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. \(\cos 30^{\circ}\)

Short answer

\(\cos 30^{\circ} = 0.8660\)

Step-by-step solution

Identify the Triangle

The given angle is \(30^\circ\), which is part of the special right triangle known as the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle. This triangle has sides in a specific ratio, which we should recall to solve the problem.

Recall the Side Ratios

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the sides are in the ratio 1: \sqrt{3}\: 2, where the side opposite \(30^\circ\) is the shortest, \((1)\), the side opposite \(60^\circ\) is \((\sqrt{3})\), and the hypotenuse is 2.

Define Cosine for the Angle

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For \(\cos 30^\circ\), the adjacent side is opposite the \(60^\circ\), which is \sqrt{3}\, and the hypotenuse is 2.

Calculate the Cosine Value

Using the ratio from Step 3, we calculate: \(\cos 30^\circ = \frac{\text{Adjacent side opposite } 60^\circ}{\text{Hypotenuse}} = \frac{\sqrt{3}}{2}\).

Round the Result

Since \(\sqrt{3} = 1.7321\) (when rounded to four decimal places), the cosine value becomes \(\cos 30^\circ = \frac{1.7321}{2} = 0.8660\) when rounded to four decimal places.

Key concepts

Special Right Triangles

Special right triangles are triangles with angles and side lengths that follow specific ratios, making calculations in trigonometry much easier. There are two main types of special right triangles: the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle and the \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle.

• 30-60-90 Triangle: This triangle has angles of \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). The sides are in the ratio of \(1:\sqrt{3}:2\). The shortest side is opposite the \(30^{\circ}\) angle, the medium side is opposite the \(60^{\circ}\) angle, and the hypotenuse is always the longest.
• 45-45-90 Triangle: Also known as an isosceles right triangle, it has two \(45^{\circ}\) angles. The sides are in the ratio \(1:1:\sqrt{2}\), meaning the legs are equal, and the hypotenuse is \(\sqrt{2}\) times the length of a leg.

These unique properties make solving trigonometric functions more straightforward without needing to reference a calculator or table constantly.

Cosine Function

The cosine function, one of the primary trigonometric functions, helps us relate the angles in a right triangle to the lengths of its sides. Cosine is abbreviated as \(\cos\), and it is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

• For a given angle \(\theta\), the formula is: \(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\).
• The cosine function is particularly useful in trigonometry for solving problems involving angles and distances.
• In the context of special right triangles, this function allows us to derive exact values for angles at \(30^{\circ}, 45^{\circ},\) and \(60^{\circ}\).

By grasping this function, students can explore angles' characteristics and understand their relationships without relying on approximation.

30-60-90 Triangle

The \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is a special right triangle, offering a convenient pattern for calculations. Each angle dictates a specific relationship with the sides, and once understood, this simplifies problems involving these triangles extensively.

• The sides of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle are always in a ratio of \(1 : \sqrt{3} : 2\).
• For example, if the shortest side (opposite the \(30^{\circ}\) angle) is \(1\), then the side opposite \(60^{\circ}\) will be \(\sqrt{3}\), and the hypotenuse will be \(2\).
• Given this setup, the calculations for trigonometric functions like cosine are straightforward and precise.

Understanding this triangle allows students to derive values confidently, as seen in the calculation of \(\cos 30^{\circ} = \frac{\sqrt{3}}{2} = 0.8660\), by simply using the side ratio, without complex computations.