Problem 30 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(\sin 23^{\circ}\)

Step-by-Step Solution

Answer

\( \sin 23^{\circ} \) is approximately 0.3917

Step 1: Identify the Trigonometric Problem

We examine the given expression: \(\sin 23^{\circ}\).

Step 2: Apply Trigonometric Identities

We apply the relevant trigonometric identities, definitions, or formulas to solve the problem.

Step 3: Simplify and Compute

We simplify the expression using known values and identities.

Step 4: State the Result

\( \sin 23^{\circ} \) is approximately 0.3917

Key Concepts

Sine Function

The sine function is a fundamental part of trigonometry, which deals with the study of triangles and the relationships between their angles and sides. 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 hypotenuse. The sine function is usually denoted as \( \sin \) in mathematical formulas.

The sine function is periodic, with a period of \( 360^{\circ} \) or \( 2\pi \) radians. This means that the function repeats its values in cycles. When using a calculator, you often see values ranging from 0 to 1 when dealing with angles between \( 0^{\circ} \) and \( 90^{\circ} \), since it's the part of the function that is positive in the first quadrant. To better understand, consider the unit circle, where the sine of an angle corresponds to the y-coordinate of the point where the line at that angle intersects the circle.

• The sine function is positive in the 1st and 2nd quadrants.
• It is negative in the 3rd and 4th quadrants.
• Sine function values are close to zero at \(0^{\circ}\) and \(180^{\circ}\).

Degree Mode

In trigonometry, angles can be measured in two primary units: degrees and radians. The degree mode is an angle measurement system in which a full circle is divided into 360 equal parts known as degrees.When working with trigonometric functions on a calculator, it's essential to ensure that the calculator is set to the correct mode. For angles given in degrees, like \( 23^{\circ} \) in our example, the calculator should be in degree mode. Otherwise, calculations will yield incorrect results.

To switch your calculator to degree mode, look for a button or a setting labeled "Deg" or "D." Each calculator is different, so refer to your calculator's manual if you're unsure how to change modes.

• Degree mode is commonly used in many real-world applications.
• Always double-check the mode before solving trigonometric problems.
• Angles in degrees are more intuitive when dealing with everyday situations.

Scientific Calculator

A scientific calculator is a powerful tool that aids in performing a wide range of mathematical operations, including trigonometric calculations such as finding the sine, cosine, and tangent of an angle.When dealing with trigonometry, a scientific calculator helps by allowing us to quickly compute trigonometric function values such as \( \sin 23^{\circ} \). To ensure accurate results, it is important to make sure the calculator is set to the correct mode corresponding to the angle's unit of measurement.

Using a scientific calculator is straightforward:

• Choose the correct angle mode (degree or radian).
• Enter the angle value (like 23).
• Press the trig function button, such as the 'sin' button for sine.

Scientific calculators are equipped with keys and functions to perform other complex calculations, making them indispensable for students and professionals in math, science, and engineering fields.

Angle Measurement

Understanding angle measurement is crucial in trigonometry since it affects how we interpret and compute trigonometric functions. Angles can be measured in degrees or radians, and choosing the right unit is vital in solving mathematical problems accurately. Degrees are prevalent in day-to-day activities, such as navigating directions or measuring squares and rectangles.

On the other hand, radians provide a more natural mathematical framework and are often used in calculus and further mathematical studies.

• One full rotation around a circle is \(360^{\circ}\) or \(2\pi\) radians.
• There are \(180^{\circ}\) in \(\pi\) radians.
• Always match the unit of angle measurement to the calculator settings.

For accurate calculations and better understanding, knowing how to convert between degrees and radians will be beneficial in advanced mathematical contexts.