Problem 41 - Algebra and Trigonometry

Question

Use a calculator to find the value of the trigonometric function to four decimal places. $$ \tan 32.7^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \tan 32.7^{\circ} \) to four decimal places is 0.6443.

Step 1: Input in Calculator

Type '32.7' in the calculator, then choose the tangent function (usually represented by the 'tan' button).

Step 2: Compute

Press the equals button ('=') on the calculator to compute the tangent of 32.7 degrees.

Step 3: Round the Result

The result needs to be rounded off to four decimal places. If the fifth decimal is 5 or more raise the fourth decimal by one, else leave it as it is.

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function that relates an angle of a right triangle to the ratio of the lengths of the opposite side to the adjacent side. It is usually expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] where \( \theta \) is the angle in question. This function is particularly useful in understanding angles and slopes in geometry and physics. The tangent function is periodic with a period of \( 180^{\circ} \) or \( \pi \) radians, meaning it repeats its values in regular intervals. Additionally, while cosine and sine are bounded between -1 and 1, the tangent function can have any real number as its output.

Degree Measurement

When working with trigonometric functions, it's crucial to recognize the unit of angle measurement being used, either degrees or radians, as it affects the calculated values. In this exercise, we're dealing with degree measurement. Degrees are a traditional unit of angular measurement where a full circle is divided into 360 equal parts. Calculators often allow users to switch between degrees and radians; ensure yours is set to degrees if working with angles like \( 32.7^{\circ} \). To check this, look for a 'DEG' symbol on the display or access the mode/settings menu. Adjusting this setting incorrectly can lead to errors in calculations and misunderstanding of the angle value.

Calculator Usage

For calculations involving trigonometric functions, a scientific calculator or a calculator app on your phone or computer can be used. These calculators simplify the process, especially for functions like tangent.

• First, ensure your calculator is in "degree" mode and not "radian." This is important for correctly interpreting the input, which is in degrees in this case.
• Next, input the angle value. Here, type '32.7' into the calculator.
• Then press the "tan" button, typically found alongside other trigonometric function keys. This button calculates the tangent of the angle input.
• Finally, press the equal button ('=') to get the result.

Though manual calculations are educational, calculators provide speed and accuracy, reducing potential errors.

Rounding Decimals

Rounding numbers is an essential part of presenting numerical answers, especially when the precision of decimal places matters, like in scientific fields. In this context, rounding to four decimal places means keeping four digits after the decimal point. To round correctly:

• Identify the fifth number after the decimal. If it is 5 or greater, increase the fourth number by one unit.
• If the fifth number is less than 5, simply leave the fourth number unchanged.

Consider an example: Suppose the calculator provides \( \tan(32.7^{\circ}) = 0.64098243 \). Here, the fifth digit (8) is more than 5. So, increase '0.6409' to '0.6410'. Precision is vital in ensuring that mathematical communication remains clear and accurate.