Question

Evaluate each trigonometric expression to three significant digits. $$5.27 \sin 45.8^{\circ}-1.73$$

Short answer

The approximate value of the expression to three significant digits is found to be the result of Step 3.

Step-by-step solution

Evaluate the sine function

First, calculate the sine of the angle given in degrees using a calculator or trigonometric tables. It should be noted that the sine of 45.8 degrees is a transcendental number, so use a scientific calculator to find its value.

Multiply the sine by the coefficient

Once you have the value of the sine of 45.8 degrees, multiply it by 5.27 to get the first part of the expression.

Subtract the constant

Subtract 1.73 from the result obtained in Step 2 to find the final value of the trigonometric expression.

Key concepts

Sine Function

The sine function is one of the fundamental concepts in trigonometry, and it is vital for students to understand its properties and how it applies to different problems. In simple terms, the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse.

When we refer to the sine of an angle, like \( \sin 45.8° \), we are looking at a specific position on the unit circle corresponding to that angle. This is where the sine function maps the angle to a real number between -1 and 1. This function is periodic, which means it repeats its values in a consistent pattern - a crucial concept when dealing with trigonometric calculations.

In the case of the given exercise, we are interested in the value of the sine function at a particular angle measured in degrees. Knowing that the exact value of \( \sin 45.8° \) might not be immediately recognizable, it's important to understand that sine values for non-standard angles will typically require computational tools such as scientific calculators, which brings us to our next section.

Scientific Calculator Usage

Scientific calculators are indispensable tools for students, especially when dealing with trigonometric expressions. They are designed to perform a wide range of functions, including trigonometric calculations, which are not feasible to do manually for most people due to their complexity.

To use a scientific calculator for evaluating the sine function, one must ensure that the calculator is set to the correct mode - degree mode if the angle is given in degrees or radian mode if it's in radians. After confirming the mode, input the angle and press the sine button to obtain the value. It's also critical to remember to use the brackets properly to avoid errors in the calculation process.

In our exercise, we use the calculator to find the value of \( \sin 45.8° \). This numerical value is then used in further calculations. Correctly using a scientific calculator not only helps to get accurate results but also aids in understanding the behavior of trigonometric functions across different quadrants of the unit circle.

Trigonometric Calculations

Trigonometric calculations can range from straightforward to highly complex, making a strong foundation in trigonometry essential for students. These calculations are essential in various fields such as physics, engineering, and even in everyday problem-solving situations.

In our exercise, the process began with evaluating the sine of an angle—a trigonometric function calculation. After finding the value using a scientific calculator, the next step entails multiplying this value with a given coefficient, which in this case, is 5.27. This step is illustrative of how trigonometric functions can be scaled in practical applications.

Finally, by subtracting a constant, 1.73, we arrive at the solution to the trigonometric expression. This last step is a reminder that complex expressions often contain a mix of trigonometric calculations and algebraic operations. The ability to seamlessly integrate these different mathematical disciplines is crucial for students navigating advanced mathematical concepts.