Problem 26 - Contemporary Precalculus

Question

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\tan \theta=69.4$$

Step-by-Step Solution

Answer

Answer: The two angles within the given range that satisfy the equation are \(\theta = 89.345^{\circ}\) and \(\theta = 269.345^{\circ}\).

Step 1: Calculate the reference angle

Find the reference angle \(\alpha\) using the given information: $$\tan \alpha = 69.4$$ We have to find the angle \(\alpha\) (in degrees) that has a tangent of \(69.4\). Use a calculator to find the arctangent (also known as the inverse tangent) of \(69.4\): $$\alpha = \arctan(69.4) \approx 89.345^{\circ}$$

Step 2: Find the principal angle

Now that we have the reference angle, we want to find the principal angle \(\theta_1\) in the range \(0^{\circ} \leq \theta<360^{\circ}\) that satisfies the given equation. Since the tangent function has a period of \(180^{\circ}\), the principal angle can be obtained by adding or subtracting a multiple of \(180^{\circ}\) to the reference angle: $$\theta_1 = \alpha + n \times 180^{\circ}$$ Where \(n\) is an integer. We will start with \(n=0\): $$\theta_1 = 89.345^{\circ} + 0 \times 180^{\circ} = 89.345^{\circ}$$ So, the first principal angle in the given range is \(\theta_1 = 89.345^{\circ}\).

Step 3: Find the second solution

Since the tangent function has a period of \(180^{\circ}\), there must be another angle in the given range that also has the same tangent value. To find this angle, we can add the period to the first principal angle: $$\theta_2 = \theta_1 + 180^{\circ}$$ $$\theta_2 = 89.345^{\circ} + 180^{\circ} \approx 269.345^{\circ}$$ Now, we have found both solutions in the given range: $$\theta = 89.345^{\circ}, 269.345^{\circ}$$

Key Concepts

Angle Measurement

In trigonometry, angle measurement is crucial for solving equations involving angles, like the one in our problem. Angles can be measured in degrees or radians, and understanding the conversion between these units is key.

• A full circle is 360 degrees or \(2\pi\) radians.
• To convert from degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
• To convert from radians to degrees, use: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

In our exercise, we're working with degrees, which is more common in practical applications like navigation and engineering.
These conversions help in performing calculations and interpreting results, especially when shifting from theoretical to practical settings.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when given a trigonometric ratio. They are the opposite operations of the regular trigonometric functions.

• The inverse tangent, or arctan, function is specifically used when you have the ratio and need the angle.
• For example, if \( \tan \alpha = 69.4 \), then \( \alpha = \arctan(69.4) \).

These functions are crucial for solving equations like the one in our problem. Using a calculator in degree mode allows us to find the specific angle that corresponds to a tangent value.
This step is vital to determine the reference angle, which can then be adjusted to find the actual angles that satisfy the given equation.

Tangent Function

The tangent function is one of the three primary trigonometric functions, relating the opposite and adjacent sides of a right triangle.

• For an angle \( \theta \), \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
• In the unit circle, it can also be expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

The tangent function has certain characteristics:

• It is undefined for angles where the cosine is zero.
• It has a period of \(180^{\circ}\) or \(\pi\) radians, meaning the values repeat every half-circle rotation.

Understanding how the tangent function behaves helps in solving trigonometric equations, as seen in finding multiple solutions within a specified range.

Periodicity of Trigonometric Functions

Periodicity is a fundamental property of trigonometric functions, especially useful in solving equations spanning specific intervals.

• The tangent function repeats every \(180^{\circ}\), unlike sine and cosine which repeat every \(360^{\circ}\).
• This property allows us to find additional solutions by adding or subtracting the function's period.

In our exercise, the periodicity of the tangent function enabled us to determine both solutions, \( \theta = 89.345^{\circ} \) and \( \theta = 269.345^{\circ} \).
By understanding periodicity, we can easily extend solutions beyond a given range or find all possible angles that meet the criteria of a trigonometric equation.