Problem 101 - Algebra and Trigonometry

Question

Give exact values for \(\cos 30^{\circ}, \sin 30^{\circ}, \cos 60^{\circ}, \sin 60^{\circ}, \cos 90^{\circ}\) and \(\sin 90^{\circ}\).

Step-by-Step Solution

Answer

\(\cos 30^{\circ}=\sqrt{3}/2, \sin 30^{\circ}=1/2, \cos 60^{\circ}=1/2, \sin 60^{\circ}=\sqrt{3}/2, \cos 90^{\circ}=0, and \sin90^{\circ}=1\).

Step 1: Apply the unit circle

The Unit Circle is a common tool in trigonometry and it is a circle with its center at the origin (0,0) of the coordinate plane and its radius is one unit. The trigonometric functions sine and cosine can be determined by considering a right triangle formed in the unit circle. For a point on the unit circle at an angle θ counterclockwise from (1,0), the x-coordinate is \(\cos\theta\) and the y-coordinate is \(\sin\theta\).

Step 2: Obtain the cosine values

Now, let's obtain the cosine values for each given angle.For \(\cos 30^{\circ}\), picture an angle of 30 degrees in the unit circle. The x-coordinate (cosine value) of this point is \(\sqrt{3}/2\).\(\cos 60^{\circ}\) would be \(1/2\) and \(\cos 90^{\circ}\), since this is on the vertical axis, the x-coordinate is 0. Thus, \(\cos 90^{\circ}=0\).

Step 3: Obtain the sine values

Now let's find the sine values for each given angle.\(\sin 30^{\circ}\) would be \(1/2\) and \(\sin 60^{\circ}\), the y-coordinate of this point is \(\sqrt{3}/2\). Finally \(\sin 90^{\circ}\), since it's on the vertical axis, the y-coordinate is 1, thus \(\sin 90^{\circ}=1\).

Key Concepts

Cosine and Sine Values

The values of cosine and sine for specific angles are essential in the study of trigonometry. These values stem from the unit circle, which is a circle with a radius of one unit centered at the origin of a coordinate plane.

When we look at the unit circle, every point on the circle represents an angle formed by drawing a line from the origin to that point. The x-coordinate of this point gives us the cosine value of the angle, and the y-coordinate gives us the sine value. For example,

\begin{itemize}\item \( \cos 30^\circ = \sqrt{3}/2 \), because at 30 degrees the point on the unit circle has an x-coordinate of \(\sqrt{3}/2\).\item \( \sin 30^\circ = 1/2 \), as the point's y-coordinate is 1/2.\item \( \cos 60^\circ = 1/2 \), matching the x-coordinate at 60 degrees.\item \( \sin 60^\circ = \sqrt{3}/2 \), reflected by the y-coordinate.\item \( \cos 90^\circ = 0 \), since at 90 degrees, we are directly above the origin on the y-axis.\item \( \sin 90^\circ = 1 \), as the point is at the top of the circle with a y-coordinate of 1.\end{itemize}
Remembering that the unit circle's radius is always one, we can understand why these sine and cosine values are simply the coordinates of the points where the terminal side of the angle intersects the circle.

Trigonometric Functions

Trigonometric functions are a key component of mathematics, and they help us understand relationships within right-angled triangles as well as circular motion.

The most fundamental trigonometric functions are sine (\(\sin\)) and cosine (\(\cos\)). As we saw when calculating sine and cosine values, these functions correspond to the y and x coordinates, respectively, of a point on the unit circle. But why does this matter? Because trigonometric functions provide a way to describe angles and distances that is consistent regardless of the size of the triangle.

This makes them incredibly useful for modeling physical phenomena such as sound waves, light waves, and many types of motion. A deeper understanding of trigonometric functions also extends into more complex functions like tangent, cotangent, secant, and cosecant, which all have unique properties and applications in both geometry and real-life scenarios.

Right Triangle in Unit Circle

A right triangle inscribed in the unit circle is a powerful visual tool for understanding trigonometry. By placing a right triangle within the unit circle—such that one vertex is at the origin, one is on the circumference, and one is perpendicular on the x-axis—we can illustrate trigonometric principles.

The hypotenuse of this right triangle is always equal to the radius of the circle, which is one. Therefore, the lengths of the other two sides of the triangle can be represented by the sine and cosine values of the angle at the origin. For example, the side opposite the angle (the height of the triangle) corresponds to the sine value, while the side adjacent to the angle (the base of the triangle) corresponds to the cosine value.

Because the hypotenuse is always one, the cosine and sine values for any angle will also be the lengths of the legs of a right triangle inscribed in the unit circle. This relationship is invaluable not just for solving trigonometric problems but also for understanding the geometry behind these functions and how they relate to the circle and angles within it.