Problem 30 - Algebra 2 and Trigonometry

Question

In \(28-43,\) for each function value, if \(0^{\circ} \leq \theta <3 60^{\circ},\) find, to the nearest degree, two values of \(\theta\) \(\tan \theta=0.3240\)

Step-by-Step Solution

Answer

The two values of \(\theta\) are 18° and 198°.

Step 1: Understand the Problem

We need to find the angle(s) between 0° and 360° such that the tangent of the angle, \(\tan \, \theta\), equals 0.3240. We should find two such angles and round them to the nearest degree.

Step 2: Use the Inverse Tangent Function

To find the first angle, use the inverse tangent function: \( heta = \tan^{-1}(0.3240)\). Solve using a calculator to find that \( heta_1 \approx 17.97^\circ\). Round this to the nearest degree to get \( heta_1 \approx 18^\circ\).

Step 3: Find the Second Solution

The tangent function is periodic with a period of 180°. Hence, the second solution is offset by 180° from the first solution. Calculate \( heta_2 = 18^\circ + 180^\circ = 198^\circ\).

Step 4: Verify the Solutions

Both \( heta_1 = 18^\circ\) and \( heta_2 = 198^\circ\) fall within the given range of \(0^\circ \leq \theta < 360^\circ\). The solutions match the condition \(\tan \theta = 0.3240\).

Key Concepts

Inverse Tangent

The inverse tangent, often written as \( \tan^{-1}(x) \) or \( \arctan(x) \), is a trigonometric function used to find an angle when the tangent value is known. Unlike the regular tangent function that takes an angle and gives a ratio, the inverse tangent takes a ratio and returns an angle. This is especially useful when you want to determine a specific angle for a given tangent value, which is what we're dealing with in this problem.
When you compute \( \tan^{-1}(0.3240) \), you're asking what angle has a tangent of 0.3240. This function does not give a direct angle in degrees unless your calculator is set to degrees. By using it, we find our first angle: \( \theta_1 \approx 18^\).

• This angle is in the first quadrant because the arctangent will initially return an angle between \(-90^\) and \(90^\) by default.
• Remember to set your calculator to degree mode to correctly find the angle in degrees.

Angle Calculation

After using the inverse tangent to find one angle, calculating additional angles becomes simpler. In our case, we found that the first principal angle is \( \theta_1 = 18^\).
Now we need to find another angle that shares the same tangent value. The periodic nature of the tangent function allows us to find one such angle by using the period of the tangent function. We do this by adding the period of \(180^\) to \( \theta_1 \). So we calculate: \( \theta_2 = 18^ + 180^ = 198^\).
This gives us two angles from 0° to 360° that have tangent values of 0.3240. It's essential to ensure both angles are within this range.

• For many problems in trigonometry, the angles found will repeat every full period.
• Always re-check that the angles lie within the specified interval for the problem.

Tangent Periodicity

The tangent function is unique among trigonometric functions because of its periodicity of 180°. This means tangent values repeat every 180°.
Understanding this periodic nature helps when solving problems like the one given. If you know one angle, you can always find another in the specified range by adding or subtracting multiples of the period. For this exercise, starting from \(18^\), we add 180° to find the second angle \(198^\).
Periodicity is key in simplifying complex trigonometric calculations:

• Always consider the periodicity when asked to find multiple angles that produce the same trigonometric value.
• The smaller the period, like with tangent (180°), the quicker it is to find the next angle.

This characteristic is vital for graphing tangent functions and understanding their behavior across different intervals.