Problem 19 - Algebra 2 and Trigonometry

Question

In \(13-24,\) find, to the nearest ten-thousandth, the radian measure \(\theta\) of a first-quadrant angle with the given function value. $$ \tan \theta=1.5277 $$

Step-by-Step Solution

Answer

\( \theta \approx 0.993 \) radians.

Step 1: Use Arctan Function

To find the angle \( \theta \) given the tangent of the angle, we use the inverse tangent function, also known as \( \arctan \) or \( \tan^{-1} \). Compute \( \theta = \arctan(1.5277) \).

Step 2: Convert Answer to Radians

The result of the inverse tangent function is given in radians by default if the calculator is in radian mode. Calculate \( \arctan(1.5277) \) using a calculator to get the radian measure approximately.

Step 3: Round to Nearest Ten-Thousandth

After getting the radian measure from Step 2, round the value to four decimal places, as it asks for the nearest ten-thousandth.

Key Concepts

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are crucial in finding the angles when the value of a trigonometric function is known. Let's say you know that the tangent of an angle equals a specific value, like in this exercise where \( \tan \theta = 1.5277 \). To determine the angle \( \theta \), you need to use the inverse tangent function, written as \( \arctan \) or \( \tan^{-1} \).
This function helps us "undo" the tangent function. With this operation, you obtain the angle whose tangent value matches the number given. In practical terms, it takes a ratio derived from a right-angled triangle and converts it back to an angle measure. Since many calculators return the result in radians, knowing how to interpret these is essential, especially when working on problems that involve angles in technology or science.
Inverse trigonometric functions include:

• \( \arcsin \) or \( \sin^{-1} \) for sine
• \( \arccos \) or \( \cos^{-1} \) for cosine
• \( \arctan \) or \( \tan^{-1} \) for tangent

These functions are vital for calculating angles when exact values or conditions are specified, bridging the gap between various fields of mathematics and practical applications.

Exploring Radian Measure

Radians are a way to measure angles, focusing on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are directly based on the circle's circumference and radius. One complete circle is \( 2\pi \) radians, which is equivalent to 360 degrees.
To visualize this, imagine wrapping the radius of a circle around its circumference. The length that covers the circle entirely equals \( 2\pi \) radius units. This unique approach makes radians particularly efficient for advanced mathematics, as they simplify many formulas and calculations, especially when dealing with periodic functions and calculus.
When solving trigonometric problems, especially those in physics or engineering, using radians is common. This method provides a natural alignment with mathematical constants like \( \pi \).

• \( 1 \text{ radian} \approx 57.2958 \text{ degrees} \)
• An angle of \( \pi/2 \) radians corresponds to 90 degrees, or a right angle.

Thus, learning to think in radians as well as degrees broadens your mathematical toolkit, helping you handle problems that require precise angular measurements.

Identifying First-Quadrant Angles

The concept of "first-quadrant angles" refers to angles positioned in the first quarter of the Cartesian coordinate plane. This quadrant ranges from \( 0 \) to \( \pi/2 \) radians, or equivalently from \( 0 \) to \( 90 \) degrees.
First-quadrant angles are distinguished by the fact that all trigonometric function values (sine, cosine, and tangent) are positive here. This characteristic serves as a foundation for understanding trigonometric functions, as their behavior alters according to the quadrant in which they reside.
Some key points about the first quadrant include:

• It is the area where initial angle exploration often happens since values are straightforward.
• All basic trigonometric functions reflect positive outputs.
• Proximity to the x-axis and y-axis helps in easy identification of certain trigonometric values like \( \sin 0 = 0 \) or \( \cos 0 = 1 \).

Being adept in determining angles and function values within this quadrant is vital, as it sets the foundation before moving on to angles in other quadrants or more intricate trigonometric phenomena.