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. \(\tan 45^{\circ}\)

Short answer

\(\tan 45^{\circ} = 1\)

Step-by-step solution

Identify the Right Triangle

The given angle, \(45^{\circ}\), is part of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle. This is an isosceles right triangle where the two non-hypotenuse sides are equal.

Determine Side Ratios

In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the ratio of the lengths of the sides opposite the \(45^{\circ}\) angles to the hypotenuse is \(1:1:\sqrt{2}\). This means both legs are equal.

Define Tangent Ratio

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. For \(\tan 45^{\circ}\), both the opposite and adjacent sides are equal.

Calculate \(\tan 45^{\circ}\)

Using the tangent ratio definition, \(\tan 45^{\circ} = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{1}{1} = 1\).

Key concepts

Special Triangles

Special triangles, such as the \(30^{\circ}-60^{\circ}-90^{\circ}\) and \(45^{\circ}-45^{\circ}-90^{\circ}\) triangles, are fundamental in trigonometry for simplifying the computation of trigonometric ratios. These triangles have easily recognizable side length ratios, which help in determining the trigonometric values of their angles without using a calculator.

• In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, it's an isosceles right triangle where the sides opposite the \(45^{\circ}\) angles are equal, and the length of the hypotenuse is \(\sqrt{2}\times\) the length of each leg. This is often remembered as the ratio \(1:1:\sqrt{2}\).
• The \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has side lengths in the ratio \(1:\sqrt{3}:2\), where the shortest side is opposite the \(30^{\circ}\) angle, and the longest side (hypotenuse) is opposite the right angle.

These ratios are key to solving problems related to these angles quickly. Recognizing these triangles helps us approximate trig values and prove properties of different geometric figures.

Tangent Function

The tangent function is one of the basic trigonometric functions, usually abbreviated as "tan." For any given angle in a right triangle, the tangent is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.
For instance, if \(\theta\) is the angle, then \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).
This function is particularly useful when you need to solve for angles or sides in right triangles when only partial information is available.

• In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, both the sides opposite and adjacent to the \(45^{\circ}\) angle are of equal length, resulting in \(\tan 45^{\circ} = \frac{1}{1} = 1\).
• This consistency makes it easy to remember that in any isosceles right triangle, the tangent of a \(45^{\circ}\) angle is always 1.

Knowing how to use the tangent function effectively allows for solving various geometry and surveying problems where direct measurements are inconvenient.

Right Triangles

Right triangles are triangles that have one 90-degree angle, known as a right angle. In trigonometry, these triangles are essential because they serve as the basis for defining the trigonometric functions-sine, cosine, and tangent.
The unique properties of right triangles simplify many geometric and trigonometric calculations. They consist of one hypotenuse, which is the longest side, and two legs.

• The legs are the two shorter sides, and one of them is called the opposite side for a chosen reference angle, and the other is the adjacent side.
• For instance, in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, both legs are equal, and the hypotenuse is \(\sqrt{2}\times\) the length of a leg. This perfect symmetry simplifies calculations, such as finding the tangent, because for a \(45^{\circ}\) angle, the ratio of opposite to adjacent sides is 1.

Understanding right triangles allows you to leverage their properties to solve practical and theoretical problems in mathematics, physics, and engineering.