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Chapter 4: Problem 63

Describe the triangle used to find the trigonometric functions of \(45^{\circ}\).

Short Answer

Expert verified
The triangle is an isosceles right-angled triangle with legs of length 1. The hypotenuse, calculated using the Pythagorean theorem, has length \( \sqrt{2} \). The trigonometric functions are therefore: \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \) and \( \tan 45^{\circ} = 1\).

Step by step solution

01

Constructing the triangle

Start by drawing a right-angled triangle where the other two angles are both \(45^{\circ}\). This formation makes it an isosceles triangle, so the legs are of equal length. For simplicity, assign the legs a length of 1 each.
02

Calculating the hypotenuse

Use the Pythagorean theorem to find the length of the hypotenuse, which is the side opposite the right angle. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. The length of the hypotenuse will thus be \( \sqrt{1^2 + 1^2} = \sqrt{2} \).
03

Finding the trigonometrical functions

Now that we have a right-angled isosceles triangle, we can find the trigonometric functions for the \(45^{\circ}\) angle. The sine of an angle \(A\) in a right-angled triangle is given by the ratio of the length of the side opposite angle \(A\) and the length of the hypotenuse. Thus, \( \sin 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\). Because the triangle is isosceles, the cosine of the angle will have the same value as the sine, \( \cos 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\). The tangent is found by the ratio of the sine and the cosine, \( \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-Angled Triangle
A right-angled triangle is a triangle that has one of its angles equal to 90 degrees. It's an essential shape in trigonometry and geometry.
The properties of a right-angled triangle are simple yet powerful in mathematical applications like trigonometric functions.
Here are some key points to know:
  • One angle is always 90 degrees, forming what we call the right angle.
  • The side opposite the right angle is known as the hypotenuse and it's always the longest side in the triangle.
  • The other two sides are known as the 'legs' of the triangle.
In trigonometric calculations, we often use the properties of right-angled triangles to determine various functions such as sine, cosine, and tangent, which describe relationships between the angles and sides of the triangle.
Isosceles Triangle
An isosceles triangle is a triangle that has two sides of equal length. This property lends simplicity and symmetry to many trigonometric problems.
In the context of a right-angled isosceles triangle, like the one used to find trigonometric functions at 45 degrees, both legs are equal.
  • When the two non-right angles are equal, they are often both 45 degrees when dealing with a right-angle.
  • The two equal sides convene at the right-angle corner, providing balance to the triangle's structure.
Creating this balance introduces the opportunity to measure angles with respect to their sides accurately, which makes the isosceles triangle a favorite tool for finding values of trigonometric functions at specific degrees.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry and trigonometry that applies to right-angled triangles. It provides a way to relate the lengths of the sides in these triangles.
The theorem is naturally expressed as: \[ a^2 + b^2 = c^2 \]Here, \(a\) and \(b\) represent the lengths of the legs in the right-angled triangle, and \(c\) represents the length of the hypotenuse.
  • This theorem confirms that the hypotenuse is always the longest side in a right-angled triangle.
  • It's utilized to determine the unknown length when two sides are known, which is essential in calculating trigonometric ratios.
Through this theorem, precise measurements of the sides can be obtained, making it pivotal for calculating sine, cosine, and tangent values precisely by using ratios of a right-angled triangle.

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