Problem 34 - 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 \(\theta\) \(\cos \theta=-0.3090\)

Step-by-Step Solution

Answer

The two values of \( \theta \) are \( 108^{\circ} \) and \( 252^{\circ} \).

Step 1: Understand the Problem

We are given the trigonometric equation \( \cos \theta = -0.3090 \) and need to find two angles \( \theta \) within the range \( 0^{\circ} \leq \theta < 360^{\circ} \).

Step 2: Recognize the Quadrants for Cosine

Cosine is negative in the second and third quadrants. We will find an angle in each of these quadrants that satisfies \( \cos \theta = -0.3090 \).

Step 3: Find the Reference Angle

Use a calculator to find the inverse cosine of the positive value: \( \theta = \cos^{-1}(0.3090) \approx 72^{\circ} \). This is the reference angle we will use to find actual solutions.

Step 4: Find the Second Quadrant Angle

Since cosine is negative in the second quadrant, the angle in this quadrant is given by \( 180^{\circ} - \text{reference angle} = 180^{\circ} - 72^{\circ} = 108^{\circ} \).

Step 5: Find the Third Quadrant Angle

In the third quadrant, the angle will be \( 180^{\circ} + \text{reference angle} = 180^{\circ} + 72^{\circ} = 252^{\circ} \).

Step 6: Verify the Solutions

Confirm that both 108° and 252° fit within the given range \(0^{\circ} \leq \theta < 360^{\circ}\) and satisfy the original equation \( \cos \theta = -0.3090 \) to the nearest degree.

Key Concepts

Trigonometric Equation

A trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, tangent, and their reciprocals. In the given problem, the equation is expressed as \( \cos \theta = -0.3090 \). Solving trigonometric equations often involves finding the angles that satisfy the equation within a specified interval. In this case, we need to find values for \( \theta \) where \( 0^{\circ} \leq \theta < 360^{\circ} \). It can be visually helpful to plot these functions on a unit circle, as the circle can show where the function values become positive or negative. Understanding where the trigonometric function changes its sign in different quadrants is key to solving these types of equations.

Reference Angle

A reference angle is always defined as the acute angle formed by the terminal side of the given angle and the horizontal axis. More precisely, it is the smallest angle between the terminal side and x-axis, ranging from 0° to 90°. In this problem, we first handle the positive value of the cosine. By finding the inverse function, \( \cos^{-1}(0.3090) \), we arrive roughly at \( 72^{\circ} \). This 72° is our reference angle. This angle is not the solution itself but simply helps to determine the actual angles in the specified quadrants where the original equation holds. Reference angles help simplify our calculations, letting us zero in on the secondary and tertiary solutions after considering both the positive and negative intervals of trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle given a trigonometric value. When we apply \( \cos^{-1} \) on a calculator with 0.3090, it returns an angle (the reference angle) of about 72°. It comes in handy as a tool to work backward from a cosine value to the angle, unlocking solutions for trigonometric equations. These functions however conventionally give results within a principal range. For cosine, \( \cos^{-1} \) will provide angles usually in the range of 0° to 180°. To check the angles given by inverse trigonometric functions, especially when dealing with negative values, it's necessary to understand the quadrant circle well enough to extend these principal range solutions accordingly.

Quadrants of Trigonometric Functions

The unit circle divides the plane into four quadrants, each playing a key role in the sign determination of trigonometric functions. Quadrant I contains angles from 0° to 90°, where all trigonometric functions are positive. However, in this context of \( \cos \theta = -0.3090 \), we focus on Quadrant II (angles between 90° to 180°) and Quadrant III (angles from 180° to 270°) since cosine is negative there. After finding the reference angle, we determine the actual angles in Quadrants II and III:

• In Quadrant II, obtain the angle as \( 180^{\circ} - \text{reference angle} \), resulting in \( 108^{\circ} \).
• In Quadrant III, calculate it as \( 180^{\circ} + \text{reference angle} \), yielding \( 252^{\circ} \).

Knowing which quadrants the function is positive or negative is essential to finding correct solutions for trigonometric equations.