Problem 27 - Algebra 2

Question

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answers to the nearest hundredth. $$ \tan \theta=-2 $$

Step-by-Step Solution

Answer

The solutions for the equation \(\tan \theta =-2\) in the interval from 0 to \(2\pi \) are \( \theta = 2.03\) radians and \( \theta = 0.89\) radians.

Step 1: Find the reference angle

To find the reference angle for \(\tan \theta = -2\), we need to disregard the negative sign and take the arctangent of 2. We then convert the angle from radians to degrees by performing the calculation: \(\arctan(2) = 1.11\) radians. Afterwards, we convert it to degrees: \(1.11 \times\frac{180}{\pi } = 63.43\) degrees.

Step 2: Determine the angle in each quadrant

Tangent is negative in the second and fourth quadrants hence we need to find the angles in those quadrants. We subtract the reference angle from 180 degrees for the second quadrant and add it to 360 degrees for the fourth quadrant. So, \(\theta =180 - 63.43 = 116.57^\circ \) for the second quadrant and \(\theta =360 + 63.43 = 423.43^\circ \) for the fourth quadrant.

Step 3: Convert the angles back to radians within the range [0,2\(\pi\)]

The final step is to convert these solutions back to radians and ensure they fall within the range [0,2\(\pi\)]. We use the conversion formula: \(1 \degree = \frac{\pi }{180} \) radians, which gives us \( \theta = 116.57 \times \frac{\pi }{180} =2.03\) radians and \( \theta = 423.43 \times \frac{\pi }{180}= 7.39\) radians. However, since the latter value doesn't fall within the given interval, we subtract \(2\pi \) from it to obtain a value within range, giving \(\theta = 7.39 - 2\pi = 0.89\) radians.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the primary functions in trigonometry. It can be defined in a right-angled triangle as the ratio of the length of the opposite side to the adjacent side. More generally for any angle \( \theta \), the tangent function is defined by the ratio of the sine to the cosine of that angle: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

• This function is periodic with a period of \( \pi \) radians (or 180 degrees).
• The tangent function is undefined wherever the cosine of an angle is zero, which occurs at odd multiples of \( \frac{\pi}{2} \).

Tangents can be either positive or negative depending on the angle's quadrant. In this problem, since \( \tan \theta = -2 \), the solution involves finding where the tangent function is negative. This occurs in the second and fourth quadrants on the unit circle.

Reference Angle

A reference angle is the acute angle that a terminal side of any given angle makes with the x-axis. It is usually measured to help in solving trigonometric equations. To find the reference angle for \( \tan \theta = -2 \), you need to first ignore the negative sign and use the arctangent function: \[ \theta_{\text{ref}} = \tan^{-1}(2) \]This will give you the smallest angle whose tangent is 2.

• Reference angles are always positive and are essential in determining the actual angles in different quadrants.
• In this problem, the reference angle was calculated to be \( 63.43^\circ \).

This angle is then used to find actual angles in the second and fourth quadrants, where the tangent function is negative.

Radians and Degrees Conversion

An essential skill in solving trigonometric equations is the conversion between radians and degrees.Radians and degrees are two measures of angle size, where \[ 2\pi \text{ radians} = 360^\circ \]Thus, one degree is equivalent to \( \frac{\pi}{180} \) radians, and vice versa.

• For converting from radians to degrees, the formula \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \) is used.
• Conversely, degrees can be converted to radians using \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

In this exercise, the reference angle was converted from radians (1.11) to degrees (63.43) to determine the angles in different quadrants, and then the final results were converted back into radians to fit within the interval \([0, 2\pi]\).

Unit Circle

The unit circle is a fundamental tool for understanding trigonometric functions. It is a circle with a radius of one centered at the origin of the coordinate plane. The angle in radians or degrees is measured from the positive x-axis.

• The coordinates of a point on the unit circle correspond to the values of the cosine and sine of the angle formed with the x-axis at that point, noted as \((\cos \theta, \sin \theta)\).
• The tangent of an angle can also be viewed as the y-coordinate divided by the x-coordinate, or equivalently \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

All trigonometric equations can be visualized using the unit circle, helping identify which quadrants have positive or negative function values. From this visualization, we understand where \( \tan \theta \) is negative, as required by the problem—the second and fourth quadrants.