Question

Use the \(30^{\circ}-60^{\circ}-90^{\circ}\) and \(45^{\circ}-45^{\circ}-90^{\circ}\) triangles to find each value. Round to four decimal places, if necessary. \(\cos 45^{\circ}\)

Short answer

\(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step-by-step solution

Identify the Triangle Type

This problem deals with a special triangle: the \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle. This is also known as an isosceles right triangle, where the two legs are of the same length, and the angles are \(45^{\circ}, 45^{\circ}, \text{and } 90^{\circ}\).

Understand the Triangle Properties

In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, if each leg is \(x\), then the hypotenuse is \(x\sqrt{2}\). This special property allows us to calculate trigonometric functions easily.

Calculate Cosine

By definition, \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\). If we use \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle properties, and let the leg (adjacent side) be \(x\), the hypotenuse is \(x\sqrt{2}\). Therefore, \(\cos 45^{\circ} = \frac{x}{x\sqrt{2}}\).

Simplify the Expression

Simplifying \(\frac{x}{x\sqrt{2}}\), we can cancel out the \(x\) from the numerator and denominator, resulting in \(\frac{1}{\sqrt{2}}\). Multiplying and dividing by \(\sqrt{2}\) to rationalize the expression gives \(\frac{\sqrt{2}}{2}\).

Final Answer

Thus, \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\).

Key concepts

Special Right Triangles

Special right triangles are often used in trigonometry for simplifying complex problems, thanks to their predictable relationships between angles and side lengths. There are two primary types of special right triangles: the \(30^{\circ}-60^{\circ}-90^{\circ}\) and the \(45^{\circ}-45^{\circ}-90^{\circ}\).

The \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, also known as the isosceles right triangle, has two equal angles of \(45^{\circ}\) and a right angle of \(90^{\circ}\). Both legs of the triangle are of equal length, and the length of the hypotenuse is \(x\sqrt{2}\) if each leg is \(x\).

On the other hand, the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has side relationships governed by the angles. The shortest side is opposite the \(30^{\circ}\) angle, the longer leg is \(x\sqrt{3}\), and the hypotenuse is twice the length of the shortest side, or \(2x\).

• \(45^{\circ}-45^{\circ}-90^{\circ}\): Equal legs, hypotenuse \(x\sqrt{2}\).
• \(30^{\circ}-60^{\circ}-90^{\circ}\): Relationships between short, long sides, and hypotenuse.

Understanding these relationships can simplify the calculation of trigonometric functions for these angles.

Cosine Function

The cosine function is an essential part of trigonometry, expressing the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Mathematically, it is defined as \(\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\).

In the context of the \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle:

• The adjacent side is one of the legs, say \(x\).
• The hypotenuse is \(x\sqrt{2}\).

Therefore, the cosine of \(45^{\circ}\) becomes \(\frac{x}{x\sqrt{2}}\), which can be simplified due to common factors to \(\frac{1}{\sqrt{2}}\). To express the cosine value in a standardized form, we rationalize the denominator, leading to \(\frac{\sqrt{2}}{2}\).

Cosine values for special angles like \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \text{and } 90^{\circ}\) are frequently used:

• \(\cos 0^{\circ} = 1\)
• \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
• \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\cos 60^{\circ} = \frac{1}{2}\)
• \(\cos 90^{\circ} = 0\)

Knowing these values helps in solving many trigonometric problems quickly.

Triangle Properties

A triangle’s properties are crucial to understanding trigonometry, particularly in calculating angles and sides using functions like sine, cosine, and tangent. Key properties that define triangles include the sum of angles, specific side relationships in right triangles, and the characteristics of special right triangles.

• In any triangle, the sum of the internal angles is always \(180^{\circ}\).
• For right triangles, one angle is always \(90^{\circ}\), leaving the sum of the other two angles to be \(90^{\circ}\).
• Special right triangles use geometric properties to maintain specific side length ratios, like \(1:1:\sqrt{2}\) in \(45^{\circ}-45^{\circ}-90^{\circ}\) triangles.

Understanding the relationship between angles and sides in these triangles is foundational for using trigonometric functions. For the \(45^{\circ}-45^{\circ}-90^{\circ}\):

• Both legs are equal, forming the basis for the unique side ratios.
• Knowing one side allows derivation of the others using the special ratio properties of the triangle.

These properties simplify the computation of trigonometric values and the application of functions like cosine, sine, and tangent to solve practical problems involving angles and distances.