Problem 62 - Algebra and Trigonometry

Question

Use a calculator to find the value of the acute angle \(\theta\) in radians, rounded to three decimal places. $$ \tan \theta=0.5117 $$

Step-by-Step Solution

Answer

To find the exact value of θ, use a calculator to compute the arctan(0.5117). Ensure the calculator is in radian mode because the answer should be in radians.

Step 1: Understand the problem

Looking at the given information, we know the value of tan(θ) which is 0.5117. We need to find the actual angle θ, in radians.

Step 2: Use the inverse tangent function

Since we know that tan(θ) is 0.5117, to find θ we need to apply the inverse tangent function or the arctan function to both sides of the equation. This gives us: θ = arctan(0.5117).

Step 3: Calculate the Angle

Using a calculator to perform this calculation and round off to three decimal places, arctan(0.5117) will give the final answer.

Key Concepts

Inverse Tangent Function

The inverse tangent function, often denoted as \( \arctan \) or \( \tan^{-1} \), is a critical concept when solving trigonometric equations involving angles. When you know the tangent of an angle and want to calculate the angle itself, you use this function. In simple terms, if \( \tan(\theta) = x \), then \( \theta = \arctan(x) \). This function is crucial in reversing the tangent operation to find the angle value.

When you use your calculator to apply \( \arctan(0.5117) \), it takes the input number and determines the angle whose tangent is 0.5117. Most scientific calculators have a specific button for this operation, which simplifies finding acute angles or other angle measures based on tangent values.

Acute Angle

Angles are measured in degrees or radians, and an acute angle is an angle that is less than 90 degrees or \( \frac{\pi}{2} \) radians. In trigonometry, calculating acute angles is vital for understanding triangles, especially right-angled triangles.

If you are given the tangent of an angle and instructed to find an acute angle specifically, you are interested in angles between 0 and \( \frac{\pi}{2} \) radians. This is where inverse trigonometric functions come into play, as they help identify these specific angle measures. Always remember: the context of the problem will dictate which angle is appropriate, and for acute angles, the value will always be in this specific range.

Radians

Radians are an alternative to degrees when measuring angles and are widely used in trigonometry. One complete revolution around a circle is \(2\pi\) radians, equivalent to 360 degrees. This unit is more natural in mathematical analysis because it relates directly to the properties of circles.

The problem asked for the angle in radians, which is a standard request in mathematics-focused problems. When using inverse trigonometric functions like \( \arctan \), calculators often return the result in radians by default. It's important to be comfortable converting between radians and degrees: remember that \( \pi \) radians is 180 degrees, so a quarter-turn, which is 90 degrees, equals \( \frac{\pi}{2} \) radians.

To fully understand solutions derived from trigonometric problems, such as \( \arctan(0.5117) \) resulting in a value in radians, ensure your calculator is set to the right mode (radian vs. degree), based on the problem's requirements.