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Chapter 4: Problem 58

In Exercises 53-68, evaluate the sine, cosine, and tangent of the angle without using a calculator. \(-840^\circ\)

Short Answer

Expert verified
The sine of -840 degrees is \(-\sqrt{3}/2\), the cosine is -0.5, and the tangent is \(\sqrt{3}\).

Step by step solution

01

Normalizing the angle

First, normalize the angle to a value between 0-360 degrees, by adding multiples of 360 degrees until the angle lies in this range. Since -840 degrees is negative, it implies we are measuring the angle clockwise. Adding \(360^\circ\) times 3 or \(3 * 360 = 1080^\circ\) will bring the angle to a positive range. Therefore, \(-840^\circ + 1080^\circ = 240^\circ\). Now, the angle is 240 degrees.
02

Evaluating trigonometric functions

Next, utilize the knowledge of the unit circle to know the sine, cosine and tangent of the 240 degree angle. From a unit circle, we know that \( cos(240^\circ) = -0.5\), \(sin(240^\circ) = -\sqrt{3}/2 \), and \( tan(240^\circ) = sin(240^\circ) / cos(240^\circ);\) which simplifies to \( tan(240^\circ) = \sqrt{3}\). Here, the positive sign indicates the direction of the angle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a powerful tool in Trigonometry. It simplifies the understanding and calculation of trigonometric functions like sine, cosine, and tangent. Imagine a circle drawn on a coordinate plane with its center at the origin (0,0) and a radius of 1. This is the unit circle. Each point on the circle represents an angle corresponding to the rotation from the positive x-axis.
If a point on the unit circle has the coordinates \(x, y\), then:
  • The x-coordinate represents the cosine value of that angle.
  • The y-coordinate represents the sine value of that angle.
  • The tangent of an angle is calculated as the sine divided by the cosine, \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \].
This relationship makes the unit circle a bridge between geometry and trigonometry. It helps visualize how sine and cosine values change with angles.
Angle Normalization
Angle normalization is the process of bringing an angle within a standard range. This is typically between \(0^{\circ}\) and \(360^{\circ}\) for degrees, or between \(0\) and \(2\pi\) when working with radians. If an angle is less than \(0^{\circ}\) or more than \(360^{\circ}\), it needs to be adjusted by adding or subtracting full rotations (multiples of \(360^{\circ}\)).
In the case of \(-840^{\circ}\), we normalize it by adding \(1080^{\circ}\) (or \(3 \times 360^{\circ}\)). This brings the angle to \(240^{\circ}\). Normalizing helps in finding an equivalent angle that is easier to use with the unit circle and trigonometric functions.
Sine
The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, it's the y-coordinate of the point where the terminal side of the angle intersects the circle.
For \(240^{\circ}\), which is in the third quadrant of the unit circle, both sine and cosine will have negative values. Therefore, \( \sin(240^{\circ}) = -\frac{\sqrt{3}}{2} \). The negative sign indicates that in the third quadrant, the angle lands in the negative y-space of the circle.
Cosine
Cosine is similar to sine; however, it represents the x-coordinate on the unit circle. For any angle, cosine is the ratio of the adjacent side to the hypotenuse in a right triangle.
At \(240^{\circ}\), which is located in the third quadrant, \( \cos(240^{\circ}) = -0.5 \). The negative sign reflects the placement in the left part of the x-axis, as the x-value is negative.
Cosine provides us information on the horizontal alignment of the angle. Just like sine, its pattern repeats every \(360^{\circ}\).
Tangent
Tangent is a trigonometric function represented as the ratio of sine to cosine. On the unit circle, it represents the slope of the line created by an angle.
  • In simpler terms, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
For \(240^{\circ}\), \( \tan(240^{\circ}) = \frac{-\frac{\sqrt{3}}{2}}{-0.5} = \sqrt{3} \). Here, tangent being positive reflects both sine and cosine are negative, cancelling each other out.
Tangent illustrates how steep a line is at an angle, which is helpful in both geometric interpretation and algebraic solutions of problems.

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Most popular questions from this chapter

In Exercises 91-96, use a graphing utility to graph the function. \(f(x)\ =\ arctan(2x-3)\)

In Exercises 67-76, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) cot(arctan \(\dfrac{1}{x}\))

DATA ANALYSIS The number of hours \(H\) of daylight in Denver, Colorado on the 15th of each month are: \(1(9.67)\), \(2(10.72)\), \(3(11.92)\), \(4(13.25)\), \(5(14.37)\), \(6(14.97)\), \(7(14.72)\), \(8(13.77)\), \(9(12.48)\), \(10(11.18)\), \(11(10.00)\), \(12(9.38)\). The month is represented by \(t\), with \(t=1\) corresponding to January. A model for the data is given by \(H(t)\ =\ 12.13\ +\ 2.77\ sin[(\pi t/6)\ -\ 1.60]\). (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.

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