the triangle a perfect model for understanding sine, cosine, and tangent functions at 45° angles.

Determining the Trigonometric Ratios

Given the properties, the key trigonometric functions for the angles in this triangle can be deduced.

Calculating Sine, Cosine, and Tangent for 45°

Since the triangle is isosceles with angles of 45°, the trigonometric ratios are well-known:

1. Sine (sin 45°):

• Definition: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)


• For \( \theta = 45° \):


• Opposite side = \( x \)


• Hypotenuse = \( x\sqrt{2} \)


• Therefore,

\[
\sin 45° = \frac{x}{x\sqrt{2}} = \frac{1}{\sqrt{2}} \approx 0.7071
\]

1. Cosine (cos 45°):

• Definition: \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)


• Similarly,

\[
\cos 45° = \frac{x}{x\sqrt{2}} = \frac{1}{\sqrt{2}} \approx 0.7071
\]

1. Tangent (tan 45°):

• Definition: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)


• Since both legs are equal,

\[
\tan 45° = \frac{x}{x} = 1
\]

Summary of key ratios:

| Trigonometric Function | Value | Explanation |
|-------------------------|-------------------|---------------------------------------------------|
| \( \sin 45° \) | \( \frac{1}{\sqrt{2}} \) | Opposite over hypotenuse |
| \( \cos 45° \) | \( \frac{1}{\sqrt{2}} \) | Adjacent over hypotenuse |
| \( \tan 45° \) | 1 | Opposite over adjacent |

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Application of Trigonometric Ratios in the Triangle

Understanding these ratios allows us to solve various problems involving right-angled triangles.

Calculating Side Lengths

Suppose you know the length of one side, you can determine the others:

• If legs (AB and AC) are of length \( x \):


• Hypotenuse (BC): \( x\sqrt{2} \)


• Conversely, if the hypotenuse is \( h \):


• Legs: \( \frac{h}{\sqrt{2}} \)

Example:

• If \( AB = AC = 7 \) units,


• Hypotenuse \( BC = 7 \times \sqrt{2} \approx 9.899 \) units.

Solving for Angles with Trigonometric Ratios

Given side lengths, you can verify the angles:

• For example, if \( AB \) and \( AC \) are known:


• \( \sin 45° = \frac{AB}{BC} \)


• If \( AB = 5 \), and \( BC \) is known:


• \( \sin 45° = \frac{5}{BC} \)


• Solving for \( BC \):

\[
BC = \frac{5}{\sin 45°} = \frac{5}{1/\sqrt{2}} = 5 \times \sqrt{2} \approx 7.07
\]

This demonstrates the utility of trigonometric ratios in practical calculations.

Real-World Applications of 45°-45°-90° Triangles and Trigonometry

The principles derived from this special triangle are widely applicable across various fields.

Engineering and Architecture

• Designing structures with specific angles.
• Calculating load distributions in frameworks.
• Ensuring precise measurements in construction.

Navigation and Geolocation

• Determining distances and angles using triangulation.
• GPS technology relies on trigonometric calculations involving angles similar to 45°.

Physics and Signal Processing

• Analyzing waveforms and oscillations.
• Calculating components of vectors in physics problems.

Mathematics Education

• Teaching fundamental concepts of angles and ratios.
• Building a foundation for understanding complex trigonometric functions.

Summary and Key Takeaways

• Triangle ABC with angles 90°, 45°, and 45° is an isosceles right triangle.
• The sides obey the ratio: legs are equal, hypotenuse is \( \sqrt{2} \) times the legs.
• The primary trigonometric ratios are:
• \( \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} \),
• \( \tan 45° = 1 \).
• These ratios simplify calculations and have numerous practical applications across various disciplines.

Understanding the properties and ratios of special triangles like the 45°-45°-90° triangle is fundamental in mastering trigonometry. Recognizing these patterns enhances problem-solving skills and provides a solid foundation for more advanced mathematical concepts.

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In conclusion, when faced with a right-angled triangle where one of the non-right angles is 45°, the trigonometric functions associated with 45° are directly applicable. The symmetry and ratios inherent in the 45°-45°-90° triangle make it a powerful tool for various mathematical and real-world applications, emphasizing the importance of mastering these core concepts.