Problem 17 - Precalculus: A Concise Course

Question

Evaluate (if possible) the sine, cosine, and tangent of the real number. $$ t=\frac{\pi}{4} $$

Step-by-Step Solution

Answer

The sine of \(\pi/4\) is \(\sqrt{2}/2\), the cosine of \(\pi/4\) is \(\sqrt{2}/2\), and the tangent of \(\pi/4\) is 1.

Step 1 - Evaluate the Sine Value

Start by finding the sine value. From the unit circle, it can be seen that the sine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\). So, \(sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\).

Step 2 - Evaluate the Cosine Value

Next, find the cosine value. By definition, the cosine value of \(\frac{\pi}{4}\) is also \(\frac{\sqrt{2}}{2}\). So, \(cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\).

Step 3 - Evaluate the Tangent Value

Finally, the tangent of a number is defined as the ratio of its sine value to its cosine value. As \(sin(\frac{\pi}{4}) = cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\), the tangent of \(\frac{\pi}{4}\) equals one. So, \(tan(\frac{\pi}{4}) = 1\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 unit, centered at the origin (0,0) in the coordinate plane. Any point on the unit circle corresponds to a right triangle whose hypotenuse is 1. This makes it incredibly useful for evaluating trigonometric functions at various angles.

Because the radius is 1, any coordinate on the unit circle (A, B) is the cosine value (A) for the angle at that point and the sine value (B) for the same angle. Furthermore, the angle formed by the line from the origin to the point and the positive x-axis is directly related to various trigonometric functions.

Sine Value

The sine value of an angle in trigonometry is a measurement of the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. In terms of the unit circle, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the circle.

For an angle of \(\frac{\pi}{4}\), as demonstrated in our example, the unit circle shows this point to be at (\(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{2}}{2}\)). Therefore, the sine value is \(\frac{\sqrt{2}}{2}\). The significance of sine can be extended to various applications, such as determining the height of an object using trigonometry.

Cosine Value

Cosine is another trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse of a right-angled triangle. For angles plotted on the unit circle, the cosine value is the x-coordinate of the point where the angle's terminal side intersects the circle.

In the example of the angle \(\frac{\pi}{4}\), the cosine value is the same as the sine value because of the angle's quarter-circle symmetry, resulting in the x and y-coordinates being equal. Thus, the cosine value here is also \(\frac{\sqrt{2}}{2}\), indicative of how sine and cosine are closely related and can often appear as mirrored functions in the first quadrant of the unit circle.

Tangent Ratio

The tangent of an angle in trigonometry is a ratio of the sine value to the cosine value of that angle. In simpler terms, it compares the y-coordinate to the x-coordinate of a point on the unit circle. The tangent ratio is particularly useful because it can define slopes of lines and angles of elevation or depression.

For the angle \(\frac{\pi}{4}\), as both sine and cosine values are \(\frac{\sqrt{2}}{2}\), dividing these equals 1. Therefore, the tangent ratio for this angle is 1. It's important for students to understand that the tangent can be seen as the slope of the terminal side of the angle, which is insightful when visualizing trigonometric concepts in geometry.