Problem 84 - Algebra and Trigonometry

Question

Describe the triangle used to find the trigonometric functions of \(30^{\circ}\) and \(60^{\circ}\)

Step-by-Step Solution

Answer

\(\sin(30^{\circ}) = \frac{1}{2}\), \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\), \(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\), \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\), \(\cos(60^{\circ}) = \frac{1}{2}\), \(\tan(60^{\circ}) = \sqrt{3}\)

Step 1: Create a 30-60-90 Triangle

First, draw a right triangle (a triangle where one angle measures \(90^{\circ}\)). Label one of the angles as \(30^{\circ}\) and the other as \(60^{\circ}\). The side opposite the \(30^{\circ}\) angle is half the length of the hypotenuse, and the side opposite the \(60^{\circ}\) angle is equal to the hypotenuse times the square root of 3, divided by 2.

Step 2: Find the Trigonometric Functions for \(30^{\circ}\)

For \(30^{\circ}\):\n\n1. Sine (\(\sin\)) is calculated by the length of the opposite side divided by the length of the hypotenuse. Therefore, \(\sin(30^{\circ}) = \frac{1}{2}\).\n\n2. Cosine (\(\cos\)) is calculated by the length of the adjacent side divided by the length of the hypotenuse. Therefore, \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\).\n\n3. Tangent (\(\tan\)) is calculated by the length of the opposite side divided by the length of the adjacent side. Therefore, \(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\).

Step 3: Find the Trigonometric Functions for \(60^{\circ}\)

For \(60^{\circ}\):\n\n1. Sine (\(\sin\)) is calculated by the length of the opposite side divided by the length of the hypotenuse. Therefore, \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\).\n\n2. Cosine (\(\cos\)) is calculated by the length of the adjacent side divided by the length of the hypotenuse. Therefore, \(\cos(60^{\circ}) = \frac{1}{2}\).\n\n3. Tangent (\(\tan\)) is calculated by the length of the opposite side divided by the length of the adjacent side. Therefore, \(\tan(60^{\circ}) = \sqrt{3}\).