Problem 15 - Algebra 2

Question

Solve each equation by finding the value of \(x\) to the nearest degree. $$ x=\cos ^{-1} 0 $$

Step-by-Step Solution

Answer

x = 90°

Step 1: Understand the Inverse Function

The equation given is \(x = \cos^{-1}(0)\). This function tells us to determine the angle whose cosine is 0. The inverse cosine function, \(\cos^{-1}\), provides the angle in the range of 0° to 180°.

Step 2: Analyze the Cosine Values

The cosine of an angle is 0 at specific angles within the specified range (0° to 180°). Recall that \( \cos(90°) = 0 \). Hence, \(x\) might be 90°.

Step 3: Check the Solution

Verify if \( \cos(90°) = 0 \), which is true. Thus, the angle we are looking for indeed results in the cosine value of 0.

Key Concepts

Angle Measurement

Angles are a fundamental part of trigonometry and geometry. They are measured in degrees or radians. In the context of trigonometric functions, degrees are commonly used for practical problems and everyday applications, while radians are prevalent in calculus and theoretical work.

An important aspect of measuring angles is understanding how units translate. For instance:

• There are 360 degrees in a full circle, making it a complete rotation.
• Alternatively, this is equivalent to 2π radians.

For problems involving inverse trigonometric functions, the angle is often sought within a specific range. In this case, we're looking for an angle where the cosine equals 0. The inverse cosine function, also known as arccosine, limits the angle to between 0° and 180°. Thus, knowing how to properly measure and locate angles helps solve trigonometric problems efficiently.

Cosine

The cosine function is one of the primary trigonometric functions, alongside sine and tangent.Cosine relates the length of the adjacent side of a right triangle to its hypotenuse.

For an angle θ in a right triangle, cosine is defined as:

• a = adjacent side
• h = hypotenuse
• o = opposite side

The formula is written as \(\cos(θ) = \frac{a}{h}\). In the unit circle approach, the cosine of an angle can be represented as the x-coordinate of the point on the circle's circumference.It ranges from -1 to 1.

In this problem, we are tasked with finding the angle whose cosine is zero, which occurs when the corresponding point on the unit circle crosses the y-axis at a height of zero.Interpreting these relationships is key to understanding trigonometric functions.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions that need to be solved for specific values, often angles.They occur frequently in physics, engineering, and geometry to model periodic phenomena like waves or oscillations.

The equation given, x = \(\cos^{-1}(0)\), illustrates an inverse trigonometric equation requiring you to find an angle whose trigonometric ratio is known. To solve:

• Recognize the inverse function, which yields the angle whose cosine is zero.
• Identify that \(\cos(90°) = 0\), thus x = 90°

To ensure problem-solving accuracy:

• Always verify your solution by plugging it back into the original equation.
• In general, other inverse trigonometric equations require careful consideration of which quadrant the resulting angle might fall into based on function characteristics.

Mastering these kinds of problems requires persistent practice and a strong grasp of the relationships between angles and their trigonometric ratios.