Problem 56 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -405^{\circ} $$

Step-by-Step Solution

Answer

Therefore, the sine of -405 degrees is -√2/2, the cosine is √2/2, and the tangent is -1.

Step 1: Determine the Coterminal Angle

First, a coterminal angle needs to be found in the range of 0 to 360 degrees, because the trigonometric values (sine, cosine, and tangent) are known and easy to remember in this range. This is accomplished by adding or subtracting multiples of 360 degrees until the angle falls within the range. In this case, since the angle is -405 degrees, we add 360 to get a coterminal angle of -45 degrees.

Step 2: Calculate the Sine Value

Draw the angle -45 degrees in the unit circle and look for the y-coordinate. This coordinate represents the sine of the angle. Since the angle -45 degrees corresponds to the angle 315 degrees in the unit circle, the sine of -45 degrees is given by \(\sin(-45^\circ) = \sin(315^\circ) = -\sqrt{2}/2\).

Step 3: Calculate the Cosine Value

For cosine, we'll look at the x-coordinate in the unit circle. Hence, the cosine of -45 degrees is \(\cos(-45^\circ) = \cos(315^\circ) = \sqrt{2}/2\).

Step 4: Calculate the Tangent Value

The tangent of an angle is the sine of the angle divided by the cosine. By using the obtained sine and cosine values: \(\tan(-45^\circ) = \sin(-45^\circ) / \cos(-45^\circ) = (-\sqrt{2}/2) / (\sqrt{2}/2) = -1\).

Key Concepts

Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. These angles can be found by adding or subtracting multiples of 360 degrees (or 2π radians if working with radians) to the original angle. This is because a full rotation around the circle equals 360 degrees or 2π radians. For instance, in the exercise where the angle is -405 degrees, you determine its coterminal angle by adding 360 degrees. Doing this, you achieve -45 degrees, another angle that lands on the same position relative to the positive x-axis as -405 degrees. In general, if you face an angle outside the typical 0 to 360-degree circle, finding a coterminal angle within that range can simplify further calculations like finding its sine, cosine, and tangent values.

The Unit Circle and Its Importance

The unit circle is a fundamental tool in trigonometry and comprises a circle with a radius of 1, centered at the origin of a coordinate plane. This makes it extremely useful in calculating the sine, cosine, and tangent of angles since every angle on the circle corresponds to a point. The x-coordinate of this point gives the cosine of the angle, while the y-coordinate provides the sine.

For angle -45 degrees, for example, the unit circle assists in determining that the sine is -√2/2 and the cosine is √2/2. Moreover, the utility of the unit circle extends to providing significant trigonometrical insights across different rotations and amplitude. Since -45 degrees and 315 degrees point to the same direction on the circle, the unit circle easily shows their trigonometric values are equal.

Exploring Trigonometric Functions

Trigonometric functions – sine, cosine, and tangent – are essential in understanding angles and distances. They provide a way to model periodic behavior and prove geometry problems.

- **Sine** captures the vertical aspect (y-coordinate) on a unit circle. At -45 degrees, the sine value is -√2/2, demonstrating how far the angle is from the horizontal axis.- **Cosine** measures the horizontal aspect (x-coordinate). At the same angle, cosine yields √2/2, showcasing how much the angle stretches across the horizontal axis.- **Tangent** represents the ratio of sine to cosine (y/x). For -45 degrees, \(-\frac{\sqrt{2}/2}{\sqrt{2}/2} = -1\), perfectly aligning the understanding of slopes of angles with real-world phenomena like the steepness of a hill.

Understanding these functions allows for practical applications in fields like engineering, physics, and architecture.