Question

Find two positive angles less than \(360^{\circ}\) whose trigonometric function is given. Round your angles to a tenth of a degree. $$\cos \theta=0.8372$$

Short answer

Principal angle is \(\cos^{-1}(0.8372)\) degrees, and the second angle is \(360^\circ - \cos^{-1}(0.8372)\) degrees.

Step-by-step solution

Use Inverse Cosine to find the principal angle

To find the angle whose cosine is 0.8372, use the inverse cosine function: \(\theta = \cos^{-1}(0.8372)\). This will give us the principal angle, which is the smallest angle between 0° and 180° that has this cosine value.

Calculate the principal angle

Use a calculator to compute the value of \(\cos^{-1}(0.8372)\). Make sure your calculator is in degree mode. The computed angle will be the first angle less than 360° with the given cosine value.

Find the second angle using cosine symmetry

Since the cosine function is positive in both the first and fourth quadrants, and is symmetric about the y-axis, the second angle \(\theta'\) can be found using \(\theta' = 360^\circ - \theta\).

Compute the second angle

Subtract the principal angle from 360° to find the second angle. This will give us the second positive angle less than 360° with the given cosine value.

Key concepts

Inverse Cosine

Understanding the inverse cosine is crucial when solving trigonometry problems involving angles and their corresponding cosine values. The inverse cosine function, denoted by \( \cos^{-1} \), takes a number (which represents a cosine value) and returns the angle in radians or degrees that corresponds to that cosine value. For instance, when you come across a problem like \( \cos \theta = 0.8372 \), you use the inverse cosine to find the corresponding angle \(\theta\). It's important to have your calculator set to the correct mode (degrees if you are dealing with degrees) before carrying out the calculation.

The inverse cosine function will only give you one angle, typically referred to as the 'principal angle', which lies within the range of 0° to 180° for a cosine value. This is because the cosine function is uniquely invertible only in this domain. Whenever you're given a cosine value and asked to find angles, remember to consider the symmetry of the cosine function to find additional solutions — but we'll touch on that concept in a later section.

Principal Angle

The 'principal angle' is a term that often comes up when dealing with trigonometric equations. By definition, the principal angle is the smallest angle, measured in a counterclockwise direction from the positive x-axis, that can represent a given trigonometric function value in a standard position. So, when using \( \cos^{-1} \) to find the angle corresponding to a given cosine value, the result is the principal angle.

For cosine values, the principal angle will always be within the range of 0° to 180°, that is the first and second quadrants of the unit circle, where the cosine values are positive and negative respectively. It's an essential concept, especially when you're asked to find multiple angles corresponding to a single trigonometric value. The principal angle serves as the starting point to discovering other possible angles through methods like using symmetry properties of trigonometric functions.

Cosine Symmetry

The symmetry of the cosine function is a fundamental characteristic of trigonometry. The cosine function exhibits even symmetry, meaning it reflects function values across the y-axis. This trait is because the cosine of an angle is equal to the cosine of its reference angle in the standard position.

When we talk about 'cosine symmetry', we're referring to the fact that \( \cos(\theta) = \cos(-\theta) \) and also that the cosine values repeat in the fourth quadrant just as they appear in the first quadrant. This symmetry is the reason why, after finding the principal angle in the first or second quadrants using the \( \cos^{-1} \) function, we can find a corresponding angle in the fourth quadrant with the same cosine value by subtracting the principal angle from 360°: \( \theta' = 360^\circ - \theta \). This method is useful in problems where you are required to provide all possible angles within a certain range, such as finding angles less than 360°, that correspond to a specific cosine value.