Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\sin 10^{\circ}\) (b) \(\cos 80^{\circ}\)

Step-by-Step Solution

Answer

The values, rounded to four decimal places, are approximately \(\sin(10^\circ) = 0.1736\) and \(\cos(80^\circ) = 0.1736\).

Step 1: Set the Calculator in Degree Mode

Ensure that the calculator is set in degree mode. This can generally be done through the settings or mode functions on most scientific calculators.

Step 2: Evaluate Sine Function for 10 Degrees

Enter \(\sin 10\) into the calculator. The sine function tells you the ratio of the length of side opposite the angle to the hypotenuse in a right-angled triangle. The calculator performs this operation and outputs a decimal.

Step 3: Evaluate Cosine Function for 80 Degrees

Enter \(\cos 80\) into the calculator. The cosine function tells you the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. The calculator performs this operation and outputs a decimal.

Key Concepts

Sine Function

When it comes to trigonometry, the sine function plays a pivotal role. It’s often introduced as part of the trio of basic trigonometric functions which include sine, cosine, and tangent. Essentially, the sine function helps us understand the relationship between an angle and the sides of a right-angled triangle. Specifically, it defines this relationship as the ratio of the length of the side opposite the given angle to the length of the hypotenuse.

For example, when calculating \(\sin 10^\circ\) using a calculator, one is essentially finding this ratio for an angle of 10 degrees. Remember, the exact value might seem abstract, but it’s really just representing this proportion in the context of a unit circle or a hypothetical right-angled triangle. Calculator solutions provide a unique convenience here, giving us the ability to quickly find accurate values for trigonometric functions and these ratios often need to be rounded to a certain number of decimal places for practicality in real-world applications.

Cosine Function

The cosine function, another cornerstone of trigonometric concepts, is deeply interconnected with the sine function. While the sine function examines the opposite side in relation to the hypotenuse, the cosine function looks at the adjacent side. To put it simply, the cosine function measures the ratio between the length of the adjacent side to that of the angle in question and the hypotenuse’s length in a right-angled triangle.

Take \(\cos 80^\circ\) for instance. When inputting this into a calculator, you are calculating the cosine of an 80-degree angle. The resulting value should be seen as the ratio of the adjacent side to the hypotenuse in a triangle with one angle measuring 80 degrees. This value is crucial in various fields such as physics, engineering, and even in certain artistic applications which require precise angles and measurements. Calculators help make these computations quick and straightforward, but it's the core concept of ratios in right triangles that gives the cosine function its significance.

Degree Mode

Understanding angle measurements is critical in trigonometry, and a common source of confusion arises between the different modes for these measurements: degree and radian. Most calculators can switch between these modes. The degree mode is what many are familiar with; it divides a full rotation into 360 equal parts, with each part being one degree. This is the mode you should be using when calculating trigonometric functions with angles provided in degrees.

Let’s delve into setting your calculator to degree mode. This is crucial because if your calculator is set to radian mode but you're inputting values in degrees, the results will be incorrect. The initial step in solving problems like \(\sin 10^\circ\) or \(\cos 80^\circ\) is to ensure the calculator's mode matches the unit of the angle. Degree mode aligns with everyday concepts of angles and is thus often a comfortable starting point for students learning trigonometry. A tip for ensuring accuracy is to always double-check the mode before starting your calculations - it can save a lot of potential headaches!