Problem 110 - Algebra and Trigonometry

Question

Exercises \(110-112\) will help you prepare for the material covered in the next section. Give exact values for \(\sin 30^{\circ}, \cos 30^{\circ}, \sin 60^{\circ},\) and \(\cos 60^{\circ}\)

Step-by-Step Solution

Answer

The exact values are \(\sin 30^{\circ} = \frac{1}{2}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), and \(\cos 60^{\circ} = \frac{1}{2}\)

Step 1: Find sin(30) and cos(30)

These values can be looked up in a unit circle or derived from a special right triangle with legs of 1 and \(\sqrt{3}\) and hypotenuse 2 (also known as a 30-60-90 triangle). The sin value of an angle in a right triangle is the length of the side opposite the angle divided by the length of the hypotenuse. The cos value of an angle in a right triangle is the length of the adjacent side divided by the length of the hypotenuse. Therefore, \(\sin 30^{\circ} = \frac{1}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).

Step 2: Find sin(60) and cos(60)

We use the same special right triangle as above, but now we're calculating sin and cos for the 60 degree angle. \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\). This is because the side opposite the 60 degree angle in the special triangle is \(\sqrt{3}\), and the adjacent side to the 60 degree angle is 1.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It’s a circle with a radius of 1, centered at the origin of a coordinate plane. Imagine a circle that snugly fits inside the coordinate grid, resting right at the center. This simple circle, even though it seems modest, is incredibly powerful in geometrical calculations.

• Each point on the unit circle corresponds to an angle's sine and cosine values.
• The x-coordinate of a point on the circle provides the cosine of the angle.
• The y-coordinate supplies the sine of the angle.

For example, if you want to find \(\sin 30^\circ\) and \(\cos 30^\circ\), you can look at the coordinates of the point on the unit circle that aligns with a 30-degree angle from the positive x-axis. The unit circle is especially handy because it simplifies the process of determining these values without relying upon calculators. It creates a direct visual that clarifies the relationship between the angles and their trigonometric counterparts.

Special Right Triangles

Special right triangles are right-angled triangles with certain fixed angle measures that make calculations straightforward. The two most famous types are the 45-45-90 triangle and the 30-60-90 triangle. Let's focus on the latter to see how it connects to trigonometric functions.

• In a 30-60-90 triangle, the sides are in a consistent ratio of 1 : \(\sqrt{3}\) : 2.
• The 30-degree angle is opposite the smallest side, which is half the hypotenuse.
• The 60-degree angle is opposite the longer leg, \(\sqrt{3}\) times the shortest side.

When you need to find \(\sin 30^\circ\), you take the side opposite the 30-degree angle (1) and divide by the hypotenuse (2), resulting in \(\sin 30^\circ = \frac{1}{2}\). Similarly, for \(\cos 30^\circ\), you divide the side adjacent to the 30-degree angle (\(\sqrt{3}\)) by the hypotenuse (2), obtaining \(\cos 30^\circ = \frac{\sqrt{3}}{2}\). These properties make these triangles incredibly useful for quick calculations and understanding the foundation of trigonometric functions.

Sine and Cosine Values

Sine and cosine are the backbone of trigonometry, each associated with angles in right triangles and their representations on the unit circle.

• For any given angle, sine is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine is the ratio of the adjacent side to the hypotenuse.

When you're asked to find \(\sin 60^\circ\) and \(\cos 60^\circ\), you return to the principles laid out by these functions:- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), because the opposite side is \(\sqrt{3}\) and the hypotenuse is 2.- \(\cos 60^\circ = \frac{1}{2}\), as the adjacent side is 1 while the hypotenuse remains the same (2).These values aren't just numbers; they are consistent patterns that recur throughout trigonometry and geometry. Knowing these basic values can significantly simplify more complex calculations as they are often used as building blocks in more complicated mathematical problems.