Problem 60 - Algebra and Trigonometry

Question

Use a calculator to find the value of the acute angle \(\theta\) in radians, rounded to three decimal places. $$ \sin \theta=0.9499 $$

Step-by-Step Solution

Answer

The acute angle \(\theta\) in radians, rounded to three decimal places, is given by \(\theta = \sin^{-1}(0.9499)\).

Step 1: Understand the required transformation

Redefine the provided equation \(\sin \theta = 0.9499\) in terms of \(\theta\). This can be achieved by applying the inverse sine function to both sides of the equation. This gives: \(\theta = \sin^{-1}(0.9499)\).

Step 2: Use a Scientific Calculator

Use a scientific calculator to evaluate \(\sin^{-1}(0.9499)\). Make sure that the calculator is in radian mode as the question asks for the angle in radians.

Step 3: Round the result

Round your result to three decimal places as per the requirement of the question.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry. It is used to relate the angle of a triangle to the lengths of its sides. Specifically, in a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. For any angle \(\theta\), the sine value ranges from -1 to 1.

• When \(\theta\) is in the first quadrant (0 to 90 degrees), \(\sin \theta\) is positive.
• As \(\theta\) increases from 0 to 90 degrees, \(\sin \theta\) also increases from 0 to 1.

The sine function helps us solve various real-world and mathematical problems, including finding unknown angles when provided with side lengths.

Radian Measure Explained

Radians are an alternate way to measure angles, often used in mathematics instead of degrees. Unlike degrees, which divide a circle into 360 parts, radians relate angles to the arc length on a unit circle.

• One full circle is equal to \(2\pi\) radians, which corresponds to 360 degrees.
• This makes \(\pi\) radians equal to 180 degrees.

Using radians makes certain mathematical equations and identities simpler, especially in calculus. When solving trigonometric equations like \(\sin \theta = 0.9499\), it's important to use radian mode to find \(\theta\) if the question specifies radians.

Scientific Calculator Usage

Using a scientific calculator is essential for solving complex mathematical problems. To find the angle \(\theta\) such that \(\sin \theta = 0.9499\), the inverse sine function, \(\sin^{-1}\), is used.

• First, ensure your calculator is in radian mode. This setting is crucial when the result needs to be in radians, as it affects the calculations.
• Input \(0.9499\) and apply the \(\sin^{-1}\) function to find \(\theta\).
• After evaluating, round the answer to three decimal places as required.
• Double-check your work to ensure the answer is in the correct mode and correctly rounded.

Mastering the use of a scientific calculator can streamline solving mathematical and scientific problems.