Problem 20 - Contemporary Precalculus

Question

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$-7 \pi / 3$$

Step-by-Step Solution

Answer

Question: Determine the exact values of sin, cos, and tan for the angle -7π/3. Answer: sin(-7π/3) = -√3/2, cos(-7π/3) = 1/2, and tan(-7π/3) = -√3.

Step 1: Determine the reference angle

Find the reference angle for -7π/3 by adding multiples of 2π until the result is between 0 and 2π: $$-7\pi/3 + 2\pi = -7\pi/3 + (6\pi/3) = -\pi/3$$ Here, we find that the reference angle is -π/3.

Step 2: Find the reference angle in the unit circle

To find the sine, cosine, and tangent values for the reference angle, recall the unit circle properties for angle π/3: $$\text{sin}(\pi/3) = \frac{\sqrt{3}}{2}, \quad \text{cos}(\pi/3) = \frac{1}{2}, \quad \text{tan}(\pi/3) = \sqrt{3}$$

Step 3: Determine the quadrant of the given angle

Since our reference angle is -π/3, it is located in the fourth quadrant. In the fourth quadrant, the sine function is negative, the cosine function is positive, and the tangent function is negative.

Step 4: Find the exact values of sin, cos, and tan for the given angle

Using our knowledge of the reference angle and the signs in the fourth quadrant, we can now find the exact values of the sine, cosine, and tangent of the angle -7π/3: $$\text{sin}(-7\pi/3) = -\text{sin}(\pi/3) = -\frac{\sqrt{3}}{2}$$ $$\text{cos}(-7\pi/3) = \text{cos}(\pi/3) = \frac{1}{2}$$ $$\text{tan}(-7\pi/3) = -\text{tan}(\pi/3) = -\sqrt{3}$$ Thus, the exact values of sine, cosine, and tangent for the angle -7π/3 are sin(-7π/3) = -√3/2, cos(-7π/3) = 1/2, and tan(-7π/3) = -√3.

Key Concepts

Sine and Cosine

Sine and cosine are fundamental trigonometric functions. They help in understanding the relationship between the angles and sides of a triangle. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Similarly, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

These functions can also be understood using the unit circle, where:

• Sine corresponds to the y-coordinate of a point on the circle.
• Cosine corresponds to the x-coordinate.

Both sine and cosine values range from -1 to 1. They repeat every 360 degrees or 2π radians. This periodic nature is critical for solving problems involving cycles.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a powerful tool for understanding trigonometric functions, including sine and cosine. By placing angles on this circle, you can visualize their sine and cosine values easily.

Every angle on the unit circle corresponds to a unique point \((x, y)\) where:

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

For example, at \(rac{\pi}{3}\), the coordinates are \(rac{1}{2}, \frac{\sqrt{3}}{2}\). Understanding these points helps solve many trigonometry problems quickly. Remember, the unit circle covers angles from 0 to 2\(\pi\) radians or 0 to 360 degrees.

Reference Angles

A reference angle is the smallest positive angle that the original angle makes with the x-axis on the unit circle. It helps simplify calculations, especially with angles that extend beyond 2π radians. To find a reference angle, subtract or add multiples of 2π until the angle falls within the first cycle (0 to 2π).

For the angle \(-\frac{7\pi}{3}\), adding \(2\pi\) puts it in the manageable range, equivalent to \(-\frac{\pi}{3}\). This reference angle is critical because its sine and cosine values can be directly used once the quadrant is known.

Reference angles bring clarity to the complexity of longer or negative angles.

Quadrants in Trigonometry

The coordinate plane in trigonometry is divided into four quadrants. Each quadrant determines the sign of the sine, cosine, and tangent values:

• First Quadrant: All functions are positive.
• Second Quadrant: Sine is positive; cosine and tangent are negative.
• Third Quadrant: Tangent is positive; sine and cosine are negative.
• Fourth Quadrant: Cosine is positive. Sine and tangent are negative.

Understanding these quadrants is crucial in determining the exact values of trigonometric functions. For \(-\frac{7\pi}{3}\), the reference angle \(-\frac{\pi}{3}\) lies in the fourth quadrant. This tells us:

• Sine is negative.
• Cosine is positive.
• Tangent is negative.

Labeling the signs helps ensure accurate trigonometric calculations.